Standard Deviation Lecture 18 Sec 5 3 4

Standard Deviation Lecture 18 Sec. 5. 3. 4 Mon, Feb 18, 2007

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

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. n Define the deviation of x to be (x – x). n 1 2 3 4 5 6 7 8 9 10

Deviations from the Mean Each unit of a sample or population deviates from the mean by a certain amount. n Define the deviation of x to be (x – x). n 1 2 3 4 5 x = 6 6 7 8 9 10

Deviations from the Mean n Each unit of a sample or population deviates from the mean by a certain amount. deviation = – 5 1 2 3 4 5 x = 4 6 7 8 9 10

Deviations from the Mean n Each unit of a sample or population deviates from the mean by a certain amount. dev = – 2 1 2 3 4 5 x = 4 6 7 8 9 10

Deviations from the Mean n Each unit of a sample or population deviates from the mean by a certain amount. dev = +1 1 2 3 4 5 x = 4 6 7 8 9 10

Deviations from the Mean n Each unit of a sample or population deviates from the mean by a certain amount. dev = +2 1 2 3 4 5 x = 4 6 7 8 9 10

Deviations from the Mean n Each unit of a sample or population deviates from the mean by a certain amount. deviation = +4 1 2 3 4 5 x = 4 6 7 8 9 10

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

Sum of Squared Deviations We will square them all first. That way, there will be no canceling. n So we compute the sum of the squared deviations, called SSX. n

Sum of Squared Deviations n Procedure ¨ Find the average ¨ Find the deviations from the average ¨ Square the deviations ¨ Add them up

Sum of Squared Deviations n SSX = sum of squared deviations n For example, if the sample is {1, 4, 7, 8, 10}, then SSX = (1 – 6)2 + (4 – 6)2 + (7 – 6)2 + (8 – 6)2 + (10 – 6)2 = (– 5)2 + (– 2)2 + (1)2 + (2)2 + (4)2 = 25 + 4 + 16 = 50.

The Population Variance of the population n The population variance is denoted by 2.

The Population Standard Deviation n The population standard deviation is the square root of the population variance.

The Sample Variance n Variance of a sample n The sample variance is denoted by s 2.

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

Example In the example, SSX = 50. n Therefore, s 2 = 50/4 = 12. 5. n

The Sample Standard Deviation n The sample standard deviation is the square root of the sample variance. n We will interpret this as being representative of deviations in the sample.

Example In our example, we found that s 2 = 12. 5. n Therefore, s = 12. 5 = 3. 536. n How does that compare to the individual deviations? n

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

Example n n n Let the sample be {1, 4, 7, 8, 10}. Then x = 30 and x 2 = 1 + 16 + 49 + 64 + 100 = 230. So SSX = 230 – (30)2/5 = 230 – 180 = 50, as before.

TI-83 – Standard Deviations n n 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. n Using the data of the previous example, we have ¨ Sx = 3. 535533906. ¨ x = 3. 16227766.

Interpreting the Standard Deviation The standard deviation is directly comparable to actual deviations. n How does 3. 536 compare to -5, -2, +1, +2, and +4? n

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

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 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 – 2 s x – s x x + s x + 2 s

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