Percentiles Two students graduating with the same class rank could be in different percentiles depending on the school population. (credit: "graduation caps" by John Walker/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Compute percentiles. Solve application problems involving percentiles. A college admissions officer is comparing two students. The first, Anna, finished 12th in her class of 235 people. The second, Brian, finished 10th in his class of 170 people. Which of these outcomes is better? Certainly 10 is less than 12, which favors Brian, but Anna’s class was much bigger. In fact, Anna beat out 223 of her classmates, which is 223 235 ≈ 95 % of her classmates, while Brian bested 160 out of 170 people, or 94%. Comparing the proportions of the data values that are below a given number can help us evaluate differences between individuals in separate populations. These proportions are called percentiles . If p % of the values in a dataset are less than a number n , then we say that n is at the p th percentile. Finding Percentiles There are some other terms that are related to "percentile" with meanings you may infer from their roots. Remember that the word percent means “per hundred.” This reflects that percentiles divide our data into 100 pieces. The word quartile has a root that means “four.” So, if a data value is at the first quantile of a dataset, that means that if you break the data into four parts (because of the quart- ), this data value comes after the first of those four parts. In other words, it’s greater than 25% of the data, placing it at the 25th percentile. Quintile has a root meaning “five,” so a data value at the third quintile is greater than three-fifths of the data in the set. That would put it at the 60th percentile. The general term for these is quantiles (the root quant – means “number”). In Mean, Median, and Mode , we defined the median as a number that is greater than no more than half of the data in a dataset and is less than no more than half of the data in the dataset. With our new term, we can more easily define it: The median is the value at the 50th percentile (or second quartile). Let’s look at some examples. Finding Percentiles Consider the dataset 5, 8, 12, 1, 2, 16, 2, 15, 20, 22. At what percentile is the value 5? What value is at the 60th percentile? Before we can answer these two questions, we must put the data in increasing order: 1, 2, 2, 5, 8, 12, 15, 16, 20, 22. There are three values (1, 2, and 2) in the set that are less than 5, and there are ten values in the set. Thus, 5 is at the 3 10 × 100 = 30 th percentile. To find the value at the 60th percentile, we note that there are ten data values, and 60% of ten is six. Thus, the number we want is greater than exactly six of the data values. Thus, the 60th percentile is 15. In each of the examples above, the computations were made easier by the fact that the we were looking for percentiles that “came out evenly” with respect to the number of values in our dataset. Things don’t always work out so cleanly. Further, different sources will define the term percentile in different ways. In fact, Google Sheets has three built-in functions for finding percentiles, none of which uses our definition. Some of the definitions you’ll see differ in the inequality that is used. Ours uses “less than or equal to,” while others use “less than” (these correspond roughly to Google Sheets’ ‘PERCENTILE.INC’ and ‘PERCENTILE.EXC’). Some of them use different methods for interpolating values. (This is what we did when we first computed medians without technology; if there were an even number of data values in our dataset, found the mean of the two values in the middle. This is an example of interpolation. Most computerized methods use this technique.) Other definitions don’t interpolate at all, but instead choose the closest actual data value to the theoretical value. Fortunately, for large datasets, the differences among the different techniques become very small. So, with all these different possible definitions in play, what will we use? For small datasets, if you’re asked to compute something involving percentiles without technology , use the technique we used in the previous example. In all other cases, we’ll keep things simple by using the built-in ‘PERCENTILE’ and ‘PERCENTRANK’ functions in Google Sheets (which do the same thing as the ‘PERCENTILE.INC’ and ‘PERCENTRANK.INC’ functions; they’re “inclusive, interpolating” definitions). Using RANK, PERCENTRANK, and PERCENTILE in Google Sheets Using Google Sheets to Compute Percentiles: Average SAT Scores The data in “AvgSAT” contains the average SAT score for students attending every institution of higher learning in the US for which data is available. What score is at the 3rd quartile? What score is at the 40th percentile? At what percentile is Albion College in Michigan (average SAT: 1132)? At what percentile is Oregon State University (average SAT: 1205)? The 3rd quartile is the 75th percentile, so we’ll use the PERCENTILE function. Click on an empty cell, and type “=PERCENTILE(“. Next, enter the data: click on the letter at the top of the column containing the average SAT scores. Then, tell the function which percentile we want; it needs to be entered as a decimal. So, type a comma (to separate the two pieces of information we’re giving this function), then type “0.75” and close the parentheses with “)”. The result should look like this (assuming the data are in column C): “=PERCENTILE(C:C, 0.75)”. When you hit the enter key, the formula will be replaced with the average SAT score at the 75th percentile: 1199. Using the PERCENTILE function, we find that an average SAT of 1100 is at the 40th percentile. Since we want to know the percentile for a particular score, we’ll use the PERCENTRANK function. Like the PERCENTILE function, we need to give PERCENTRANK two pieces of information: the data, and the value we care about. So, click on an empty cell, type “=PERCENTRANK(“, and then click on the letter at the top of the column containing the data. Next, type a comma and then the value we want to find the percentile for; in this case, we’ll type “, 1132”. Finally, close the parentheses with “)” and hit the enter key. The formula will be replaced with the information we want: an average SAT of 1132 is at the 54th percentile. Using the PERCENTRANK function, an average SAT of 1205 is at the 76.1th percentile. Using Google Sheets to Compute Percentiles: In-State Tuition The dataset "InState" contains the in-state tuitions of every college and university in the country that reported that data to the Department of Education. Use that dataset to answer these questions. What tuition is at the second quintile? What tuition is at the 95th percentile? At what percentile is Walla Walla University in Washington (in-state tuition: $28,035)? At what percentile is the College of Saint Mary in Nebraska (in-state tuition: $20,350)? The second quintile is the 40th percentile; using PERCENTILE in Google Sheets, we get $8,400. Using PERCENTILE again, we get $44,866. Using PERCENTRANK, we get the 81.6th percentile. Using PERCENTRANK, we get the 74.8th percentile. Check Your Understanding Key Terms percentile quartile quintile quantile Key Concepts The percentile rank of a data value is the percentage of all values in the dataset that are less than or equal to the given value. Videos Using RANK, PERCENTRANK, and PERCENTILE in Google Sheets

Math


The Normal Distribution Symmetric, bell-shaped distributions arise naturally in many different situations, including coin flips. Learning Objectives After completing this section, you should be able to: Describe the characteristics of the normal distribution. Apply the 68-95-99.7 percent groups to normal distribution datasets. Use the normal distribution to calculate a z -score. Find and interpret percentiles and quartiles. Many datasets that result from natural phenomena tend to have histograms that are symmetric and bell-shaped. Imagine finding yourself with a whole lot of time on your hands, and nothing to keep you entertained but a coin, a pencil, and paper. You decide to flip that coin 100 times and record the number of heads. With nothing else to do, you repeat the experiment ten times total. Using a computer to simulate this series of experiments, here’s a sample for the number of heads in each trial: 54, 51, 40, 42, 53, 50, 52, 52, 47, 54 It makes sense that we’d get somewhere around 50 heads when we flip the coin 100 times, and it makes sense that the result won’t always be exactly 50 heads. In our results, we can see numbers that were generally near 50 and not always 50, like we thought. Moving Toward Normality Let’s take a look at a histogram for the dataset in our section opener: This is interesting, but the data seem pretty sparse. There were no trials where you saw between 43 and 47 heads, for example. Those results don’t seem impossible; we just didn’t flip enough times to give them a chance to pop up. So, let’s do it again, but this time we'll perform 100 coin flips 100 times. Rather than review all 100 results, which could be overwhelming, let's instead visualize the resulting histogram. From the histogram, we see that most of the trials resulted in between, say, 44 and 56 heads. There were some more unusual results: one trial resulted in 70 heads, which seems really unlikely (though still possible!). But we’re starting to maybe get a sense of the distribution. More data would help, though. Let’s simulate another 900 trials and add them to the histogram! We can still see that 70 is a really unusual observation, though we came close in another trial (one that had 68 heads). Now, the distribution is coming more into focus: It looks quite symmetric and bell-shaped. Let’s just go ahead and take this thought experiment to an extreme conclusion: 10,000 trials. The distribution is pretty clear now. Distributions that are symmetric and bell-shaped like this pop up in all sorts of natural phenomena, such as the heights of people in a population, the circumferences of eggs of a particular bird species, and the numbers of leaves on mature trees of a particular species. All of these have bell-shaped distributions. Additionally, the results of many types of repeated experiments generally follow this same pattern, as we saw with the coin-flipping example; this fact is the basis for much of the work done by statisticians. It’s a fact that’s important enough to have its own name: the Central Limit Theorem. John Kerrich Having enough time on your hands to actually perform this coin-flipping experiment may sound far-fetched, but the English mathematician John Kerrich found himself in just such a situation. While he was studying abroad in Denmark in 1940, that country was invaded by the Germans. Kerrich was captured and placed in an internment camp, where he remained for the duration of the war. Kerrich knew that he had all kinds of time on his hands. He also studied statistics, so he knew what should happen theoretically if he flipped a coin many, many times. He also knew of nobody who had ever tested that theory with an actual, large-scale experiment. So, he did it: While he was incarcerated, Kerrich flipped a regular coin 10,000 times and recorded the results. Sure enough, the theory held up! The Normal Distribution In the coin flipping example above, the distribution of the number of heads for 10,000 trials was close to perfectly symmetric and bell-shaped: Because distributions with this shape appear so often, we have a special name for them: normal distributions . Normal distributions can be completely described using two numbers we’ve seen before: the mean of the data and the standard deviation of the data. You may remember that we described the mean as a measure of centrality; for a normal distribution, the mean tells us exactly where the center of the distribution falls. The peak of the distribution happens at the mean (and, because the distribution is symmetric, it’s also the median). The standard deviation is a measure of dispersion; for a normal distribution, it tells us how spread out the histogram looks. To illustrate these points, let’s look at some examples. Identifying the