Standard Standard DeviationDeviation

Lecture 18Lecture 18

Sec. 5.3.4Sec. 5.3.4

Tue, Feb 15, 2005Tue, Feb 15, 2005

Deviations from the Deviations from the MeanMean

Each unit of a sample or population Each unit of a sample or population deviates from the mean by a certain deviates from the mean by a certain amount.amount.

Define the Define the deviationdeviation of of xx to be ( to be (xx – –xx).).

x = 40 1 2 3 5 6 7 8

Deviations from the Deviations from the MeanMean

Each unit of a sample or population Each unit of a sample or population deviates from the mean by a certain deviates from the mean by a certain amount.amount.

x = 40 1 2 3 5 6 7 8

deviation = –4

Deviations from the Deviations from the MeanMean

Each unit of a sample or population Each unit of a sample or population deviates from the mean by a certain deviates from the mean by a certain amount.amount.

x = 40 1 2 3 5 6 7 8

dev = 1

Deviations from the Deviations from the MeanMean

Each unit of a sample or population Each unit of a sample or population deviates from the mean by a certain deviates from the mean by a certain amount.amount.

x = 40 1 2 3 5 6 7 8

deviation = 3

Sum of Squared Sum of Squared DeviationsDeviations

We want to add up all the deviations, We want to add up all the deviations, but to keep the negative ones from but to keep the negative ones from canceling the positive ones, we canceling the positive ones, we square them all first.square them all first.

So we compute the sum of the So we compute the sum of the squared deviations, called squared deviations, called SSXSSX..

ProcedureProcedure Find the deviationsFind the deviations Square them allSquare them all Add them upAdd them up

Sum of Squared Sum of Squared DeviationsDeviations

SSXSSX = sum of squared deviations = sum of squared deviations

For example, if the sample is {0, 5, 7}, For example, if the sample is {0, 5, 7}, thenthen

SSXSSX = (0 – 4) = (0 – 4)22 + (5 – 4) + (5 – 4)22 + (7 – 4) + (7 – 4)22

= (-4)= (-4)22 + (1) + (1)22 + (3) + (3)22

= 16 + 1 + 9= 16 + 1 + 9

= 26.= 26.

2xxSSX

The Population VarianceThe Population Variance

Variance of the populationVariance of the population – The – The average squared deviation for the average squared deviation for the population.population.

The population variance is denoted The population variance is denoted by by 22..

N

SSX

N

x

22

The Sample VarianceThe Sample Variance

Variance of a sampleVariance of a sample – The average – The average squared deviation for the sample, except squared deviation for the sample, except that we divide by that we divide by nn – 1 instead of – 1 instead of nn..

The sample variance is denoted by The sample variance is denoted by ss22..

This formula for This formula for ss22 makes a better makes a better estimator of estimator of 22 than if we had divided by than if we had divided by nn..

11

22

n

SSX

n

xxs

ExampleExample

In the example, In the example, SSXSSX = 26. = 26. Therefore,Therefore,

ss22 = 26/2 = 13. = 26/2 = 13.

The Standard DeviationThe Standard Deviation

Standard deviationStandard deviation – The square root – The square root of the variance of the sample or of the variance of the sample or population.population.

The standard deviation of the The standard deviation of the populationpopulation is denoted is denoted ..

The standard deviation of a The standard deviation of a samplesample is denoted is denoted ss..

ExampleExample

In our example, we found that In our example, we found that ss22 = = 13.13.

Therefore, Therefore, ss = = 13 = 3.606.13 = 3.606.

ExampleExample

Example 5.10, p. 293.Example 5.10, p. 293. Use Excel to compute the mean and Use Excel to compute the mean and

standard deviation of the height and standard deviation of the height and weight data.weight data. HeightWeight.xlsHeightWeight.xls.. Use basic operations.Use basic operations. Use special functions.Use special functions.

Alternate Formula for Alternate Formula for the Standard Deviationthe Standard Deviation

An alternate way to compute An alternate way to compute SSXSSX is is to computeto compute

Note that only the second term is Note that only the second term is divided by divided by nn..

Then, as beforeThen, as before

n

xxSSX

22

12

n

SSXs

ExampleExample

Let the sample be {0, 5, 7}.Let the sample be {0, 5, 7}. Then Then xx = 12 and = 12 and

xx22 = 0 + 25 + 49 = 74. = 0 + 25 + 49 = 74. SoSo

SSXSSX = 74 – (12) = 74 – (12)22/3 /3

= 74 – 48 = 74 – 48

= 26,= 26,

as before.as before.

TI-83 – Standard TI-83 – Standard DeviationsDeviations

Follow the procedure for computing Follow the procedure for computing the mean.the mean.

The display shows Sx and The display shows Sx and x.x. SxSx is the is the samplesample standard deviation. standard deviation. xx is the is the populationpopulation standard deviation. standard deviation.

Using the data of the previous Using the data of the previous example, we haveexample, we have Sx = 3.605551275.Sx = 3.605551275. x = 2.943920289.x = 2.943920289.

Interpreting the Interpreting the Standard DeviationStandard Deviation

Both the standard deviation and the Both the standard deviation and the variance are measures of variation in a variance are measures of variation in a sample or population.sample or population.

The standard deviation is measured in The standard deviation is measured in the same units as the measurements in the same units as the measurements in the sample.the sample.

Therefore, the standard deviation is Therefore, the standard deviation is directly comparable to actual directly comparable to actual deviations.deviations.

Interpreting the Interpreting the Standard DeviationStandard Deviation

The variance is not comparable to The variance is not comparable to deviations.deviations.

The most basic interpretation of the The most basic interpretation of the standard deviation is that it is standard deviation is that it is roughlyroughly the average deviation. the average deviation.

Interpreting the Interpreting the Standard DeviationStandard Deviation

Observations that deviate fromObservations that deviate fromxx by by much more than s are unusually far much more than s are unusually far from the mean.from the mean.

Observations that deviate fromObservations that deviate fromxx by by much less than s are unusually close much less than s are unusually close to the mean.to the mean.

Interpreting the Interpreting the Standard DeviationStandard Deviation

xx

Interpreting the Interpreting the Standard DeviationStandard Deviation

xx

s s

Interpreting the Interpreting the Standard DeviationStandard Deviation

xx x + sx + sx – sx – s

s s

Interpreting the Interpreting the Standard DeviationStandard Deviation

xx x + sx + sx – sx – s

Closer than normal toxx

Interpreting the Interpreting the Standard DeviationStandard Deviation

xx x + sx + sx – sx – s

Farther than normal fromxx

Interpreting the Interpreting the Standard DeviationStandard Deviation

xx x + sx + sx – sx – s

Unusually far fromxx

x – x – 22ss x + x + 22ss

Let’s Do It!Let’s Do It!

Let’s do it! 5.13, p. 295 – Increasing Let’s do it! 5.13, p. 295 – Increasing Spread.Spread.

Let’s do it! 5.14, p. 297 – Variation Let’s do it! 5.14, p. 297 – Variation in Scores.in Scores.