Standard Deviation Lecture 18 Sec. 5.3.4 Wed, Oct 4, 2006

Preview Later in the course, we will use a sample proportion to estimate the population proportion. And we will also use a sample mean to estimate a population mean. In both cases, we will need to know how close the estimate is likely to be to the true value.

Preview For example, if an opinion poll shows that one candidate has 43% of the vote, that is the sample proportion. How close is it likely to be to the true proportion? This is where statisticians get the notion of the “margin of error.” For opinion polls, the margin of error is typically 3% with 95% probability.

Preview The margin of error is an expression of the variability of the estimator.

Variability To find the margin of error associated withx , we need to measure its variability. It turns out that to measure the variability inx, we must measure the variability in the population from which we are sampling. 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 deviations 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 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.

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.

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

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.

Example Use Excel to compute the mean and standard deviation of the sample {0, 3, 6, 7}. Do it once using basic operations. Do it again using special functions.

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 The variance is not comparable to 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