Mean of a Normal Distribution This graph shows three normal distributions. What are their means? Step 1: Take a look at the three curves on the graph. Since the mean of a normal distribution occurs at the peak, we should look for the highest point on each distribution. Let’s draw a line from each curve's peak down to the axis, so we can see where these peaks occur: Step 2: The peak of the red (leftmost) distribution occurs over the number 1 on the horizontal axis. Thus, the mean of the red distribution is 1. Similarly, the mean of the blue (middle) distribution is 2, and the mean of the yellow (rightmost) distribution is 3. Identifying the Standard Deviation of a Normal Distribution This graph shows three distributions, all with mean 2. What are their standard deviations? Step 1: Identifying the standard deviation from a graph can be a little bit tricky. Let’s focus in on the yellow (lowest peaked) curve: Step 2: Notice that the graph curves downward in the middle, and curves upwards on the ends. Highlighted in red is the part that curves downward and in green, the part that curves upward: Step 3: The places where the graph changes from curving up to curving down (or vice versa) are called inflection points . Let’s identify where those occur by dropping a line straight down from each: Step 4: We can estimate that the inflection points occur at x = - 1 and x = 5 ; the mean is at x = 2 (as shown by the middle dotted line). The difference between the mean and the location of either inflection point is the standard deviation; since 5 - 2 = 2 - ( - 1 ) = 3 , we conclude that the standard deviation of the green distribution is 3. Step 5: Now, looking at the other two graphs, let's first identify the inflection points: Step 6: The red (tallest peaked) distribution has inflection points at 1 and 3, and the mean is 2. Thus, the standard deviation of the red distribution is 3 - 2 = 2 - 1 = 1 . The blue (lower peaked) distribution has inflection points at 0 and 4, and its mean is also 2. So, the standard deviation of the blue distribution is 2. Let’s put it all together to identify a completely unknown normal distribution. Identifying the Mean and Standard Deviation of a Normal Distribution Using the graph, identify the mean and standard deviation of the normal distribution. Step 1: Let’s start by putting dots on the graph at the peak and at the inflection points, then drop lines from those points straight down to the axis: Step 2: From the red (middle) line, we can see that the mean of this distribution is 55. The blue (outermost) lines are each 3 units away from the mean (at 52 and 58), so the standard deviation is 3. Properties of Normal Distributions: The 68-95-99.7 Rule The most important property of normal distributions is tied to its standard deviation. If a dataset is perfectly normally distributed, then 68% of the data values will fall within one standard deviation of the mean. For example, suppose we have a set of data that follows the normal distribution with mean 400 and standard deviation 100. This means 68% of the data would fall between the values of 300 (one standard deviation below the mean: 400 - 100 = 300 ) and 500 (one standard deviation above the mean: 400 + 100 = 500 ). Looking at the histogram below, the shaded area represents 68% of the total area under the graph and above the axis: Since 68% of the area is in the shaded region, this means that 100 % − 68 % = 32 % of the area is found in the unshaded regions. We know that the distribution is symmetric, so that 32% must be divided evenly into the two unshaded tails: 16% in each. Of course, datasets in the real world are never perfect; when dealing with actual data that seem to follow a symmetric, bell-shaped distribution, we’ll give ourselves a little bit of wiggle room and say that approximately 68% of the data fall within one standard deviation of the mean. The rule for one standard deviation can be extended to two standard deviations. Approximately 95% of a normally distributed dataset will fall within 2 standard deviations of the mean. If the mean is 400 and the standard deviation is 100, that means 95% calculation describes the way we compute standardized scores. (two standard deviations below the mean: 400 – 2 × 100 = 200 ) and 600 (two standard deviations above the mean: 400 + 2 × 100 = 600 ). We can visualize this in the following histogram: As before, since 95% of the data are in the shaded area, that leaves 5% of the data to go into the unshaded tails. Since the histogram is symmetric, half of the 5% (that’s 2.5%) is in each. We can even take this one step further: 99.7% of normally distributed data fall within 3 standard deviations of the mean. In this example, we’d see 99.7% of the data between 100 (calculated as 400 – 3 × 100 = 100 ) and 700 (calculated as 400 + 3 × 100 = 700 ). We can see this in the histogram below, although you may need to squint to find the unshaded bits in the tails! This observation is formally known as the 68-95-99.7 Rule . Using the 68-95-99.7 Rule to Find Percentages If data are normally distributed with mean 8 and standard deviation 2, what percent of the data falls between 4 and 12? If data are normally distributed with mean 25 and standard deviation 5, what percent of the data falls between 20 and 30? If data are normally distributed with mean 200 and standard deviation 15, what percent of the data falls between 155 and 245? Let’s look at a table that sets out the data values that are even multiples of the standard deviation (SD) above and below the mean: mean - 3 × SD mean - 2 × SD mean - 1 × SD Mean mean + 1 × SD mean + 2 × SD mean + 3 × SD 2 4 6 8 10 12 14 Since 4 and 12 represent two standard deviations above and below the mean, we conclude that 95% of the data will fall between them. Let’s build another table: mean - 3 × SD mean - 2 × SD mean - 1 × SD Mean mean + 1 × SD mean + 2 × SD mean + 3 × SD 10 15 20 25 30 35 40 We can see that 20 and 30 represent one standard deviation above and below the mean, so 68% of the data fall in that range. Let’s make one more table: mean - 3 × SD mean - 2 × SD mean − 1 × SD Mean mean + 1 × SD mean + 2 × SD mean + 3 × SD 155 170 185 200 215 230 245 Since 155 and 245 are three standard deviations above and below the mean, we know that 99.7% of the data will fall between them. Using the 68-95-99.7 Rule to Find Data Values If data are distributed normally with mean 100 and standard deviation 20, between what two values will 68% of the data fall? If data are distributed normally with mean 0 and standard deviation 15, between what two values will 95% of the data fall? If data are distributed normally with mean 14 and standard deviation 2, between what two values will 99.7% of the data fall? The 68-95-99.7 Rule tells us that 68% of the data will fall within one standard deviation of the mean. So, to find the values we seek, we’ll add and subtract one standard deviation from the mean: 100 - 1 × 20 = 80 and 100 + 1 × 20 = 120 . Thus, we know that 68% of the data fall between 80 and 120. Using the 68-95-99.7 Rule again, we know that 95% of the data will fall within 2 standard deviations of the mean. Let’s add and subtract two standard deviations from that mean: 0 - 2 × 15 = - 30 and 0 + 2 × 15 = 30 . So, 95% of the data will fall between -30 and 30. Once again, the 68-95-99.7 Rule tells us that 99.7% of the data will fall within three standard deviations of the mean. So, let’s add and subtract three standard deviations from the mean: and 14 + 3 × 2 = 20 . Thus, we conclude that 99.7% of the data will fall between 8 and 20. There are more problems we can solve using the 68-95-99.7 Rule. but first we must understand what the rule implies. Remember, the rule says that 68% of the data falls within one standard deviation of the mean. Thus, with normally distributed data with mean 100 and standard deviation 10, we have this distribution: Since we know that 68% of the data lie within one standard deviation of the mean, the implication is that 32% of the data must fall beyond one standard deviation away from the mean. Since the histogram is symmetric, we can conclude that half of the 32% (or 16%) is more than one standard deviation above the mean and the other half is more than one standard deviation below the mean: Further, we know that the middle 68% can be split in half at the peak of the histogram, leaving 34% on either side: So, just the “68” part of the 68-95-99.7 Rule gives us four other proportions in addition to the 68% in the rule. Similarly, the “95” and “99.7” parts each give us four more proportions: We can put all these together to find even more complicated proportions. For example, since the proportion between 100 and 120 is 47.5% and the proportion between 100 and 110 is 34%, we can subtract to find that the proportion between 110 and 120 is 47.5 - 34 = 13.5 % : Finding Other Proportions Using the 68-95-99.7 Rule Assume that we have data that are normally distributed with mean 80 and standard deviation 3. What proportion of the data will be greater than 86? What proportion of the data will be between 74 and 77? What proportion of the data will be between 74 and 83? Before we can answer these questions, we must mark off sections that are multiples of the standard deviation away from the mean: To figure out what proportion of the data will be greater than 86, let's start by shading in the area of data that are above 86 in our figure, or the data more than two standard deviations above the mean. We saw in that this proportion is 2.5%. To figure out what proportion of the data will be between 74 and 77, let's start by shading in that area of data. These are data that are more than one but less than two standard deviations below the mean. From , we know that the proportion of data less than two standard deviations below the mean is 47.5%. And, from YOUR TURN 8.33 , we know that 34% of the data is less than one standard deviation below the mean: Subtracting, we see that the proportion of data between 74 and 77 is 13.5%. To figure out what proportion of the data will be between 74 and 83, let's start by shading in that area of data in our figure. Next, we'll break this region into two pieces at the mean: From , we know the blue (leftmost) region represents 47.5% of the data. And, using YOUR TURN 8.33 , we get that the red (rightmost) region covers 34% of the data. Adding those together, the proportion we want is 81.5%. Standardized Scores When we want to apply the 68-95-99.7 Rule, we must first figure out how many standard deviations above or below the mean our data fall. This calculation is common enough that it has its own name: the standardized score . Values above the mean have positive standardized scores, while those below the mean have negative standardized scores. Since it's common to use the letter z to represent a standard score, this value is also often referred to as a z -score . So far, we’ve only really considered z -scores that are whole numbers, but in general they can be any number at all. For example, if we have data that are normally distributed with mean 80 and standard deviation 6, the value 85 is five units above the mean, which is less than one standard deviation. Dividing by the standard deviation, we get 5 6 . Since 85 is 5 6 of one standard deviation above the mean, we’d say that the standardized score for 85 is z = 5 6 (which is positive, since 85 > 80 ). This calculation describes the way we compute standardized scores. If x is a member of a normally distributed dataset with mean µ and standard deviation σ , then the standardized score for x is z = x - µ σ . If you know a z -score but not the original data value x , you can find it by solving the previous equation for x : x = µ + z × σ . The symbols μ and σ are the Greek letters mu and sigma . They are the analogues of the English letters m and s , which stand for mean and standard deviation . If you convert every data value in a dataset into its z -score, the resulting set of data will have mean 0 and standard deviation 1. This is why we call these standardized scores : the normal distribution with mean 0 and standard deviation 1 is often called the standard normal distribution . Standardizing Data Suppose we have data that are normally distributed with mean 50 and standard deviation 6. Compute the standardized scores (rounded to three decimal places) for these data values: 52 40 68 For each of these, we’ll plug the given values into the formula. Remember, the mean is μ = 50 and the standard deviation is σ = 6 : z = x - µ σ = 52 - 50 6 = 0.333 z = x - µ σ = 40 - 50 6 = - 1.667 z = x - µ σ = 68 - 50 6 = 3 Converting Standardized Scores to Original Values Suppose we have data that are normally distributed with mean 10 and standard deviation 2. Convert the following standardized scores into data values. 