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where formula_18 are the observed values of the sample items, and formula_19 is the mean value of these observations, while the denominator "N" stands for the size of the sample: this is the square root of the sample variance, which is the average of the squared deviations about the sample mean.
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This is a consistent estimator (it converges in probability to the population value as the number of samples goes to infinity), and is the maximum-likelihood estimate when the population is normally distributed. However, this is a biased estimator, as the estimates are generally too low. The bias decreases as sample si...
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If the "biased sample variance" (the second central moment of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of the population's standard deviation, the result is
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Here taking the square root introduces further downward bias, by Jensen's inequality, due to the square root's being a concave function. The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question.
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An unbiased estimator for the "variance" is given by applying Bessel's correction, using "N" − 1 instead of "N" to yield the "unbiased sample variance," denoted "s":
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This estimator is unbiased if the variance exists and the sample values are drawn independently with replacement. "N" − 1 corresponds to the number of degrees of freedom in the vector of deviations from the mean, formula_23
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Taking square roots reintroduces bias (because the square root is a nonlinear function which does not commute with the expectation, i.e. often formula_24), yielding the "corrected sample standard deviation," denoted by "s:"
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As explained above, while "s" is an unbiased estimator for the population variance, "s" is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviat...
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For unbiased estimation of standard deviation, there is no formula that works across all distributions, unlike for mean and variance. Instead, "s" is used as a basis, and is scaled by a correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by "s"/"c", where the c...
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This arises because the sampling distribution of the sample standard deviation follows a (scaled) chi distribution, and the correction factor is the mean of the chi distribution.
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The error in this approximation decays quadratically (as 1/"N"), and it is suited for all but the smallest samples or highest precision: for "N" = 3 the bias is equal to 1.3%, and for "N" = 9 the bias is already less than 0.1%.
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For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation:
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where "γ" denotes the population excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.
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The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by the confidence interval or CI.
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To show how a larger sample will make the confidence interval narrower, consider the following examples:
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A small population of "N" = 2 has only 1 degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD; the factors here are as follows:
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where formula_34 is the "p"-th quantile of the chi-square distribution with "k" degrees of freedom, and formula_35 is the confidence level. This is equivalent to the following:
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With "k" = 1, formula_37 and formula_38. The reciprocals of the square roots of these two numbers give us the factors 0.45 and 31.9 given above.
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A larger population of "N" = 10 has 9 degrees of freedom for estimating the standard deviation. The same computations as above give us in this case a 95% CI running from 0.69 × SD to 1.83 × SD. So even with a sample population of 10, the actual SD can still be almost a factor 2 higher than the sampled SD. For a sample ...
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These same formulae can be used to obtain confidence intervals on the variance of residuals from a least squares fit under standard normal theory, where "k" is now the number of degrees of freedom for error.
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For a set of "N" > 4 data spanning a range of values "R", an upper bound on the standard deviation "s" is given by "s = 0.6R".
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An estimate of the standard deviation for "N" > 100 data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values "R" represents four standard deviation...
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The standard deviation is invariant under changes in location, and scales directly with the scale of the random variable. Thus, for a constant "c" and random variables "X" and "Y":
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The standard deviation of the sum of two random variables can be related to their individual standard deviations and the covariance between them:
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The calculation of the sum of squared deviations can be related to moments calculated directly from the data. In the following formula, the letter E is interpreted to mean expected value, i.e., mean.
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which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value.
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See computational formula for the variance for proof, and for an analogous result for the sample standard deviation.
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A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean.
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For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. These standard deviations have the same...
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Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements ...
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While the standard deviation does measure how far typical values tend to be from the mean, other measures are available. An example is the mean absolute deviation, which might be considered a more direct measure of average distance, compared to the root mean square distance inherent in the standard deviation.
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The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average (mean).
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For example, in industrial applications the weight of products coming off a production line may need to comply with a legally required value. By weighing some fraction of the products an average weight can be found, which will always be slightly different from the long-term average. By using standard deviations, a mini...
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In experimental science, a theoretical model of reality is used. Particle physics conventionally uses a standard of "5 sigma" for the declaration of a discovery. A five-sigma level translates to one chance in 3.5 million that a random fluctuation would yield the result. This level of certainty was required in order to ...
