Standard Deviation Lecture 20 Sec. 5.3.4 Fri, Sep 28, 2007

Variability Our ability to estimate a parameter accurately depends on the variability of the population. What do we mean by variability in the population? How do we measure it?

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 = 4

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. deviation = –4 1 2 3 4 5 6 7 8 x = 4

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. dev = –1 1 2 3 4 5 6 7 8 x = 4

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. dev = +2 1 2 3 4 5 6 7 8 x = 4

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. deviation = +3 1 2 3 4 5 6 7 8 x = 4

Deviations from the Mean How do we obtain one number that is representative of the whole set of individual deviations? Normally we use an average to summarize a set of numbers. Why will the average not work in this case?

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 average Find the deviations from the average Square them all Add them up

Sum of Squared Deviations SSX = sum of squared deviations For example, if the sample is {0, 3, 6, 7}, then SSX = (0 – 4)2 + (3 – 4)2 + (6 – 4)2 + (7 – 4)2 = (–4)2 + (–1)2 + (2)2 + (3)2 = 16 + 1 + 4 + 9 = 30.

The Population Variance Variance of 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.

The Sample Variance Variance of a sample The sample variance is denoted by s2.

The Sample Variance Theory shows that if we divide by n – 1 instead of n, we get a better estimator of 2. Therefore, we do it.

Example In the example, SSX = 30. Therefore, s2 = 30/3 = 10.

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 = 10. Therefore, s = 10 = 3.162. How does that compare to the individual deviations?

Alternate Formula for the Standard Deviation An alternate way to compute SSX is to compute Then, as before

Example Let the sample be {0, 3, 6, 7}. Then  x = 16 and So SSX = 94 – (16)2/4 = 94 – 64 = 30, 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 = 3.16227766. x = 2.738612788.

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

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