One feature of the standard deviation that distinguishes it from a variance is that the standard deviation is expressed in the same units as the raw data, whereas the variance is expressed in those units squared. The meaning of standard deviation is more readily understood from its use. Although the standard deviation and the variance are closely related and can be computed from each other, differentiating between them is important, because both are widely used in statistics. What is a standard deviation? What does it do, and what does it mean?
The most precise way to define standard deviation is by reciting the formula used to compute it. However, insight into the concept of standard deviation can be gleaned by viewing the manner in which it is applied. Two ways of applying the standard deviation are the empirical rule and Chebyshev’s theorem. •The empirical rule is an important rule of thumb that is used to state r/. v approximate per-centage of values that lie within a given number of standard deviations from the mean of a set of data if the data are normally distributed.
The empirical rule is used only for three numbers of standard deviations: 1? , 2? , and 3?. •The empirical rule applies only when data are known to be approximately normally distributed. What do researchers use when data are not normally distributed or when the-shape of the distribution is unknown? Chebyshev’s theorem applies to all distributions regardless of their shape and thus can be used whenever the data distribution shape is unknown or is non-normal.
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Even though Chebyshev’s theorem can in theory be applied to data that are normally distributed, the empirical rule is more widely known and is preferred whenever appropriate. Chebyshev’s theorem is not a rule of thumb, as is the empirical rule, but rather it is presented in formula format and therefore can be more widely applied. Chebyshev’s theorem states that at least 1—1/K2 values M ill fall within ± k standard deviations of the mean regardless of the shape of the distribution. The sample variance is denoted by s2 and the sample standard deviation by s.
The main use for sample variances and standard deviations is as estimators of population variances and standard deviations. Because of this, computation of the sample variance and standard deviation differs slightly from computation of the population variance and standard deviation. Both the sample variance and sample standard deviation use n – 1 in the denominator instead of n because using n in the denominator of a sample variance results in a statistic that tends to underestimate the population variance. Reference link: http://classof1. com/homework-help/engineering-homework-help
