text stringlengths 2 2.33k | source stringclasses 826
values |
|---|---|
Categorical data places units into groups (categories), while quantitative data is a numerical measure of a property of a unit. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
The sampling method for a study depends on the way that randomization is used to select units for the sample. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
Frequency distributions help to summarize data by counting the number of units that fall into a particular category or range of quantitative values. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
Categorical data can be visualized using pie charts or bar charts; quantitative data can be visualized using stem-and-leaf plots or histograms. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
Areas in pie charts and bar charts represent proportions of the data falling into a particular category, while areas in histograms represent proportions of the data that fall into a given range of data values (or âbinsâ). Stem-and-leaf plots are visual representations of entire datasets. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
By manipulating the axes, changing widths of bars, or making bad choices for bins, we can create data visualizations that misrepresent the distribution of data. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
The mode of a dataset is the value that appears the most frequently. The median is a value that is greater than or equal to no more than 50% of the data and less than or equal to no more than 50% of the data. The mean is the sum of all the data values, divided by the number of units in the dataset. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
The median of a dataset is not affected by outliers, but the mean will be biased toward outliers. This distinction might affect which measure of centrality is used to summarize a dataset. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
The range of a dataset is the difference between its largest and smallest values. The standard deviation is approximately the mean difference (in absolute value) that individual units fall from the mean of the dataset. | https://openstax.org/books/contemporary-mathematics/pages/8-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. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
Normally distributed data follow a bell-shaped, symmetrical distribution. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
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. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
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. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
Thezz-score for a data value is the number of standard deviations that value falls above (or below, if thezz-score is negative) the mean. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
We can use the normal distribution to estimate percentiles. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
We can usezz-scores to compare data values from different datasets. | https://openstax.org/books/contemporary-mathematics/pages/8-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. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
Scatter plots place a point in thexyxy-plane for each unit in the dataset. Thexx-value is the value of the explanatory variable, and theyy-value is the value of the response variable. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
The correlation coefficientrrgives us information about the strength and direction of the relationship between two variables. Ifrris positive, the relationship is positive: an increase in the value of the explanatory variable tends to correspond to anincreasein the value of the response variable. Ifrris negative, the r... | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
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. | https://openstax.org/books/contemporary-mathematics/pages/8-key-concepts |
You can convert between unit sizes with the same base unit using the conversion factors shown inFigure 9.4. | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
To determine the area of rectangular-shaped objects: | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
Area=length(l)Ãwidth(w)Area=length(l)Ãwidth(w) | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
You can convert between metric area units using the conversion factors shown inFigure 9.7. | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
To determine the volume of a rectangular prism: | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
You can convert between metric volume units and metric capacity units using the relationships shown inTable 9.2. | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
You can convert between metric weight units using the conversion factors shown inFigure 9.13. | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
To convert temperature from Fahrenheit to Celsius: | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
To convert temperature from Celsius to Fahrenheit: | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
To estimate temperature from Fahrenheit to Celsius: | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
To estimate temperature from Celsius to Fahrenheit: | https://openstax.org/books/contemporary-mathematics/pages/9-formula-review |
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. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
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. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
The most used prefixes are listed inTable 9.1. An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonicKing Henry Died from Drinking Chocolate Milk. | https://openstax.org/books/contemporary-mathematics/pages/9-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. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
Area is measured in square meters units; in the metric system the base unit for area is square meters (m2m2). | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
Volume is a measure of the space contained within or occupied by three-dimensional objects. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
Volume is measured in cubic units; the base unit for volume in the metric system is cubic meters (m3) | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
The liter (L) is a metric unit of capacity but is often used to express the volume of liquids. One liter is equivalent to one cubic decimeter, which is the volume of a cube measuring10cmÃ10cmÃ10cm10cmÃ10cmÃ10cm. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
Mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The mass of an object never changes; the weight of an object changes depending on the force of gravity. An object with the same mass would weigh less on the moon than on Earth because the moonâs gravity is less than... | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
In the metric system, weight and mass are often used interchangeably and are expressed in terms in grams or kilograms. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
In the metric system, temperature is measured using the Celsius (°C) scale. | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
The Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to anotherâfrom solid (ice) to liquid (water) to gas (water vapor). | https://openstax.org/books/contemporary-mathematics/pages/9-key-concepts |
To translate an angle measured in degrees to radians, multiply byÏ180.Ï180. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
To translate an angle measured in radians to degrees, multiply by180Ï.180Ï. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The formula for the perimeterPPof a rectangle isP=2L+2WP=2L+2W, twice the lengthLLplus twice the widthWW. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The sum of the interior angles of a polygon withnnsides is given by | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The measure of each interior angle of a regular polygon withnnsides is given by | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
