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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 than from any other point.
For example, if the mean height in a population of 21-year-old men is 1.75 meters, and one randomly chosen man is 1.80 meters tall, then the "error" is 0.05 meters; if the randomly chosen man is 1.70 meters tall, then the "error" is −0.05 meters.
Bias in standard deviation for autocorrelated data. The figure shows the ratio of the estimated standard deviation to its known value (which can be calculated analytically for this digital filter), for several settings of α as a function of sample size n. Changing α alters the variance reduction ratio of the filter, which is known to be
For an approximately normal data set, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%. Shown percentages are rounded theoretical probabilities intended only to approximate the empirical ...
¯ is the sample mean; σ 2 is the population variance; s n 2 is the biased sample variance (i.e., without Bessel's correction) s 2 is the unbiased sample variance (i.e., with Bessel's correction) The standard deviations will then be the square roots of the respective variances.
From DasGupta's inequality it follows that for a normal distribution at least 95% lies within approximately 2.582 standard deviations of the mean. This is less sharp than the true figure (approximately 1.96 standard deviations of the mean). DasGupta has determined a set of best possible bounds for a normal distribution for this inequality. [43]
Since the null hypothesis for Tukey's test states that all means being compared are from the same population (i.e. μ 1 = μ 2 = μ 3 = ... = μ k), the means should be normally distributed (according to the central limit theorem) with the same model standard deviation σ, estimated by the merged standard error, , for all the samples; its ...