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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.
The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter . [10] The alternative form of the Greek letter phi, φ {\textstyle \varphi } , is also used quite often.
In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value.
In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic.For an arbitrarily large number of samples where each sample, involving multiple observations (data points), is separately used to compute one value of a statistic (for example, the sample mean or sample variance) per sample, the sampling distribution is ...
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 ...
The sample extrema can be used for a simple normality test, specifically of kurtosis: one computes the t-statistic of the sample maximum and minimum (subtracts sample mean and divides by the sample standard deviation), and if they are unusually large for the sample size (as per the three sigma rule and table therein, or more precisely a Student ...
For example, the parameter (the expectation) can be estimated by the mean of the data and the parameter (the variance) can be estimated from the standard deviation of the data. The mean is found as = /, where is the data value and the number of data, while the standard deviation is calculated as = (). With these parameters many distributions, e ...
The weighted sample mean, ¯, is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one).