To find the measure of an exterior angle of a regular polygon withnnsides we use the formula | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The circumference of a circle is found using the formulaC=Ïd,C=Ïd,whereddis the diameter of the circle, orC=2Ïr,C=2Ïr,whererris the radius. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The area of a triangle is given asA=12bh,A=12bh,wherebbrepresents the base andhhrepresents the height. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The formula for the area of a square isA=sâsA=sâsorA=s2.A=s2. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The area of a rectangle is given asA=lw.A=lw. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The area of a parallelogram isA=bh.A=bh. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The formula for the area of a trapezoid is given asA=12h(a+b).A=12h(a+b). | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The area of a rhombus is found using one of these formulas: | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
A=d1d22,A=d1d22,whered1d1andd2d2are the diagonals. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
A=12bh,A=12bh,wherebbis the base andhhis the height. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The area of a regular polygon is found with the formulaA=12ap,A=12ap,whereaais the apothem andppis the perimeter. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The area of a circle is given asA=Ïr2,A=Ïr2,whererris the radius. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The formula for the surface area of a right prism is equal to twice the area of the base plus the perimeter of the base times the height,SA=2B+ph,SA=2B+ph,whereBBis equal to the area of the base and top,ppis the perimeter of the base, andhhis the height. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The formula for the volume of a rectangular prism, given in cubic units, is equal to the area of the base times the height,V=Bâh,V=Bâh,whereBBis the area of the base andhhis the height. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The surface area of a right cylinder is given asSA=2Ïr2+2Ïrh.SA=2Ïr2+2Ïrh. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The volume of a right cylinder is given asV=Ïr2h.V=Ïr2h. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
The Pythagorean Theorem states | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
whereaaandbbare two sides (legs) of a right triangle andccis the hypotenuse. | https://openstax.org/books/contemporary-mathematics/pages/10-formula-review |
Modern-day geometry began in approximately 300 BCE with EuclidâsElements, where he defined the principles associated with the line, the point, and the plane. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The union of two sets,AAandBB, contains all points that are in both sets and is symbolized asAâªB.AâªB. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The intersection of two setsAAandBBincludes only the points common to both sets and is symbolized asAâ©B.Aâ©B. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
Angles are classified as acute if they measure less than90â,90â,obtuse if they measure greater than90â90âand less than180â,180â,right if they measure exactly90â,90â,and straight if they measure exactly180â.180â. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
If the sum of angles equals90â90â, they are complimentary angles. If the sum of angles equals180â180â, they are supplementary. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The sum of the interior angles of a triangle equals180â.180â. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same... | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The perimeter of a polygon is the measure of the outline of the shape. We determine a shapeâs perimeter by calculating the sum of the lengths of its sides. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The sum of the interior angles of a regular polygon withnnsides is found using the formulaS=(nâ2)180â.S=(nâ2)180â.The measure of a single interior angle of a regular polygon withnnsides is determined using the formulaa=(nâ2)180ân.a=(nâ2)180ân. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The sum of the exterior angles of a regular polygon is360â.360â.The measure of a single exterior angle of a regular polygon withnnsides is found using the formulab=360ân.b=360ân. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The circumference of a circle isC=2Ïr,C=2Ïr,whererris the radius, orC=Ïd,C=Ïd,andddis the diameter. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no gaps or overlaps. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
Properties of tessellations include rigid motions of the shapes called transformations. Transformations refer to translations, rotations, reflections, and glide reflections. Shapes are transformed in such a way to create a pattern. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The areaAAof a triangle is found with the formulaA=12bh,A=12bh,wherebbis the base andhhis the height. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The area of a parallelogram is found using the formulaA=bh,A=bh,wherebbis the base andhhis the height. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The area of a rectangle is found using the formulaA=lw,A=lw,wherellis the length andwwis the width. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The area of a trapezoid is found using the formulaA=12h(b1+b2),A=12h(b1+b2),wherehhis the height,b1b1is the length of one base, andb2b2is the length of the other base. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The area of a rhombus is found using the formulaA=d1d22,A=d1d22,whered1d1is the length of one diagonal andd2d2is the length of the other diagonal. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The area of a regular polygon is found using the formulaA=12ap,A=12ap,whereaais the apothem andppis the perimeter. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The area of a circle is found using the formulaA=Ïr2,A=Ïr2,whererris the radius. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
A right prism is a three-dimensional object that has a regular polygonal face and congruent base such that that lateral sides form a90â90âangle with the base and top. The surface areaSASAof a right prism is found using the formulaSA=2B+ph,SA=2B+ph,whereBBis the area of the base,ppis the perimeter of the base, andhh... | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
A right cylinder is a three-dimensional object with a circle as the top and a congruent circle is the base, and the side forms a90â90âangle to the base and top. The surface area of a right cylinder is found using the formulaSA=2Ïr2+2Ïrh,SA=2Ïr2+2Ïrh,whererris the radius andhhis the height. The volume is found u... | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The Pythagorean Theorem is applied to right triangles and is used to find the measure of the legs and the hypotenuse according the formulaa2+b2=c2,a2+b2=c2,wherecis the hypotenuse. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
To find the measure of the sides of aspecialangle, such as a30â-60â-90â30â-60â-90âtriangle, use the ratiox:x3:2x,x:x3:2x,where each of the three sides is associated with the opposite angle and 2xxis associated with the hypotenuse, opposite the90â90âangle. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
To find the measure of the sides of the secondspecialtriangle, the45â-45â-90â45â-45â-90âtriangle, use the ratiox:x:x2,x:x:x2,where each of the three sides is associated with the opposite angle andx2x2is associated with the hypotenuse, opposite the90â90âangle. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
The primary trigonometric functions aresinθ=opphyp,sinθ=opphyp,cosθ=adjhyp,cosθ=adjhyp,andtanθ=oppadj.tanθ=oppadj. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
Trigonometric functions can be used to find either the length of a side or the measure of an angle in a right triangle, and in applications such as the angle of elevation or the angle of depression formed using right triangles. | https://openstax.org/books/contemporary-mathematics/pages/10-key-concepts |
LetAAbe a particular item andBBanother such that there is a constant ratio ofAAtoBB. | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.