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Standard Deviation Lecture 18 Sec. 5.3.4 Fri, Oct 8, 2004 Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. Define the deviation of x to be (x –x). 0 1 2 3 x = 4 5 6 7 8 Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. deviation = –4 0 1 2 3 x = 4 5 6 7 8 Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. dev = 1 0 1 2 3 x = 4 5 6 7 8 Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. deviation = 3 0 1 2 3 x = 4 5 6 7 8 Sum of Squared Deviations We want to add up all the deviations, but to keep the negative ones from canceling the positive ones, so we square them all first. Then we compute the sum of the squared deviations. We call this quantity SSX. Sum of Squared Deviations SSX = sum of squared deviations SSX x x 2 For example, if the sample is {0, 5, 7}, then SSX = (0 – 4)2 + (5 – 4)2 + (7 – 4)2 = (-4)2 + (1)2 + (3)2 = 16 + 1 + 9 = 26. The Population Variance Variance of the population – The average squared deviation for the population. The population variance is denoted by 2. x N 2 2 The Sample Variance Variance of a sample – The average squared deviation for the sample, except that we divide by n – 1 instead of n. The sample variance is denoted by s2. x x s n 1 This formula for s2 makes a better estimator of 2 than if we had divided by n. 2 2 Example In the example, SSX = 26. Therefore, s2 = 26/2 = 13. The Standard Deviation Standard deviation – The square root of the variance of the sample or population. The standard deviation of the population is denoted . The standard deviation of a sample is denoted s. Example In our example, we found that s2 = 13. Therefore, s = 13 = 3.606. Example Example 5.10, p. 293. Use Excel to compute the mean and standard deviation of {0, 5, 7}. Use basic operations. Use special functions. Alternate Formula for the Standard Deviation An alternate way to compute SSX is to compute x 2 SSX x 2 n Note that only the second term is divided by n. Then, as before SSX s n 1 2 Example Let the sample be {0, 5, 7}. Then x = 12 and x2 = 0 + 25 + 49 = 74. So SSX = 74 – (12)2/3 = 74 – 48 = 26, as before. TI-83 – Standard Deviations Follow the instructions for computing the mean. The display shows Sx and x. Sx is the sample standard deviation. x is the population standard deviation. Using the data of the previous example, we have Sx = 3.605551275. x = 2.943920289. Interpreting the Standard Deviation Both the standard deviation and the variance are measures of variation in a sample or population. The standard deviation is measured in the same units as the measurements in the sample. Therefore, the standard deviation is directly comparable to actual deviations. Interpreting the Standard Deviation The variance is not comparable to deviations. The most basic interpretation of the standard deviation is that it is roughly the average deviation. Interpreting the Standard Deviation Observations that deviate fromx by much more than s are unusually far from the mean. Observations that deviate fromx by much less than s are unusually close to the mean. Interpreting the Standard Deviation x Interpreting the Standard Deviation s s x Interpreting the Standard Deviation s x – s s x x + s Interpreting the Standard Deviation Closer than normal tox x – s x x + s Interpreting the Standard Deviation Farther than normal fromx x – s x x + s Interpreting the Standard Deviation Extraordinarily far fromx x – 2s x – s x x + s x + 2s Let’s Do It! Let’s do it! 5.13, p. 295 – Increasing Spread. Let’s do it! 5.14, p. 297 – Variation in Scores.