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As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum ...
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In finance, standard deviation is often used as a measure of the risk associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets (actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently mana...
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For example, assume an investor had to choose between two stocks. Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20 percentage points (pp) and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk...
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Calculating the average (or arithmetic mean) of the return of a security over a given period will generate the expected return of the asset. For each period, subtracting the expected return from the actual return results in the difference from the mean. Squaring the difference in each period and taking the average give...
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Population standard deviation is used to set the width of Bollinger Bands, a technical analysis tool. For example, the upper Bollinger Band is given as formula_46 The most commonly used value for "n" is 2; there is about a five percent chance of going outside, assuming a normal distribution of returns.
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Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling use of statistical to...
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To gain some geometric insights and clarification, we will start with a population of three values, "x", "x", "x". This defines a point "P" = ("x", "x", "x") in R. Consider the line "L" = {("r", "r", "r") : "r" ∈ R}. This is the "main diagonal" going through the origin. If our three given values were all equal, then th...
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A little algebra shows that the distance between "P" and "M" (which is the same as the orthogonal distance between "P" and the line "L") formula_56 is equal to the standard deviation of the vector ("x", "x", "x"), multiplied by the square root of the number of dimensions of the vector (3 in this case).
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An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table.
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The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of
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where "μ" is the expected value of the random variables, "σ" equals their distribution's standard deviation divided by "n", and "n" is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the normalizing constan...
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If a data distribution is approximately normal, then the proportion of data values within "z" standard deviations of the mean is defined by:
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where formula_59 is the error function. The proportion that is less than or equal to a number, "x", is given by the cumulative distribution function:
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If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, "μ" ± "σ", where "μ" is the arithmetic mean), about 95 percent are within two standard deviations ("μ" ± 2"σ"), and about 99.7 percent lie within three standard deviatio...
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For various values of "z", the percentage of values expected to lie in and outside the symmetric interval, CI = (−"zσ", "zσ"), are as follows:
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The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller tha...
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Using calculus or by completing the square, it is possible to show that "σ"("r") has a unique minimum at the mean:
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Variability can also be measured by the coefficient of variation, which is the ratio of the standard deviation to the mean. It is a dimensionless number.
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Often, we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by:
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where "N" is the number of observations in the sample used to estimate the mean. This can easily be proven with (see basic properties of the variance):
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In order to estimate the standard deviation of the mean formula_68 it is necessary to know the standard deviation of the entire population formula_69 beforehand. However, in most applications this parameter is unknown. For example, if a series of 10 measurements of a previously unknown quantity is performed in a labora...
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The following two formulas can represent a running (repeatedly updated) standard deviation. A set of two power sums "s" and "s" are computed over a set of "N" values of "x", denoted as "x", ..., "x":
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Given the results of these running summations, the values "N", "s", "s" can be used at any time to compute the "current" value of the running standard deviation:
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In a computer implementation, as the two "s" sums become large, we need to consider round-off error, arithmetic overflow, and arithmetic underflow. The method below calculates the running sums method with reduced rounding errors. This is a "one pass" algorithm for calculating variance of "n" samples without the need to...
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When the values "x" are weighted with unequal weights "w", the power sums "s", "s", "s" are each computed as:
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And the standard deviation equations remain unchanged. "s" is now the sum of the weights and not the number of samples "N".
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The incremental method with reduced rounding errors can also be applied, with some additional complexity.
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The above formulas become equal to the simpler formulas given above if weights are taken as equal to one.
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The term "standard deviation" was first used in writing by Karl Pearson in 1894, following his use of it in lectures. This was as a replacement for earlier alternative names for the same idea: for example, Gauss used "mean error".
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In two dimensions, the standard deviation can be illustrated with the standard deviation ellipse (see "Multivariate normal distribution § Geometric interpretation").
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The Dunning–Kruger effect is a cognitive bias whereby people with low ability, expertise, or experience regarding a certain type of task or area of knowledge tend to overestimate their ability or knowledge. Some researchers also include in their definition the opposite effect for high performers: their tendency to unde...
