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In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or ''s'') is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A standard deviation close to 0 indicates that the data points tend to be very close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values. The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same units as the data. There are also other measures of deviation from the norm, including mean absolute deviation, which provide different mathematical properties from standard deviation.〔Gorard, Stephen. (Revisiting a 90-year-old debate: the advantages of the mean deviation ). Department of Educational Studies, University of York〕 In addition to expressing the variability of a population, the standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. The reported margin of error is typically about twice the standard deviation—the half-width of a 95 percent confidence interval. In science, researchers commonly report the standard deviation of experimental data, and only effects that fall much farther than two standard deviations away from what would have been expected are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from causal variation. The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment. When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data or to a modified quantity that is a better estimate of the population standard deviation (the standard deviation of the entire population). ==Basic examples== For a finite set of numbers, the standard deviation is found by taking the square root of the average of the squared deviations of the values from their average value. For example, the marks of a class of eight students (that is, a population) are the following eight values: : These eight data points have the mean (average) of 5: : First, calculate the deviations of each data point from the mean, and square the result of each: : The variance is the mean of these values: : and the ''population'' standard deviation is equal to the square root of the variance: : This formula is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some larger parent population (for example, they were 8 marks randomly chosen from a class of 20), then we would have divided by instead of in the denominator of the last formula, and then the quantity thus obtained would be called the ''sample'' standard deviation. Dividing by ''n''−1 gives a better estimate of the population standard deviation for the larger parent population than dividing by ''n'', which gives a result which is correct for the sample only. This is known as ''Bessel's correction''. As a slightly more complicated real-life example, the average height for adult men in the United States is about 70 inches, with a standard deviation of around 3 inches. This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67–73 inches) – one standard deviation – and almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches) – two standard deviations. If the standard deviation were zero, then all men would be exactly 70 inches tall. If the standard deviation were 20 inches, then men would have much more variable heights, with a typical range of about 50–90 inches. Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal (bell-shaped). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「standard deviation」の詳細全文を読む スポンサード リンク
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