1.4 −0.9 3.5 We’ll use the formula previously introduced to convert z -scores into x -values. In this case, the mean is µ = 10 and the standard deviation is σ = 2 : x = µ + z × σ = 10 + 1.4 × 2 = 12.8 x = µ + z × σ = 10 + ( - 0.9 ) × 2 = 8.2 x = µ + z × σ = 10 + 3.5 × 2 = 17 Using Google Sheets to Find Normal Percentiles The 68-95-99.7 Rule is great when we’re dealing with whole-number z -scores. However, if the z -score is not a whole number, the Rule isn’t going to help us. Luckily, we can use technology to help us out. We’ll talk here about the built-in functions in Google Sheets, but other tools work similarly. Let’s say we’re working with normally distributed data with mean 40 and standard deviation 7, and we want to know at what percentile a data value of 50 would fall. That corresponds to finding the proportion of the data that are less than 50. If we create our histogram and mark off whole-number multiples of the standard deviation like we did before, we’ll see why the 68-95-99.7 Rule isn’t going to help: Since 50 doesn’t line up with one of our lines, the 68-95-99.7 Rule fails us. Looking back at and , the best we can say is that 50 is between the 84th and 99.5th percentiles, but that’s a pretty wide range. Google Sheets has a function that can help; it’s called NORM.DIST. Here’s how to use it: Click in an empty cell in your worksheet. Type “=NORM.DIST(“ Inside the parentheses, we must enter a list of four things, separated by commas: the data value, the mean, the standard deviation, and the word “TRUE”. These have to be entered in this order! Close the parentheses, and hit Enter. The result is then displayed in the cell; convert it to a percent to get the percentile. So, for our example, we should type “=NORM.DIST(50, 40, 7, TRUE)” into an empty cell, and hit Enter. The result is 0.9234362745; converting to a percent and rounding, we can conclude that 50 is at the 92nd percentile. Let’s walk through a few more examples. Using Google Sheets to Find Percentiles Suppose we have data that are normally distributed with mean 28 and standard deviation 4. At what percentile do each of the following data values fall? 30 23 35 By entering “=NORM.DIST(30, 28, 4, TRUE)” we find that 30 is at the 69th percentile. By entering “=NORM.DIST(23, 28, 4, TRUE)” we find that 23 is at the 11th percentile. By entering “=NORM.DIST(35, 28, 4, TRUE)” we find that 35 is at the 96th percentile. Google Sheets can also help us go the other direction: If we want to find the data value that corresponds to a given percentile, we can use the NORM.INV function. For example, if we have normally distributed data with mean 150 and standard deviation 25, we can find the data value at the 30th percentile as follows: Click on an empty cell in your worksheet. Type “=NORM.INV(“ Inside the parentheses, we’ll enter a list of three numbers, separated by commas: the percentile in question expressed as a decimal, the mean, and the standard deviation. These must be entered in this order! Close the parentheses and hit Enter. The desired data value will be in the cell! In our example, we want the 30th percentile; converting 30% to a decimal gives us 0.3. So, we’ll type “=NORM.INV(0.3, 150, 25)” to get 136.8899872; let’s round that off to 137. Using Google Sheets to Find the Data Value Corresponding to a Percentile Suppose we have data that are normally distributed with mean 47 and standard deviation 9. Find the data values (rounded to the nearest tenth) corresponding to these percentiles: 75th (that’s the third quartile) 12th 90th By entering “=NORM.INV(0.75, 47, 9)” we find that 53.1 is at the 75th percentile. By entering “=NORM.INV(0.12, 47, 9)” we find that 36.4 is at the 12th percentile. By entering “=NORM.INV(0.9, 47, 9)” we find that 58.5 is at the 90th percentile. Check Your Understanding normal distribution 68-95-99.7 Rule standardized score ( z -score) Key Concepts Normally distributed data follow a bell-shaped, symmetrical distribution. The mean of normally distributed data falls at the peak of the distribution. The standard deviation of normally-distributed data is the distance from the peak to either of the inflection points. Data that are normally distributed follow the 68-95-99.7 Rule, which says that approximately 68% of the data fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. The z -score for a data value is the number of standard deviations that value falls above (or below, if the z -score is negative) the mean. We can use the normal distribution to estimate percentiles. Formulas If x is a member of a normally distributed dataset with mean µ and standard deviation σ , then the standardized score for x is z = x - µ σ . If you know a z -score but not the original data value x , you can find it by solving the previous equation for x : x = µ + z × σ .

Math


Applications of the Normal Distribution Standardized test results generally adhere to the normal distribution. (credit: “Taking a Test” by Marco Verch Professional Photographer/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Apply the normal distribution to real-world scenarios. As we saw in The Normal Distribution , the word “standardized” is closely associated with the normal distribution. This is why tests like college entrance exams, state achievement tests for K–12 students, and Advanced Placement tests are often called “standardized tests”: scores are assigned in a way that forces them to follow a normal distribution, with a mean and standard deviation that are consistent from year to year. Standardization also allows people like college admissions officers to directly compare an applicant who took the ACT (a college entrance exam) to an applicant who instead chose to take the SAT (a different college entrance exam). Standardization allows us to compare individuals from different groups; this is among the most important applications of the normal distribution. We’ll explore this and other real-world uses of the normal distribution in this section. College Entrance Exams There are two good ways to compare two data values from different groups: using z -scores and using percentiles. The two methods will always give consistent results (meaning that we won’t find, for example, that the first value is better using z -scores but the second value is better using percentiles), so use whichever method is more comfortable for you. Evaluating College Entrance Exam Scores According to the Digest of Education Statistics , composite scores on the SAT have mean 1060 and standard deviation 195, while composite scores on the ACT have mean 21 and standard deviation 5. At what percentile would an SAT score of 990 fall? What is the z -score of an ACT score of 27? Which is better: a score of 1450 on the SAT or 29 on the ACT? Using Google Sheets, we can answer this question with the formula “=NORM.DIST(990, 1060, 195, TRUE)”. A score of 990 would fall at the 36th percentile. Using the formula z = x - µ s , we get z = 27 - 21 5 = 1.2 . Let’s compare the values using both percentiles and z -values: Percentiles: Using “=NORM.DIST(1450, 1060, 195, TRUE)” we find that an SAT score of 1450 is at the 98th percentile. Meanwhile, by entering “=NORM.DIST(29, 21, 5, TRUE)” we see that an ACT score of 29 is around the 95th percentile. Since it’s at a higher percentile, we can conclude that an SAT score of 1450 is better than an ACT score of 29. z -scores: Using the formula, we see that the z -score for an SAT score of 1450 is z = 1450 - 1060 195 = 2 , while the z -score for an ACT score of 29 is z = 29 - 21 5 = 1.6 . Since it has a higher z -score, an SAT score of 1450 is better than an ACT score of 29. Coin flipping In the opening of The Normal Distribution , we saw that the number of heads we get when we flip a coin 100 times is distributed normally. It can be shown that if n is the number of flips, then the mean of that distribution is n 2 and the standard deviation is n 2 (as long as n ≥ 20 ). So, for 100 flips, the mean of the distribution is 50 and the standard deviation is 5. In that opening example, one of our early runs gave us 70 heads in 100 flips, which we noted seemed unusual. Using the normal distribution, we can identify exactly how unusual that really is. Using Google Sheets, the formula “=NORM.DIST(70, 50, 5, TRUE)” gives us 0.999968, which is the 99.997th percentile! How is that useful? Suppose you need to test whether a coin is fair, and so you flip it 100 times. While we might be suspicious if we get 70 heads out of the 100 flips, we now have a numerical measure for how unusual that is: If the coin were fair, we would expect to see 70 heads (or more) only 100 - 99.9968 = 0.0032 % of the time. That’s really unlikely! Analysis like this is related to hypothesis testing , an important application of statistics in the sciences and social sciences. Flipping a Coin Let’s say we flip a coin 64 times and count the number of heads. What would be the mean of the corresponding distribution? What would be the standard deviation of the corresponding distribution? Suppose we got 25 heads, which seems a little low. At what percentile would 25 heads fall? Since n = 64 , the mean is 64 2 = 32 . Again using n = 64 , we get a standard deviation of 64 2 = 4 . Using “=NORM.DIST(25, 32, 4, TRUE)”, we see that 25 heads is at the 4th percentile. Reading and Interpreting Scatter Plots Analyzing Data That Are Normally Distributed Whenever we’re working with a dataset that has a distribution that looks symmetric and bell-shaped, we can use techniques associated with the normal distribution to analyze the data. Using Normal Techniques to Analyze Data The data in “AvgSAT” contains the average SAT score for students attending every institution of higher learning in the United States for which data is available. In , we created a histogram for these data: (data source: https://data.ed.gov) This distribution is fairly symmetric (it’s just a little right-skewed) and bell-shaped, so we can use normal distribution techniques to analyze the data. What is the mean of these average SAT scores? What is the standard deviation of these SAT scores? Using the answers to the previous two questions, use NORM.DIST in Google Sheets to estimate at what percentile the University at Buffalo in New York (average SAT: 1250) falls. Use PERCENTRANK to find the actual percentile of the University at Buffalo, and see how close the estimate in the previous question came. Using the AVERAGE function in Google Sheets, we find that the mean is 1141.174. Using the STDEV function, we get that the standard deviation is 125.517. Entering “=NORM.DIST(1250, 1141, 125.517, TRUE)” into Google Sheets, we estimate that the University at Buffalo is at the 81st percentile. Using PERCENTRANK, we find that the actual percentile is the 84th. These are close! Political Meddling Exposed The normal distribution pops up in some unusual places. Recently, a team at Duke University has been using statistics to help identify partisan gerrymandering, where electoral districts have been carefully drawn in a way that benefits one political party over another. In their analysis, they found that hypothetical election results in randomly drawn districts are normally distributed. By using techniques similar to the ones we used above, they can quantify precisely how biased a particular electoral map is by finding the percentile rank of the actual election result on the normal distribution of the hypothetical results. You can find out more about their work at the "Quantifying Gerrymandering" site here. Check Your Understanding Key Concepts We can use z -scores to compare data values from different datasets.

