Standard Deviation Lecture 18 Sec Tue, Feb 15, 2005

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. Each unit of a sample or population deviates from the mean by a certain amount. Define the deviation of x to be (x –  x). Define the deviation of x to be (x –  x).  x =

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. Each unit of a sample or population deviates from the mean by a certain amount.  x = deviation = –4

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. Each unit of a sample or population deviates from the mean by a certain amount.  x = dev = 1

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. Each unit of a sample or population deviates from the mean by a certain amount.  x = deviation = 3

Sum of Squared Deviations We want to add up all the deviations, but to keep the negative ones from canceling the positive ones, we square them all first. We want to add up all the deviations, but to keep the negative ones from canceling the positive ones, we square them all first. So we compute the sum of the squared deviations, called SSX. So we compute the sum of the squared deviations, called SSX. Procedure Procedure Find the deviations Find the deviations Square them all Square them all Add them up Add them up

Sum of Squared Deviations SSX = sum of squared deviations SSX = sum of squared deviations For example, if the sample is {0, 5, 7}, then 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 = = 26.

The Population Variance Variance of the population – The average squared deviation for the population. Variance of the population – The average squared deviation for the population. The population variance is denoted by  2. The population variance is denoted by  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. 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 s 2. The sample variance is denoted by s 2. This formula for s 2 makes a better estimator of  2 than if we had divided by n. This formula for s 2 makes a better estimator of  2 than if we had divided by n.

Example In the example, SSX = 26. In the example, SSX = 26. Therefore, Therefore, s 2 = 26/2 = 13.

The Standard Deviation Standard deviation – The square root of the variance of the sample or population. 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 the population is denoted . The standard deviation of a sample is denoted s. The standard deviation of a sample is denoted s.

Example In our example, we found that s 2 = 13. In our example, we found that s 2 = 13. Therefore, s =  13 = Therefore, s =  13 =

Example Example 5.10, p Example 5.10, p Use Excel to compute the mean and standard deviation of the height and weight data. Use Excel to compute the mean and standard deviation of the height and weight data. HeightWeight.xls. HeightWeight.xls. HeightWeight.xls Use basic operations. Use basic operations. Use special functions. Use special functions.

Alternate Formula for the Standard Deviation An alternate way to compute SSX is to compute An alternate way to compute SSX is to compute Note that only the second term is divided by n. Note that only the second term is divided by n. Then, as before Then, as before

Example Let the sample be {0, 5, 7}. Let the sample be {0, 5, 7}. Then  x = 12 and Then  x = 12 and  x 2 = = 74. So So SSX = 74 – (12) 2 /3 = 74 – 48 = 26, as before.

TI-83 – Standard Deviations Follow the procedure for computing the mean. Follow the procedure for computing the mean. The display shows Sx and  x. The display shows Sx and  x. Sx is the sample standard deviation. Sx is the sample standard deviation.  x is the population standard deviation.  x is the population standard deviation. Using the data of the previous example, we have Using the data of the previous example, we have Sx = Sx =  x =  x =

Interpreting the Standard Deviation Both the standard deviation and the variance are measures of variation in a sample or population. 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. 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. Therefore, the standard deviation is directly comparable to actual deviations.

Interpreting the Standard Deviation The variance is not comparable to deviations. The variance is not comparable to deviations. The most basic interpretation of the standard deviation is that it is roughly the average deviation. The most basic interpretation of the standard deviation is that it is roughly the average deviation.

Interpreting the Standard Deviation Observations that deviate from  x by much more than s are unusually far from the mean. Observations that deviate from  x by much more than s are unusually far from the mean. Observations that deviate from  x by much less than s are unusually close to the mean. Observations that deviate from  x by much less than s are unusually close to the mean.

Interpreting the Standard Deviation xxxx

xxxx ss

xxxx  x + s  x – s ss

Interpreting the Standard Deviation xxxx  x + s  x – s  x Closer than normal to  x

Interpreting the Standard Deviation xxxx  x + s  x – s  x Farther than normal from  x

Interpreting the Standard Deviation xxxx  x + s  x – s  x Unusually far from  x  x – 2s  x + 2s

Let’s Do It! Let’s do it! 5.13, p. 295 – Increasing Spread. Let’s do it! 5.13, p. 295 – Increasing Spread. Let’s do it! 5.14, p. 297 – Variation in Scores. Let’s do it! 5.14, p. 297 – Variation in Scores.