Section 3.2 Measures of Spread 109 We can now determine that the last deviation must be 6, in order to make the sum of all four deviations equal to 0. When we know the first three deviations, we can determine the fourth one. Thus, for a sample of size 4, there are 3 degrees of freedom. In general, for a sample of size n, there will be n − 1 degrees of freedom. Objective 3 Compute the standard deviation of a population and a sample The Standard Deviation There is a problem with using the variance as a measure of spread. Because the variance is computed using squared deviations, the units of the variance are the squared units of the data. For example, in Example 3.12, the units of the data are hours, and the units of variance are squared hours. In most situations, it is better to use a measure of spread that has the same units as the data. We do this simply by taking the square root of the variance. The quantity thus obtained is called the standard deviation. The standard deviation of a sample is denoted s, and the standard deviation of a population is denoted σ. DEFINITION • The sample standard deviation s is the square root of the sample variance s2. s = √ s2 • The population standard deviation σ is the square root of the population variance σ2. σ = √ σ2 EXAMPLE 3.13 Computing the standard deviation CAUTION Don’t round off the variance when computing the standard deviation. The lifetimes, in hours, of six batteries (first presented in Example 3.12) were 3, 4, 6, 5, 4, and 2. Find the standard deviation of the battery lifetimes. Solution In Example 3.12, we computed the sample variance to be s2 = 2. The sample standard deviation is therefore s = √ s2 = √ 2 = 1.414 EXAMPLE 3.14 Computing standard deviations with technology Compute the standard deviation for the data in Example 3.13. Solution Following is the display from the TI-84 Plus. The TI-84 Plus calculator denotes the sample standard deviation by Sx. The display shows that the sample standard deviation is equal to 1.414213562. The calculator does not know whether the data set represents a sample or an entire population. Therefore, the display also presents the population standard deviation, which is denoted σx.
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