Math


Scatter Plots, Correlation, and Regression Lines A scatter plot is a visualization of the relationship between quantitative dataset. Learning Objectives After completing this section, you should be able to: Construct a scatter plot for a dataset. Interpret a scatter plot. Distinguish among positive, negative and no correlation. Compute the correlation coefficient. Estimate and interpret regression lines. One of the most powerful tools statistics gives us is the ability to explore relationships between two datasets containing quantitative values, and then use that relationship to make predictions. For example, a student who wants to know how well they can expect to score on an upcoming final exam may consider reviewing the data on midterm and final exam scores for students who have previously taken the class. It seems reasonable to expect that there is a relationship between those two datasets: If a student did well on the midterm, they were probably more likely to do well on the final than the average student. Similarly, if a student did poorly on the midterm, they probably also did poorly on the final exam. Of course, that relationship isn’t set in stone; a student’s performance on a midterm exam doesn’t cement their performance on the final! A student might use a poor result on the midterm as motivation to study more for the final. A student with a really good grade on the midterm might be overconfident going into the final, and as a result doesn’t prepare adequately. The statistical method of regression can find a formula that does the best job of predicting a score on the final exam based on the student’s score on the midterm, as well as give a measure of the confidence of that prediction! In this section, we’ll discover how to use regression to make these predictions. First, though, we need to lay some graphical groundwork. Relationships Between Quantitative Datasets Before we can evaluate a relationship between two datasets, we must first decide if we feel that one might depend on the other. In our exam example, it is appropriate to say that the score on the final depends on the score on the midterm, rather than the other way around: if the midterm depended on the final, then we’d need to know the final score first , which doesn’t make sense. Here’s another example: if we collected data on home purchases in a certain area, and noted both the sale price of the house and the annual household income of the purchaser, we might expect a relationship between those two. Which depends on the other? In this case, sale price depends on income: people who have a higher income can afford a more expensive house. If it were the other way around, people could buy a new, more expensive house and then expect a raise! (This is very bad advice.) It's worth noting that not every pair of related datasets has clear dependence. For example, consider the percent of a country’s budget devoted to the military and the percent earmarked for public health. These datasets are generally related: as one goes up, the other goes down. However, in this case, there’s not a preferred choice for dependence, as each could be seen as depending on the other. When exploring the relationship between two datasets, if one set seems to depend on the other, we’ll say that dataset contains values of the response variable (or dependent variable ). The dataset that the response variable depends on contains values of what we call the explanatory variable (or independent variable ). If no dependence relationship can be identified, then we can assign either dataset to either role. Identifying Explanatory and Response Variables For each of the following pairs of related datasets, identify which (if any) should be assigned the role of response variable and which should be assigned to be the explanatory variable. A person’s height and weight A professional basketball player’s salary and their average points scored per game (which is a measure of how good they are at basketball) The length and width of leaves on a tree As people get taller, their weight tends to increase. But if a person goes on a diet and loses weight, we don’t expect them to also get shorter. So, weight depends on height. That means we’ll say that the response variable is weight and the explanatory variable is height . The more points a basketball player scores, the more money they should make. But if a basketball player gets a raise, we wouldn’t expect them to get better at basketball as a result. So, the response is salary and the explanatory is points per game . The way that the length and width of leaves are connected isn’t clear. It seems reasonable that as the width goes up, so would the length. But the other direction is also plausible: as the length goes up, so does the width. Without a clear dependence relationship, we’re free to declare either to be the response and the other to be the explanatory. Once we’ve assigned roles to our two datasets, we can take the first step in visualizing the relationship between them: creating a scatter plot. Creating Scatter Plots A scatter plot is a visualization of the relationship between two quantitative sets of data. The scatter plot is created by turning the datasets into ordered pairs: the first coordinate contains data values from the explanatory dataset, and the second coordinate contains the corresponding data values from the response dataset. These ordered pairs are then plotted in the xy -plane. Let's return to our exam example to put this into practice. Creating Scatter Plots Without Technology Students are exploring the relationship between scores on the midterm exam and final exam in their math course. Here are some of the scores reported by their classmates: Name Midterm grade Final grade Student 1 88 84 Student 2 71 80 Student 3 75 77 Student 4 94 95 Student 5 68 73 Create a scatter plot to visualize the data. Step 1: Since it makes more sense to think of the final exam score as being dependent on the midterm exam score, we’ll let the final grade be the response. So, let’s think of these two datasets as a set of ordered pairs, midterm first, final second: Name (Midterm, Final) Student 1 (88, 84) Student 2 (71, 80) Student 3 (75, 77) Student 4 (94, 95) Student 5 (68, 73) Step 2: Next, let’s make the axes. On the horizontal axis, make sure the range of values is sufficient to cover all of the explanatory data. For our data, that’s 68 to 94. Similarly, the vertical axis should cover all of the response data (73 to 95): Step 3: Our first point is (88, 84). So, we’ll locate 88 on the horizontal axis, 84 on the vertical axis, and identify the point that’s directly above the first location and horizontally level with the second: Step 4: Repeat this process to place the other four points on the graph: Step 5: Finally, label the axes: For large datasets, it’s impractical to create scatter plots manually. Luckily, Google Sheets automates this process for us. Making Scatter Plots in Google Sheets Creating Scatter Plots in Google Sheets The dataset “NHL19” gives the results of the 2018–2019 National Hockey League season. The columns are team, wins (W), losses (L), overtime losses (OTL), total points (PTS), goals scored by the team (GF), goals scored against the team (GA), and goal differential (the difference in GF and GA). Use Google Sheets to create a scatter plot for GF vs. GA. When we talk about plotting one set versus another, the first is the response and the second is explanatory. Step 1: Open the dataset in Google Sheets, and click and drag to select the data we want to visualize (in this case, we want the columns for GF and GA; make sure you include those labels in the selection). Step 2: Next, click on the “Insert” menu, then click “Chart.” Sheets will automatically choose a chart format; if the result isn’t a scatter plot, click on the drop-down menu under “Chart type” in the Chart Editor on the right side of the window and select “Scatter chart.” Step 3: Next, check that the correct choices were made for the horizontal and vertical axes. In this case, we want to see GF on the vertical axis and GA on the horizontal axis. If Sheets made the wrong choice, we can fix it in the Chart Editor by clicking on the name of the dataset under “X-axis” to open up a dropdown menu, then selecting the variable that should go on the horizontal axis (GA in this case). Step 4: Now, click on the variable under “Series” and select the one that should go on the vertical axis (GF). If you had to make that change, the axis labels in the graph may also need changing; those labels can be fixed using the “Customize” tab in the Chart Editor under “Chart & axis titles.” Your result should look like this: (data source: www.nhl.com) Reading and Interpreting Scatter Plots Scatter plots give us information about the existence and strength of a relationship between two datasets. To break that information down, there are a series of questions we might ask to help us. First: Is there a curved pattern in the data? If the answer is “yes,” then we can stop; none of the linear regression techniques from here to the end of this section are appropriate. and show several examples of scatter plots that can help us identify these curved patterns. Curved pattern No curved pattern Curved pattern No curved pattern Once we have confirmed that there is no curved pattern in our data, we can move to the next question: Is there a linear relationship? To answer this, we must look at different values of the explanatory variable and determine whether the corresponding response values are different, on average . It's important to look at the values “on average” because, in general, our scatter plots won’t include just one corresponding response point for each value of the explanatory variable (i.e., there may be multiple response values for each explanatory value). So, we try to look for the center of those points. Let’s look again at , but consider some different values for the explanatory variable. Let’s highlight the points whose x -values are around 50 and those that are around 80: Now, we can estimate the middle of each group of points. Let's add our estimated averages to the plot as starred points: Since those two starred points occur at different heights, we can conclude that there’s likely a relationship worth exploring. Here’s another example using a different set of data: Let’s look again at the points near 50 and near 80, and estimate the middles of those clusters: Notice that there’s not much vertical distance between our two starred points. This tells us that there’s not a strong relationship between these two datasets. Positive and Negative Linear Relationships Another way to assess whether there is a relationship between two datasets in a scatter plot is to see if the points seem to be clustered around a line (specifically, a line that’s not horizontal). The stronger the clustering around that line is, the stronger the relationship. Once we’ve established that there’s a relationship worth exploring, it’s time to start quantifying that relationship. Two datasets have a positive linear relationship if the values of the response tend to increase, on average, as the values of the explanatory variable increase. If the values of the response decrease with increasing values of the explanatory variable, then there is a negative linear relationship between the two datasets. The strength of the relationship is determined by how closely the scatter plot follows a single straight line: the closer the points are to that line, the stronger the relationship. The scatter plots in to depict varying strengths and directions of linear relationships. Perfect negative relationship Strong negative relationship Weak negative relationship No relationship Weak positive relationship Strong positive relationship Perfect positive relationship The strength and direction (positive or negative) of a linear relationship can also be measured with a statistic called the correlation coefficient (denoted r ). Positive values of r indicate a positive relationship, while negative values of r indicate a negative relationship. Values of r close to 0 indicate a weak relationship, while values close to ± 1 correspond to a very strong relationship. Looking again at to , the correlation coefficients for each, in sequential order, are: ‒1, ‒0.97, ‒0.55, ‒0.03, 0.61, 0.97, and 1. There’s no firm rule that establishes a cutoff value of r to divide strong relationships from weak ones, but ± 0.7 is often given as the dividing line (i.e., if r > 0.7 or r < - 0.7 the relationship is strong, and if - 0.7 < r < 0.7 the relationship is weak). The formula for computing r is very complicated; it’s almost never done without technology. Google Sheets will do the computation for you using the CORREL function. The syntax works like this: if your explanatory values are in cells A2 to A50 and the corresponding response values are in B2 to B50, then you can find the correlation coefficient by entering “=CORREL(A2:A50, B2:B50)”. (Note that the order doesn’t matter for correlation coefficients; “=CORREL(B2:B50, A2:A50)” will give the same result.) Let’s put all of this together in an example. Interpreting Scatter Plots Consider the four scatter plots below: For each of these, answer the following questions: Is there a curved pattern in the data? If yes, stop here. If no, continue to part b. Classify the strength and direction of the relationship. Make a guess at the value of r . Yes, there is a curved pattern. No, there’s no curved pattern. Since the points tend upward as we move from left to right, this is a positive relationship. The points seem pretty closely grouped around a line, so it’s fairly strong. Comparing this scatter plot to those in to , we can see that the relationship is stronger than the one in ( r = 0.61 ) but not as strong as the one in ( r = 0.97 ). So, the value of the correlation coefficient is somewhere between the two. We might guess that r = 0.9 . No, there’s no curved pattern. Since the points tend downward as we move from left to right, this is a negative relationship. The points are not tightly grouped around a line, but the pattern is clear. It looks like it has approximately the same strength as the plot in , just with the opposite sign. So, we might guess that r = - 0.6 . No, there’s no curved pattern. Since the points don’t really tend upward or downward as we move from left to right, there is no real relationship here. Thus, r ≈ 0 . Finding the Correlation Coefficient The data that were plotted in the previous example can be found in the dataset “correlationcoefficient1” . All of them share the same values for the explanatory variable x . The four responses are labeled y 1 through y 4 . Compute the correlation coefficients for each, if appropriate, using Google Sheets. Round to the nearest hundredth. Step 1: There is a curved pattern in the data, so the correlation coefficient isn’t meaningful. Step 2: Using “=CORREL(A2:A101, C2:C101)” we get r = 0.89 . Step 3: Using “=CORREL(A2:A101, D2:D101)” we get r = - 0.66 . Step 4: Using “=CORREL(A2:A101, E2:E101)” we get r = - 0.04 . Winning with Statistics Billy Beane, the former general manager of the Oakland A’s baseball team, famously took his low budget team to unprecedented heights by using statistics to identify undervalued players; his story is recounted in the book Moneyball (which was later made into a movie, with Brad Pitt playing Beane). You can do the same thing: Take a look at team statistics in the sport of your choice and try to identify a statistic that’s most closely related to winning (meaning that it has the highest correlation coefficient with team wins). Linear Regression The final step in our analysis of the relationship between two datasets is to find and use the equation of the regression line. For a given set of explanatory and response data, the regression line (also called the least-squares line or line of best fit ) is the line that does the best job of approximating the data. What does it mean to say that a particular line does the “best job” of approximating the data? The way that statisticians characterize this “best line” is rather technical, but we’ll include it for the sake of satisfying your curiosity (and backing up the claim of "best"). Imagine drawing a line that looks like it does a pretty good job of approximating the data. Most of the points in the scatter plot will probably not fall exactly on the line; the distance above or below the line a given point falls is called that point’s residual . We could compute the residuals for every point in the scatter plot. If you take all those residuals and square them, then add the results together, you get a statistic called the sum of squared errors for the line (the name tells you what it is: “sum” because we’re adding, “squared” because we’re squaring, and “errors” is another word for “residuals”). The line that we choose to be the “best” is the one that has the smallest possible sum of squared errors. The implied minimization (“smallest”) is where the “least” in “least squares” comes from; the “squares” comes from the fact that we’re minimizing the sum of squared errors. This is very similar to the process we outlined in the "game" that we used to introduce the mean. Both the regression line and the mean are designed to minimize a sum of squared errors. Here ends the super technical part. Finding the Equation of the Regression Line So, how do we find the equation of the regression line? Recall the point-slope form of the equation of a line: If a line has slope m and passes through a point ( x 0 , y 0 ) , then the point-slope form of the equation of the line is: y = m ( x - x 0 ) + y 0 The regression line has two properties that we can use to find its equation. First, it always passes through the point of means . If x ¯ and y ¯ are the means of the explanatory and response datasets, respectively, then the point of means is ( x ¯ , y ¯ ) . We’ll use that as the point in the point-slope form of the equation. Second, if s x and s y are the standard deviations of the explanatory and response datasets, respectively, and if r is the correlation coefficient, then the slope is m = r × s y s x . Putting all that together with the point-slope formula gives us this: Suppose x and y are explanatory and response datasets that have a linear relationship. If their means are x ¯ and y ¯ respectively, their standard deviations are s x and s y respectively, and their correlation coefficient is r , then the equation of the regression line is: y = r ( s y s x ) ( x - x ¯ ) + y ¯ . Let's walk through an example. Finding the Equation of the Regression Line from Statistics Suppose you have datasets x and y with the following statistics: x has mean 21 and standard deviation 4, y has mean 8 and standard deviation 2, and their correlation coefficient is −0.4. What’s the equation of the regression line? Step 1: We’re given x ¯ = 21 , s x = 4 , y ¯ = 8 , s y = 2 , and r = - 0.4 . Let's start with the formula for the equation of the regression line: y = r ( s y s x ) ( x - x ¯ ) + y ¯ Step 2: Plugging in our values gives us: y = - 0.4 ( 2 4 ) ( x - 21 ) + 8 Step 3: Our final regression line equation is: y = - 0.2 x + 12.2 As you can see, finding the equation of the regression line involves a lot of steps if you have to find all of the values of the needed quantities yourself. But, as usual, technology comes to our rescue. This video (which you actually watched earlier when learning how to create scatter plots) covers the regression line at around the 3:30 mark. Note that Google Sheets calls it the "trendline." Let's put this into practice. Finding the Equation of the Regression Line Using Google Sheets In , we considered the relationship between goals scored (GF) and goals against (GA) using the dataset “NHL19” . Recreate the scatter plot in Google Sheets, and use it to find the equation of the regression line. Once we have recreated the scatter plot, we find the equation of the regression line by clicking the three dots at the top right of the plot, selecting “Edit chart,” then clicking on “Customize” and “Series.” We add the regression line by checking the box next to “Trendline,” and then we show the equation by selecting “Use Equation” in the drop-down menu under “Label.” The equation of the tangent line is y = - 0.0554 x + 261 . Using the Equation of the Regression Line Once we’ve found the equation of the regression line, what do we do with it? We’ll look at two possible applications: making predictions and interpreting the slope. We can use the equation of the regression line to predict the response value y for a given explanatory value x . All we have to do is plug that explanatory value into the formula and see what response value results. This is useful in two ways: first, it can be used to make a guess about an unknown data value (like one that hasn’t been observed yet). Second, it can be used to evaluate performance (meaning, we can predict an outcome given a particular event). In , we created a scatter plot of final exam scores vs. midterm exam scores using this data: Name Midterm Grade Final Grade Allison 88 84 Benjamin 71 80 Carly 75 77 Daniel 94 95 Elmo 68 73 The equation of the regression line is y = 0.687 x + 27.4 , where y is the final exam score and x is the midterm exam score. If Frank scored 85 on the midterm, then our prediction for his final exam score is 0.687 × 85 + 27.4 = 85.795 . To use the regression line to evaluate performance, we use a data value we’ve already observed. For example, Allison scored 88 on the midterm. The regression line predicts that someone who scores an 88 on the midterm will get 0.687 × 88 + 27.4 = 87.856 on the final. Allison actually scored 84 on the final, meaning she underperformed expectations by almost 4 points ( 87.856 - 84 ) . The second application of the equation of the regression line is interpreting the slope of the line to describe the relationship between the explanatory and response datasets. For the exam data in the previous paragraph, the slope of the regression line is 0.687. Recall that the slope of a line can be computed by finding two points on the line and dividing the difference in the y -values of those points by the difference in the x -values. Keeping that in mind, we can interpret our slope as 0.687 = difference in final scores difference in midterm scores . Multiplying both sides of that equation by the denominator of the fraction, we get 0.687 × difference in midterm scores = difference in final scores . Thus, a one-point increase in the midterm score would result in a predicted increase in the final score of 0.687 points. A ten-point drop in the midterm score would give us a decrease in the predicted final score of 6.87 points. In general, the slope gives us the predicted change in the response that corresponds to a one unit increase in the explanatory variable. Applying the Equation of the Regression Line The data in “MLB2019Off” gives offensive team stats for the 2019 Major League Baseball season. Use that dataset to answer the following questions: What is the equation of the regression line for runs (R) vs. hits (H)? How many runs would we expect a team to score if the team got 1500 hits in a season? Did the Kansas City Royals (KCR) overperform or underperform in terms of runs scored, based on their hit total? By how much? Write a sentence to interpret the slope of the regression line. Using Google Sheets, we find that the regression line equation is y = 0.884 x – 456 , where y is the number of runs scored and x is the number of hits. Plugging 1500 into the equation of the regression line, we get 0.884 × 1500 - 456 = 870 . We would predict that a team with 1500 hits would score 870 runs. The Royals had 1356 hits, so we would predict their run total to be 0.884 × 1356 - 456 ≈ 743 . They actually scored 691 runs, so they underperformed expectations by 52 runs ( 743 – 691 ) . The slope gives us the predicted change in the response that corresponds to a one unit increase in the explanatory variable. So, we expect one additional hit to result in 0.884 more runs. Since 0.884 runs doesn’t really make sense, we can get a better interpretation by multiplying through by ten or one hundred: Ten additional hits will result in almost nine additional runs, or a hundred additional hits will yield on average just over 88 additional runs. Math and the Movies Statistics and regression are used by Hollywood movie producers to decide what movies to make, and to