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The Dunning–Kruger effect is usually measured by comparing self-assessment with objective performance. For example, the participants in a study may be asked to complete a quiz and then estimate how well they performed. This subjective assessment is then compared with how well they actually performed. This can happen ei...
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The initial study was published by David Dunning and Justin Kruger in 1999. It focused on logical reasoning, grammar, and social skills. Since then, various other studies have been conducted across a wide range of tasks, including skills from fields such as business, politics, medicine, driving, aviation, spatial memor...
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The Dunning–Kruger effect is usually explained in terms of metacognitive abilities. This approach is based on the idea that poor performers have not yet acquired the ability to distinguish between good and bad performances. They tend to overrate themselves because they do not see the qualitative difference between thei...
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Many debates surrounding the Dunning–Kruger effect and criticisms of it focus on the metacognitive explanation without denying the empirical findings. The statistical explanation interprets these findings as statistical artifacts. Some theorists hold that the way low and high performers are distributed makes assessing ...
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The Dunning–Kruger effect has been described as relevant for various practical matters, but disagreements exist about the magnitude of its influence. Inaccurate self-assessment can lead people to make bad decisions, such as choosing a career for which they are unfit or engaging in dangerous behavior. It may also inhibi...
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The Dunning–Kruger effect is defined as the tendency of people with low ability in a specific area to give overly positive assessments of this ability. This is often understood as a cognitive bias, i.e. as a systematic tendency to engage in erroneous forms of thinking and judging. Biases are systematic in the sense tha...
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Some researchers emphasize the metacognitive component in their definition. In this view, the Dunning–Kruger effect is the thesis that those who are incompetent in a given area tend to be ignorant of their incompetence, i.e. they lack the metacognitive ability to become aware of their incompetence. This definition lend...
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The Dunning–Kruger effect is usually defined specifically for the self-assessments of people with a low level of competence. Some definitions, though, do not restrict it to the bias of people with low skill, and instead see it as pertaining to false self-evaluations on different skill levels. So it is sometimes claimed...
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The most common approach to measuring the Dunning–Kruger effect is to compare self-assessment with objective performance. The self-assessment is sometimes called "subjective ability" in contrast to the "objective ability" corresponding to the actual performance. The self-assessment may be done before or after the perfo...
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The main point of interest for researchers is usually the correlation between subjective and objective ability. To provide a simplified form of analysis of the measurements, objective performances are often divided into four groups, starting from the bottom quartile of low performers to the top quartile of high perform...
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The Dunning–Kruger effect has been studied across a wide range of tasks. The initial study focused on logical reasoning, grammar skills, and social abilities, such as emotional intelligence and judging which jokes are funny. While many studies are conducted in laboratories, others take place in real-world settings. The...
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David Dunning and Justin Kruger published the initial study in 1999 under the title "Unskilled and Unaware of It: How Difficulties in Recognizing One's Own Incompetence Lead to Inflated Self-Assessments". It examines the performance and self-assessment of undergraduate students of introductory courses in psychology in ...
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Dunning, Kruger, and various other researchers published many subsequent studies. In the 2003 paper "Why People Fail to Recognize Their Own Incompetence", the relation between incorrect self-assessment of competence and the person's ignorance of a given activity's standards of performance is discussed. The 2003 study "...
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The 2006 study "Skilled or Unskilled, but Still Unaware of It: How Perceptions of Difficulty Drive Miscalibration in Relative Comparisons" tries to show that it is not true of all activities that poor performers give more inaccurate self-assessments than strong performers. The study investigates 13 different tasks and ...
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Various explanations have been proposed to account for the Dunning–Kruger effect. The initial and most common account is based on metacognitive abilities. It rests on the assumption that part of acquiring a skill consists in learning to distinguish between good and bad performances of this skill. Since people with low ...
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Not everyone agrees with the assumptions on which the metacognitive account is based. Many criticisms of the Dunning–Kruger effect have the metacognitive account as their main focus, but agree with the empirical findings themselves. This line of argument usually proceeds by providing an alternative approach that promis...
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Another account, sometimes given by theorists with an economic background, focuses on the fact that participants in the corresponding studies usually lack incentive to give accurate self-assessments. In such cases, intellectual laziness or a desire to look good to the experimenter may motivate participants to give over...
