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## Standard Deviation

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**Standard Deviation**Lecture 18 Sec. 5.3.4 Tue, Oct 4, 2005**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 5 6 7 8 4 x = 3.5**Deviations from the Mean**• Each unit of a sample or population deviates from the mean by a certain amount. deviation = -3.5 0 1 2 3 5 6 7 8 4 x = 3.5**Deviations from the Mean**• Each unit of a sample or population deviates from the mean by a certain amount. dev = -1.5 0 1 2 3 5 6 7 8 4 x = 3.5**Deviations from the Mean**• Each unit of a sample or population deviates from the mean by a certain amount. dev = +1.5 0 1 2 3 5 6 7 8 4 x = 3.5**Deviations from the Mean**• Each unit of a sample or population deviates from the mean by a certain amount. deviation = +3.5 0 1 2 3 5 6 7 8 4 x = 3.5**Deviations from the Mean**• How do we obtain one number that is representative of the set of individual deviations? • If we add them up to get the average, the positive deviations will cancel with the negative deviations, leaving a total of 0. • That’s no good.**Sum of Squared Deviations**• We will square them all first. That way, there will be no canceling. • So we compute the sum of the squared deviations, called SSX. • Procedure • Find the deviations • Square them all • Add them up**Sum of Squared Deviations**• SSX = sum of squared deviations • For example, if the sample is {0, 2, 5, 7}, then SSX = (0 – 3.5)2 + (2 – 3.5)2 + (5 – 3.5)2 + (7 – 3.5)2 = (-3.5)2 + (-1.5)2 + (1.5)2 + (3.5)2 = 12.25 + 2.25 + 2.25 + 12.25 = 29.**The Population Variance**• Variance of the population – The average squared deviation for the population. • The population variance is denoted by 2.**The Population Standard Deviation**• The population standard deviation is the square root of the population variance. • We will interpret this as being representative of deviations in the population (hence the name “standard”).**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. • This formula for s2 makes a better estimator of 2 than if we had divided by n.**Example**• In the example, SSX = 29. • Therefore, s2 = 29/3 = 9.667.**The Sample Standard Deviation**• The sample standard deviation is the square root of the sample variance. • We will interpret this as being representative of deviations in the sample.**Example**• In our example, we found that s2 = 9.667. • Therefore, s = 9.667 = 3.109.**Example**• Use Excel to compute the mean and standard deviation of the sample {0, 2, 5, 7}. • Do it once using basic operations. • Do it again using special functions. • Then compute the mean and standard deviation for the on-time arrival data. • OnTimeArrivals.xls.**Alternate Formula for the Standard Deviation**• An alternate way to compute SSXis to compute • Note that only the second term is divided by n. • Then, as before**Example**• Let the sample be {0, 2, 5, 7}. • Then x = 14 and x2 = 0 + 4 + 25 + 49 = 78. • So SSX = 78 – (14)2/4 = 78 – 49 = 29, as before.**TI-83 – Standard Deviations**• Follow the procedure 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 = 3.109126351. • x = 2.692582404.**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**s s x – s x x + s**Interpreting the Standard Deviation**A little closer than normal tox but not unusual x – s x x + s**Interpreting the Standard Deviation**Unusually close tox x – s x x + s**Interpreting the Standard Deviation**A little farther than normal fromx but not unusual x – 2s x – s x x + s x + 2s**Interpreting the Standard Deviation**Unusually far fromx x – 2s x – s x x + s x + 2s**Let’s Do It!**• Let’s Do It! 5.13, p. 329 – Increasing Spread. • Example 5.10, p. 329 – There Are Many Measures of Variability.