predict how much money they’ll earn at the box office. According to the American Statistical Association , not only do producers use statistics to identify the next potential blockbuster, but they’ve also pinned down how much money awards add to the bottom line. (An Academy Award is worth about $3 million!) In addition, studios use their streaming services to gather data about their customers and the types of movies they watch; this data helps them learn what kinds of entertainment their customers want more of. Collecting and Analyzing Your Own Data This section has demonstrated many pairs of related quantitative datasets. Think about some quantitative variables that you can ask your classmates about, which might be related. Once you have some ideas, collect the data from your classmates. Then analyze the data by creating a scatter plot, finding the equation of the regression line (if appropriate), and interpreting it. Extrapolation A very common misuse of regression techniques involves extrapolation , which involves making a prediction about something that doesn't belong in the dataset. More Applying the Equation of the Regression Line The data in “WNBA2019” gives team statistics from the 2019 WNBA season. Use that dataset to answer these questions about team wins (W) and the proportion of team field goals made (FG%, the number of shots made divided by the number of shots attempted. Even though this column is labeled using a percent sign, the values are not expressed as percentages): What is the equation of the regression line for wins vs. proportion of made field goals? How many wins would we expect for a team that makes 42% of its shots? Did the New York Liberty overperform or underperform in terms of wins, based on the team’s proportion of made field goals? Write a sentence to interpret the slope of the regression line. Using Google Sheets, we get the equation y = 178.097 x - 58.543 , where x is the proportion of field goals made and y is the number of wins. Since 42% corresponds to a proportion of 0.42, we’ll plug 0.42 into the regression equation for x , which gives us 178.097 × 0.42 - 58.543 ≈ 16.26 . We would predict that a team that makes 42% of its shots would win about 16 games. The New York Liberty made 41.4% of their shots, so we expect they would have 178.097 × 0.414 - 58.543 ≈ 15.19 wins. In fact, they had only 10 wins, so they underperformed expectations by over 5 wins. Step 1: The slope gives us the expected increase in the response that corresponds to a one unit increase in the explanatory variable. If we simply go with that interpretation, we would get a sentence like “We expect an increase in proportion of field goals made of 1 would result in an additional 178 wins.” However, that sentence doesn't make much sense. Let's consider why. Step 2: First, proportions must be between 0 and 1, the proportion of made field goals can’t be increased by 1 and still make sense. Second, the total number of games played is only 34, so no team could get an additional 178 wins! So, we’ll have to change the units. Step 3: Since the proportions of made field goals are often expressed as a percentage, we could try to use that. If we express the slope as a fraction with 1 in the denominator (remember, the denominator represents the proportion of field goals made), then convert the denominator to a percentage and simplify, we get 178.097 1 = 178.097 100 % ≈ 1.78 1 % . Step 4: So, an increase in field goal percentage of 1% would result in an expected increase of 1.78 wins. Correlation Does Not Imply Causation One of the most common fallacies about statistics has to do with the relationship between two datasets. In the dataset “Public” , we find that the correlation coefficient between the 75th percentile math SAT score and the 75th percentile verbal SAT score is 0.92, which is really strong. The slope of the regression line that predicts the verbal score from the math score is 0.729, which we might interpret as follows: “If the 75th percentile math SAT score goes up by 10 points, we’d expect the corresponding verbal SAT score to go up by just over 7 points.” Does the increasing math score cause the increase in the verbal score? Probably not. What’s really going on is that there’s a third variable that’s affecting them both: To raise the SAT math score by 10 points, a school will recruit students who do better on the SAT in general; these students will also naturally have higher SAT verbal scores. This third variable is sometimes called a lurking variable or a confounding variable . Unless all possible lurking variables are ruled out, we cannot conclude that one thing causes another. Dr. Talithia Williams A photo of Dr. Williams (credit: Used by permission of Talithia Williams) Dr. Talithia Williams is a statistician on the faculty of Harvey Mudd College, and the first Black woman to achieve tenure at this university. She advocates for more women to become involved in the fields of engineering and science, and is on the board of directors for the EDGE Foundation, an organization that helps women obtain advanced degrees in mathematics (EDGE standing for Enhancing Diversity in Graduate Education). In 2018, Dr. Williams published the book Power in Numbers: The Rebel Women of Mathematics, a retrospective look at historical female figures who have contributed to the development of the field of mathematics. Dr. Williams earned a Master’s degree in Mathematics from Howard University and a Master’s in Statistics from Rice University, and also went on to earn her Ph.D. in Statistics from Rice. She has held research appointments at the Jet Propulsion Laboratory, the National Security Agency, and NASA. Her research focuses on the environmental and medical applications of statistics. In 2014, she gave a popular TED talk titled “Own Your Body’s Data” that discussed the potential insights to be gained from collecting personal health data. She was even recently a host for the NOVA Wonders documentary series and a narrator for the NOVA Universe series on PBS. To stay up-to-date on Dr. Williams’s accomplishments, you can follow her on Twitter or her Facebook account . Statistics and Eugenics Some of the brightest minds in the history of statistics unfortunately decided to use their considerable intellects to further a pseudoscience known as eugenics. Eugenicists took Charles Darwin’s theories of evolution and ruthlessly applied them to the human race. Francis Galton (1822–1911), a cousin of Darwin and also the mathematician who invented the formula for standard deviation, claimed that people in the British upper classes possessed higher intelligence due to their superior breeding. Karl Pearson (1857–1936), who derived the formula for the correlation coefficient, argued in National Life from the Standpoint of Science , that, instead of providing social welfare programs, nations could better improve the fortunes of the poor by waging “war with inferior races.” Ronald Fisher (1890–1962) was possibly the most important statistician of the 20th century, having invented several new techniques (including the ubiquitous analysis of variance), and yet he also founded the Cambridge University Undergraduates Eugenics Society, whose self-prescribed goal was to evangelize “not by precept only, but by example, the doctrine of a new natural ability of worth and blood.” When eugenics took hold in the United States, it was used to justify terrible acts by the government, including the forced sterilization of individuals with mental illness, epilepsy, a physical impairment (like blindness), or a criminal history. The Nazi regime took these ideas to their ultimate, terrible conclusion: killing people who had mental or physical disabilities, or who were born into an “inferior” race. Over six million people died in this Holocaust, one of the darkest events in human history. To learn more, watch this video about Francis Galton and the legacy of eugenics Check Your Understanding Key Terms response variable (dependent variable) explanatory variable (independent variable) scatter plot positive linear relationship negative linear relationship correlation coefficient regression line (least-squares line, line of best fit) Key Concepts If one variable affects the value of another variable, we say the first is an explanatory variable and the second is a response variable. Scatter plots place a point in the xy -plane for each unit in the dataset. The x -value is the value of the explanatory variable, and the y -value is the value of the response variable. The correlation coefficient r gives us information about the strength and direction of the relationship between two variables. If r is positive, the relationship is positive: an increase in the value of the explanatory variable tends to correspond to an increase in the value of the response variable. If r is negative, the relationship is negative: an increase in the value of the explanatory variable tends to correspond to a decrease in the value of the response variable. Values of r that are close to 0 indicate weak relationships, while values close to –1 or indicate strong relationships. The regression line for a relationship between two variables is the line that best represents the data. It can be used to predict values of the response variable for a given value of the explanatory variable. Formulas If a line has slope m and passes through a point ( x 0 , y 0 ) , then the point-slope form of the equation of the line is: y = m ( x - x 0 ) + y 0 Suppose x and y are explanatory and response datasets that have a linear relationship. If their means are x ¯ and y ¯ respectively, their standard deviations are s x and s y respectively, and their correlation coefficient is r , then the equation of the regression line is: y = r ( s y s x ) ( x - x ¯ ) + y ¯ . Videos Making Scatter Plots in Google Sheets Projects Browse through some news websites to find five stories that report on data and include data visualizations. Can you tell from the report how the data were collected? Was randomization used? Are the visualizations appropriate for the data? Are the visualizations presented in a way that might bias the reader? We discussed three measures of centrality in this chapter: the mode, the median, and the mean. In a broader context, the mean as we discussed it is more properly called the arithmetic mean , to distinguish it from other types of means. Examples of these include the geometric mean, harmonic mean, truncated mean, and weighted mean. How are these computed? How do they compare to the arithmetic mean? In what situations would each of these be preferred to the arithmetic mean? Simpson’s Paradox is a statistical phenomenon that can sometimes appear when we observe a relationship within several subgroups of a population, but when the data for all thegroups are analyzed all together, the opposite relationship appears. Find some examples of Simpson’s Paradox in real-world situations, and write a paragraph or two that would explain the concept to someone who had never studied statistics before. Chapter Review Gathering and Organizing Data Visualizing Data Mean, Median and Mode Range and Standard Deviation Percentiles The Normal Distribution Applications of the Normal Distribution Scatter Plots, Correlation, and Regression Lines Chapter Test

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Introduction A road sign in Finland, a country that uses the metric system. (credit: modification of work by Anna Järvenpää/Flickr, Public Domain) You are planning a road trip from your home state to the sunny beaches of Mexico and need to prepare a budget. While in the United States, gasoline is sold in gallons and distances are measured in miles, but in almost any other country you will find that gasoline is sold in liters and distance is measured in kilometers. Whether you’re traveling, baking, watching an international sporting event, working with machine tools, or using scientific equipment, it's important to understand the metric system , or the International System of Units (SI). The metric system is a decimal measuring system that uses meters, liters, and grams to quantify length, capacity, and mass. It is used in all but three countries in the world, including the United States.