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A different approach, further removed from psychological explanations, sees the Dunning–Kruger effect as mainly a statistical artifact. It is based on the idea that the statistical effect known as regression toward the mean accounts for the empirical findings. In the case of the quality of performances, this effect res...
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Most researchers acknowledge that regression toward the mean is a relevant statistical effect that must be taken into account when interpreting the empirical findings. This can be achieved by various methods. But such adjustments do not eliminate the Dunning–Kruger effect, which is why the view that regression toward t...
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Opponents of this approach have argued that this explanation can account for the Dunning–Kruger effect only when assessing one's ability relative to one's peer group, not when the self-assessment is relative to an objective standard. But even proponents of this explanation agree that this does not explain the empirical...
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Various claims have been made about the Dunning–Kruger effect's practical significance or why it matters. They often focus on how it causes the affected people to make decisions that lead to dire consequences for them or others. This is especially relevant for decisions that have long-term effects. For example, it can ...
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Another implication concerns fields in which self-assessments play an essential role in evaluating skills. They are commonly used, for example, in vocational counseling or to estimate the information literacy skills of students and professionals. The Dunning–Kruger effect indicates that such self-assessments often do n...
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Not all accounts of the Dunning–Kruger effect focus on its negative sides. Some also concentrate on its positive side, e.g., ignorance can sometimes be bliss. In this sense, optimism can lead people to experience their situation more positively, and overconfidence may help them achieve even unrealistic goals. To distin...
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In 2000, Kruger and Dunning were awarded a satiric Ig Nobel Prize in recognition of the scientific work recorded in "their modest report".
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"The Dunning–Kruger Song" is part of "The Incompetence Opera", a mini-opera that premiered at the Ig Nobel Prize ceremony in 2017. The mini-opera is billed as "a musical encounter with the Peter principle and the Dunning–Kruger Effect".
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Space Exploration Technologies Corp. (SpaceX) is an American spacecraft manufacturer, launcher, and a satellite communications corporation headquartered in Hawthorne, California. It was founded in 2002 by Elon Musk with the stated goal of reducing space transportation costs to enable the colonization of Mars. The compa...
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SpaceX is developing a satellite internet constellation named Starlink to provide commercial internet service. In January 2020, the Starlink constellation became the largest satellite constellation ever launched, and as of December 2022 comprises over 3,300 small satellites in orbit.
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The company is also developing Starship, a privately funded, fully reusable, super heavy-lift launch system for interplanetary and orbital spaceflight. It is intended to become SpaceX's primary orbital vehicle once operational, supplanting the existing Falcon 9, Falcon Heavy, and Dragon fleet. It will have the highest ...
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SpaceX's achievements include the first privately developed liquid-propellant rocket to reach orbit around Earth; the first private company to successfully launch, orbit, and recover a spacecraft; the first private company to send a spacecraft to the International Space Station; the first vertical take-off and vertical...
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In early 2001, Elon Musk donated $100,000 to the Mars Society and joined its board of directors for a short time. He was offered a plenary talk at their convention where he announced "Mars Oasis", a project to land a miniature experimental greenhouse and grow plants on Mars, to revive public interest in space explorati...
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When Musk returned to Moscow, Russia with Michael Griffin (who led the CIA's venture capital arm In-Q-Tel), they found the Russians increasingly unreceptive. On the flight home Musk announced that he could start a company to build the affordable rockets they needed instead. By applying vertical integration, using cheap...
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In early 2002, Musk started to look for staff for his new space company, soon to be named SpaceX. Musk approached rocket engineer Tom Mueller (later SpaceX's CTO of propulsion) and invited him to become his business partner. Mueller agreed to work for Musk, and thus SpaceX was born. SpaceX was first headquartered in a ...
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Musk has stated that one of his goals with SpaceX is to decrease the cost and improve the reliability of access to space, ultimately by a factor of ten.
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The total development cost of Falcon 1 was approximately US$90 million to US$100 million. The Falcon name was adopted from the DARPA Falcon Project, part of the Prompt Global Strike program of the US military.
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In 2005, SpaceX announced plans to pursue a human-rated commercial space program through the end of the decade, a program that would later become the Dragon spacecraft.
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