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). 1 2 3 4 5 6 7 8 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 1 2 3 4 5 6 7 8 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 1 2 3 4 5 6 7 8 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 1 2 3 4 5 6 7 8 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 1 2 3 4 5 6 7 8 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 SSX is 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 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 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.

Interpreting the Standard Deviation x

Interpreting the Standard Deviation x

Interpreting the Standard Deviation x – s x x + s

Interpreting the Standard Deviation A little closer than normal tox but not unusual x – s x x + s

Interpreting the Standard Deviation Unusually close tox x – s x x + s

Interpreting the Standard Deviation A little farther than normal fromx but not unusual x – 2s x – s x x + s x + 2s

Interpreting the Standard Deviation Unusually far fromx 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.