The Standard deviation is a measure of the variation (or spread) of a data set. For a variable x, the standard deviation of all possible observations for the entire population is called the population standard deviation or standard deviation of the variable x. It is denoted σx or, when no confusion will arise, simply σ. Suppose that we want to obtain information about a population standard deviation. If the population is small, we can often determine σ exactly by first taking a census and then computing σ from the population data. However, if the population is large, which is usually the case, a census is generally not feasible, and we must use inferential methods to obtain the required information about σ.
In this section, we describe how to perform hypothesis tests and construct confidence intervals for the standard deviation of a normally distributed variable. Such inferences are based on a distribution called the chi-square distribution. Chi is a Greek letter whose lowercase form is χ. A variable has a chi-square distribution if its distribution has the shape of a special type of right-skewed curve, called a chi-square (χ2) curve. Actually, there are infinitely many chi-square distributions, and we identify the chi-square distribution (and χ2-curve) in question by its number of degrees of freedom. Basic Properties of χ2-Curves are:
Property 1: The total area under a χ2-curve equals 1.
Property 2: A χ2-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis as it does so. Property 3: A χ2-curve is right skewed.
Property 4: As the number of degrees of freedom becomes larger, χ2- curves look increasingly like normal curves.
Percentages (and probabilities) for a variable having a chi-square distribution are equal to areas under its associated χ2-curve. The one-standard-deviation χ2-test is also known as the χ2-test for one population standard deviation. This test is often formulated in terms of variance instead of standard deviation. Unlike the z-tests and t-tests for one and two population means, the one-standard deviation χ2-test is not robust to moderate violations of the normality assumption. In fact, it is so non robust that many statisticians advice against its use unless there is considerable evidence that the variable under consideration is normally distributed or very nearly so.
The non-parametric procedures, which do not require normality, have been developed to perform inferences for a population standard deviation. If you have doubts about the normality of the variable under consideration, you can often use one of those procedures to perform a hypothesis test or find a confidence interval for a population standard deviation. The one-standard-deviation χ2-interval procedure is also known as the χ2-interval procedure for one population standard deviation. This confidence-interval procedure is often formulated in terms of variance instead of standard deviation.
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