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The Metric System A scale that measures weight in both metric and customary units. (credit: “Weighing the homemade cheddar” by Ruth Hartnup/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Identify units of measurement in the metric system and their uses. Order the six common prefixes of the metric system. Convert between like unit values. Even if you don’t travel outside of the United States, many specialty grocery stores utilize the metric system. For example, if you want to make authentic tamales you might visit the nearest Hispanic grocery store. While shopping, you discover that there are two brands of masa for the same price, but one bag is marked 1,200 g and the other 1 kg. Which one is the better deal? Understanding the metric system allows you to understand that 1,200 grams is equivalent to 1.2 kg, so the 1,200 g bag is the better deal. Units of Measurement in the Metric System Units of measurement provide common standards so that regardless of where or when an object or substance is measured, the results are consistent. When measuring distance, the units of measure might be feet, meters , or miles. Weight might be expressed in terms of pounds or grams . Volume or capacity might be measured in gallons, or liters . Understanding how metric units of measure relate to each other is essential to understanding the metric system: The metric unit for distance is the meter (m). A person’s height might be written as 1.8 meters (1.8 m). A meter slightly longer than a yard (3 feet), while a centimeter is slightly less than half an inch. The metric unit for area is the square meter (m 2 ). The area of a professional soccer field is 7,140 square meters (7,140 m 2 ). The metric base unit for volume is the cubic meter (m 3 ). However, the liter (L), which is a metric unit of capacity, is used to describe the volume of liquids. Soda is often sold in 2-liter (2 L) bottles. The gram (g) is a metric unit of mass but is commonly used to express weight. The weight of a paper clip is approximately 1 gram (1 g). The metric unit for temperature is degrees Celsius (°C). The temperature on a warm summer day might be 24 °C. While the U.S. Customary System of Measurement uses ounces and pounds to distinguish between weight units of different sizes, in the metric system a base unit is combined with a prefix, such as kilo– in kilogram, to identify the relationship between smaller or larger units. When using abbreviations to represent metric measures, always separate the quantity and the units with a space, with no spaces between the letters or symbols in the units. For example, 7 millimeters is written as 7 mm, not 7mm. It is important to be able to convert between the U.S. Customar System of Measurement and the metric system. However, in this chapter we’ll focus on converting units within the metric system. Why? Typing “200 centimeters in inches” into any browser search bar will instantly convert those measures for you (Figure 9.3). You’ll have an opportunity in the Projects to work between measurement systems. (credit: Screenshot/Google) Let’s be honest. Most of us use computers or smartphones to perform many of the calculations and conversions we were taught in math class. But there is value in understanding the metric system since it exists all around us, and most importantly, knowing how the different metric units relate to each other allows you to compare prices, find the right tool in a workshop, or acclimate when in another country. No matter the circumstance, you cannot avoid the metric system. Determining the Correct Base Unit Which base unit would be used to express the following? amount of water in a swimming pool length of an electrical wire weight of one serving of peanuts Liquid volume is generally expressed in units of liters (L). Length is measured in units of meters (m). Weight is commonly expressed in units of grams (g). While there are other base units in the metric system, our discussions in this chapter will be limited to units used to express length, area, volume, weight, and temperature. Metric Prefixes Unlike the U.S. Customary System of Measurement in which 12 inches is equal to 1 foot and 3 feet are equal to 1 yard, the metric system is structured so that the units within the system get larger or smaller by a power of 10. For example, a centimeter is 10 − 2 , or 100 times smaller than a meter, while the kilometer is 10 3 , or 1,000 times larger than a meter. The metric system combines base units and unit prefixes reasonable to the size of a measured object or substance. The most used prefixes are shown in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died From Drinking Chocolate Milk. Prefix kilo– hecto– deca– base unit deci– cent– milli– Abbreviation k h da d c m Magnitude 10 3 10 2 10 1 10 0  or  1 10 − 1 10 − 2 10 − 3 Metric Prefixes Ordering the Magnitude of Units Order the measures from smallest unit to largest unit. centimeter, millimeter, decimeter Looking at the metric prefixes, we can see that the prefix order from smallest unit to largest unit is milli-, centi-, deci-, so the order of the units from smallest to largest is millimeter, centimeter, decimeter. Determining Reasonable Values for Length What is a reasonable value for the length of a person’s thumb: 5 meters, 5 centimeters, or 5 millimeters? Given that a meter is slightly longer than a yard, 5 meters is not a reasonable value for the length of a person’s thumb. Since a millimeter is 10 times smaller than a centimeter, which is approximately 1 2 inch, 5 millimeters is not a reasonable estimate for the length of a person’s thumb. The correct answer is 5 centimeters. Converting Metric Units of Measure Imagine you order a textbook online and the shipping detail indicates the weight of the book is 1 kg. By attaching the letter “k” to the base unit of gram (g), the unit used to express the measure is 10 3 or 1,000 times greater than a gram. One kilogram is equivalent to 1,000 grams. The tip of a highlighter measures approximately 1 cm. The letter “c” attached to the base unit of meter (m) means the unit used to express the measure is 10 − 2 or 1 100 of a meter. One meter is equivalent to 100 centimeters. A conversion factor is used to convert from smaller metric units to bigger metric units and vice versa. It is a number that when used with multiplication or division converts from one metric unit to another, both having the same base unit. In the metric system, these conversion factors are directly related to the powers of 10. The most common used conversion factors are shown in . Common Metric Conversion Factors for (a) Meters, (b) Liters, and (c) Grams Converting Metric Distances Using Multiplication The firehouse is 13.45 km from the library. How many meters is it from the firehouse to the library? When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from kilometer to the base unit of meters is 1,000. 13.45 × 1,000 = 13,450 So, the firehouse is 13,450 meters away from the firehouse. Converting Metric Capacity Using Division How many liters is 3,565 milliliters? When converting from a smaller unit to a larger unit, use division. The conversion factor from milliliter to the base unit of liters is 1,000. 3,565 ÷ 1,000 = 3.565 So, 3,565 milliliters is 3.565 liters. Converting Metric Units of Mass to Solve Problems Caroline and Aiyana are working on a chemistry experiment together and must perform calculations using measurements taken during the experiment. Due to miscommunication, Caroline took measurements in centigrams and Aiyana used milligrams. Convert Caroline’s measurement of 125 centigrams to milligrams. When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from centigrams to the milligrams is 10. 125 × 10 = 1,250 So, the 125 centigrams are 1,250 milligrams. Converting Metric Units of Volume to Solve Problems A bottle contains 500 mL of juice. If the juice is packaged in 24-bottle cases, how many liters of juice does the case contain? Step 1: Multiply the amount of juice in each bottle by the number of bottles. 500  mL × 24 = 12,000  mL Step 2: Divide by 1,000 to convert from milliliters to liters. 12,000 mL 1,000 = 12 L So, there are 12 liters of juice in each case. Valerie Antoine In the 1970s, people were told that they must learn the metric system because the United States was soon going to convert to using metric measurements. Children and young adults probably watched educational cartoons about the metric system on Saturday mornings. In 1975, President Gerald Ford signed the Metric Conversion Act and created a board of 17 people commissioned to coordinate the voluntary switch to the metric system in the United States. Among those 17 people was Valerie Antoine, an engineer who made it her life’s work to push for this change. Despite President Ronald Reagan dissolving the board in 1982, effectively killing the move to the metric system at the time, Antoine continued the movement out of her own home as the executive director of the U.S. Metric Association. Reagan’s decision followed intensive lobbying by American businesses whose factories used machinery designed to use customary measurements by workers trained in customary measurements. There was also intense public pressure from American citizens who didn’t want to go through the time consuming and expensive process of changing the country’s entire infrastructure. Fueled by a Congressional mandate in 1992 that required all federal agencies make the switch to the metric system, Antoine never gave up hope that the metric system would trickle down from the government and find its way into American schools, homes, and everyday life. U.S. Office of Education: Metric Education Converting Grams to Solve Problems The nutrition label on a jar of spaghetti sauce indicates that one serving contains 410 mg of sodium. You have poured two servings over your favorite pasta before recalling your doctor’s advice about keeping your sodium consumption below 1 g per meal. Have you followed your doctor’s recommendation? Step 1: Multiply the number of servings by the amount of sodium in each serving. 410  mg × 2 = 820  mg Step 2: Divide by 1,000 to convert from milligrams to grams. 820  mg 1,000 = 0.82  g You have followed doctor’s recommendation because 0.82 g is less than 1 gram. Comparing Different Units A student carefully measured 0.52 cg of copper for a science experiment, but their lab partner said they need 6 mg of copper total. How many more centigrams of copper does the student need to add? Step 1: Convert these two measurements into a common unit. Since the question asks for the number of centigrams, convert 6 mg to centigrams, which is 0.6 cg. Step 2: Find the difference by subtracting 0.6 − 0.52 which is 0.08 cg. This means the student must add another 0.08 cg of copper. The United States and the Metric System Did you know that the metric system pervades daily life in the United States already? While Americans still may purchase gallons of milk and measure house sizes in square feet, there are many instances of the metric system. Photographers buy 35 mm film and use 50 mm lenses. When you have a headache, you might take 600 mg of ibuprofen. And if you are eating a low-carb diet you probably restrict your carb intake to fewer than 20 g of carbs daily. Did you know even the dollar is metric? In the video, Neil DeGrasse Tyson and comedian co-host Chuck Nice provide an amusing perspective on the metric system. The International System of Units (SI) is the current international standard metric system and is the most widely used system around the world. In most English-speaking countries SI units such as meter, liter, and metric ton are spelled metre, litre, and tonne. Neil deGrasse Tyson Explains the Metric System Get to Know the Metric System Just how much is the metric system a part of your life now? Probably more than you think. For the next 24 hours, take notice as you move through your daily activities. When you are shopping, are the package sizes provided in metric units? Change the weather app on your phone to display the temperature in degrees Celsius. Are you able to tell what kind of day it will be now? While the United States is not officially using the metric system, you will still find the metric system all around you. Check Your Understanding Key Terms metric system meter (m) gram (g) liter (L) square meter (m 2 ) cubic meter (m 3 ) degrees Celsius (°C) conversion factor Key Concepts The metric system is decimal system of weights and measures based on base units of meter, liter, and gram. The system was first proposed in 1670 and has since been adopted as the International System of Units and used in nearly every country in the world. Each successive unit on the metric scale is 10 times larger than the previous one. To convert between units with the same base unit, you must either multiply or divide by a power of 10. The most used prefixes are listed in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died from Drinking Chocolate Milk. Videos U.S. Office of Education: Metric Education Neil deGrasse Tyson Explains the Metric System Formulas You can convert between unit sizes with the same base unit using the conversion factors shown in .

