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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. 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.
If is a standard normal deviate, then = + will have a normal distribution with expected value and standard deviation . This is equivalent to saying that the standard normal distribution Z {\textstyle Z} can be scaled/stretched by a factor of σ {\textstyle \sigma } and shifted by μ {\textstyle \mu } to yield a different normal distribution ...
Considerations of the shape of a distribution arise in statistical data analysis, where simple quantitative descriptive statistics and plotting techniques such as histograms can lead on to the selection of a particular family of distributions for modelling purposes. The normal distribution, often called the "bell curve" Exponential distribution
where μ is the mean, σ is the standard deviation, E is the expectation operator, μ 3 is the third central moment, and κ t are the t-th cumulants. It is sometimes referred to as Pearson's moment coefficient of skewness , [ 5 ] or simply the moment coefficient of skewness , [ 4 ] but should not be confused with Pearson's other skewness ...
For example, 50 − 25 = 25 is not the same distance as 60 − 35 = 25 because of the bell-curve shape of the distribution. Some percentile ranks are closer to some than others. Percentile rank 30 is closer on the bell curve to 40 than it is to 20. If the distribution is normally distributed, the percentile rank can be inferred from the ...
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 normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the central limit theorem : every variable that can be modelled as a sum of many small independent, identically distributed variables with finite mean and variance is approximately normal.
Figure 1. A simple bimodal distribution, in this case a mixture of two normal distributions with the same variance but different means. The figure shows the probability density function (p.d.f.), which is an equally-weighted average of the bell-shaped p.d.f.s of the two normal distributions.