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Measuring Area A painter uses an extension roller to paint a wall. (credit: "Paint Rollers are effective" by WILLPOWER STUDIOS/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Identify reasonable values for area applications. Convert units of measures of area. Solve application problems involving area. Area is the size of a surface. It could be a piece of land, a rug, a wall, or any other two-dimensional surface with attributes that can be measured in the metric unit for distance-meters. Determining the area of a surface is important to many everyday activities. For example, when purchasing paint, you’ll need to know how many square units of surface area need to be painted to determine how much paint to buy. Square units indicate that two measures in the same units have been multiplied together. For example, to find the area of a rectangle, multiply the length units and the width units to determine the area in square units. 1 cm × 1 cm = 1 cm 2 Note that to accurately calculate area, each of the measures being multiplied must be of the same units. For example, to find an area in square centimeters, both length measures (length and width) must be in centimeters. The formula used to determine area depends on the shape of that surface. Here we will limit our discussions to the area of rectangular-shaped objects like the one in Figure 9.6. Given this limitation, the basic formula for area is: Area = length ( l ) × width ( w ) or A = l × w Rectangle with Length ( l ) and Width ( w ) Labeled Reasonable Values for Area Because area is determined by multiplying two lengths, the magnitude of difference between different square units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a square meter ( m 2 ) is 100 × 100 , or 10,000 times greater in area than a square centimeter ( cm 2 ) . The relationships between benchmark metric area units are shown in the following table. Units Relationship Conversion Rate km 2 to m 2 km × km = km 2 1 km = 1,000 m 1,000 m × 1,000 m = 1,000,000 m 2 1  km 2 = 1,000,000 m 2 m 2 to cm 2 m × m = m 2 1 m = 100 cm 100 cm × 100 cm = 10,000  cm 2 1  m 2 = 10,000  cm 2 cm 2 to mm 2 cm × cm = cm 2 1 cm = 10 mm 10 mm × 10 mm = 100 mm 2 1 cm 2 = 100 mm 2 An essential understanding of metric area is to identify reasonable values for area. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given area is reasonable for the situation. Determining Reasonable Units for Area Which unit of measure is most reasonable to describe the area of a sheet of paper: km 2 , cm 2 , or mm 2 ? In the U.S. Customary System of Measurement, the length and width of paper is usually measured in inches. In the metric system centimeters are used for measures usually expressed in inches. Thus, the most reasonable unit of measure to describe the area of a sheet of paper is square centimeters. Square kilometers is too large a unit and square millimeters is too small a unit. Determining Reasonable Values for Area You want to paint your bedroom walls. Which represents a reasonable value for the area of the walls: 100 cm 2 , 100 m 2 , or 100 km 2 ? An area of 100 cm 2 is equivalent to a surface of 10 cm × 10 cm , which is much too small for the walls of a bedroom. An area of 100 km 2 is equivalent to a surface of 10 km × 10 km , which is much too large for the walls of a bedroom. So, a reasonable value for the area of the walls is 100 m 2 . Explaining Reasonable Values for Area A landscaper is hired to resod a school’s football field. After measuring the length and width of the field they determine that the area of the football field is 5,350 km 2 . Does their calculation make sense? Explain your answer. No, kilometers are used to determine longer distances, such as the distance between two points when driving. A football field is less than 1 kilometer long, so a more reasonable unit of value would be m 2 . An area of 5,350 km 2 can be calculated using the dimensions 53.5 by 100, which are reasonable dimensions for the length and width of a football field. So, a more reasonable value for the area of the football field is 5,350 m 2 . Converting Units of Measures for Area Just like converting units of measure for distance, you can convert units of measure for area. However, the conversion factor, or the number used to multiply or divide to convert from one area unit to another, is not the same as the conversion factor for metric distance units. Recall that the conversion factor for area is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are shown in . Common Conversion Factors for Metric Area Units Converting Metric Units of Area Converting Units of Measure for Area Using Division A plot of land has an area of 237,500,000 m 2 . What is the area in square kilometers? Use division to convert from a smaller metric area unit to a larger metric area unit. To convert from m 2 to km 2 , divide the value of the area by 1,000,000. 237,500,000 1,000,000 = 237.5 The plot of land has an area of 237.5 km 2 . Converting Units of Measure for Area Using Multiplication A plot of land has an area of 0.004046 km 2 . What is the area of the land in square meters? Use multiplication to convert from a larger metric area unit to a smaller metric area unit. To convert from km 2 to m 2 , multiply the value of the area by 1,000,000. 0.004046 × 1,000,000 = 4,046 The plot of land has an area of 4,046 m 2 . Determining Area by Converting Units of Measure for Length First A computer chip measures 10 mm by 15 mm. How many square centimeters is the computer chip? Step 1: Convert the measures of the computer chip into centimeters 10 mm = 1 cm 15 mm = 1.5 cm Step 2: Use the area formula to determine the area of the chip. 1 × 1.5 = 1.5 The computer chip has an area of 1.5 cm 2 . Solving Application Problems Involving Area While it may seem that solving area problems is as simple as multiplying two numbers, often determining area requires more complex calculations. For example, when measuring the area of surfaces, you may need to account for portions of the surface that are not relevant to your calculation. Solving for the Area of Complex Surfaces One side of a commercial building is 12 meters long by 9 meters high. There is a rolling door on this side of the building that is 4 meters wide by 3 meters high. You want to refinish the side of the building, but not the door, with aluminum siding. How many square meters of aluminum siding are required to cover this side of the building? Step 1: Determine the area of the side of the building. 12  m × 9  m = 106  m 2 Step 2: Determine the area of the door. 4  m × 3  m = 12  m 2 Step 3: Subtract the area of the door from the area of the side of the building. 106  m 2 − 12  m 2 = 92  m 2 So, you need to purchase 92  m 2 of aluminum siding. When calculating area, you must ensure that both distance measurements are expressed in terms of the same distance units. Sometimes you must convert one measurement before using the area formula . Solving for Area with Distance Measurements of Different Units A national park has a land area in the shape of a rectangle. The park measures 2.2 kilometers long by 1,250 meters wide. What is the area of the park in square kilometers? Step 1: Use a conversion fraction to convert the information given in meters to kilometers. 1,250  m × 1  km 1,000  m = 1.25  km Step 2: Multiply to find the area. 2.2  km × 1.25  km = 2.75  km 2 The park has an area of 2.75 km 2 . When calculating area, you may need to use multiple steps, such as converting units and subtracting areas that are not relevant. Solving for Area Using Multiple Steps A kitchen floor has an area of 15 m 2 . The floor in the kitchen pantry is 100 cm by 200 cm. You want to tile the kitchen and pantry floors using the same tile. How many square meters of tile do you need to buy? Step 1: Determine the area of the pantry floor in square centimeters. 100  cm × 200  cm = 20,000  cm 2 Step 2: Divide the area in cm 2 by the conversion factor to determine the area in m 2 since the other measurement for the kitchen floor is in m 2 . 20,000 10,000 = 2 The area of the kitchen pantry floor is 2 m 2 . Step 3: Add the two areas of the pantry and the kitchen floors together. 15  m 2 + 2  m 2 = 17  m 2 So, you need to buy 17 m 2 of tile. The Origin of the Metric System The metric system is the official measurement system for every country in the world except the United States, Liberia, and Myanmar. But did you know it originated in France during the French Revolution in the late 18th century? At the time there were over 250,000 different units of weights and measures in use, often determined by local customs and economies. For example, land was often measured in days, referring to the amount of land a person could work in a day. Why the Metric System Matters Check Your Understanding Key Terms area square units Key Concepts Area describes the size of a two-dimensional surface. It is the amount of space contained within the lines of a two-dimensional space. Area is measured in square meters units; in the metric system the base unit for area is square meters ( m 2 ). Videos Converting Metric Units of Area Why the Metric System Matters Formulas To determine the area of rectangular-shaped objects: Area = length ( l ) × width ( w ) or A = l × w Rectangle with Length ( l ) and Width ( w ) Labeled You can convert between metric area units using the conversion factors shown in .

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