standard deviation

Standard Standard DeviationDeviation

Lecture 18Lecture 18Sec. 5.3.4Sec. 5.3.4

Tue, Oct 4, 2005Tue, Oct 4, 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 = 3.5

0 1 2 3 5 6 7 84



0 1 2 3 5 6 7 8

deviation = -3.5

x = 3.5

4



0 1 2 3 5 6 7 8

dev = -1.5

x = 3.5

4



0 1 2 3 5 6 7 8

dev = +1.5

x = 3.5

4



0 1 2 3 5 6 7 8

deviation = +3.5

x = 3.5

4


How do we obtain one number that How do we obtain one number that is representative of the set of is representative of the set of individual deviations?individual deviations?

If we add them up to get the If we add them up to get the average, the positive deviations will average, the positive deviations will cancel with the negative deviations, cancel with the negative deviations, leaving a total of 0.leaving a total of 0.

That’s no good.That’s no good.

Sum of Squared Sum of Squared DeviationsDeviations

We will square them all first. That We will square them all first. That way, there will be no canceling.way, there will be no canceling.

So we compute the sum of the So we compute the sum of the squaredsquared deviations, called 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, 2, 5, 7}, For example, if the sample is {0, 2, 5, 7}, thenthenSSXSSX = (0 – 3.5) = (0 – 3.5)22 + (2 – 3.5) + (2 – 3.5)22 + (5 – 3.5) + (5 – 3.5)22 + (7 – + (7 – 3.5)3.5)22 = (-3.5)= (-3.5)22 + (-1.5) + (-1.5)22 + (1.5) + (1.5)22 + (3.5) + (3.5)22

= 12.25 + 2.25 + 2.25 + 12.25= 12.25 + 2.25 + 2.25 + 12.25= 29.= 29.

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..

NSSX

Nx

2

2

The Population Standard The Population Standard DeviationDeviation

The The population standard deviationpopulation standard deviation is the is the square root of the population variance.square root of the population variance.

We will interpret this as being We will interpret this as being representative of deviations in the representative of deviations in the population (hence the name “standard”).population (hence the name “standard”).

N

xxN

SSX2

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

nSSX

nxxs

ExampleExample In the example, In the example, SSXSSX = 29. = 29. Therefore,Therefore,

ss22 = 29/3 = 9.667. = 29/3 = 9.667.

The Sample Standard The Sample Standard DeviationDeviation

The The sample standard deviationsample standard deviation is the is the square root of the sample variance.square root of the sample variance.

We will interpret this as being We will interpret this as being representative of deviations in the representative of deviations in the sample.sample.

11

2

n

xxnSSXs

ExampleExample In our example, we found that In our example, we found that ss22 = =

9.667.9.667. Therefore, Therefore, ss = = 9.667 = 3.109.9.667 = 3.109.

ExampleExample Use Excel to compute the mean and Use Excel to compute the mean and

standard deviation of the sample {0, standard deviation of the sample {0, 2, 5, 7}.2, 5, 7}. Do it once using basic operations.Do it once using basic operations. Do it again using special functions.Do it again using special functions.

Then compute the mean and standard Then compute the mean and standard deviation for the on-time arrival data. deviation for the on-time arrival data. OnTimeArrivals.xlsOnTimeArrivals.xls..

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

nxxSSX

22

12

nSSXs

ExampleExample Let the sample be {0, 2, 5, 7}.Let the sample be {0, 2, 5, 7}. Then Then xx = 14 and = 14 and

xx22 = 0 + 4 + 25 + 49 = 78. = 0 + 4 + 25 + 49 = 78. SoSo

SSXSSX = 78 – (14) = 78 – (14)22/4 /4 = 78 – 49 = 78 – 49 = 29,= 29,

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 = 3.109126351.Sx = 3.109126351. x = 2.692582404.x = 2.692582404.

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 deviations.directly comparable to actual deviations.


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.


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

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


xx


xx

s s


xx x + sx + sx – sx – s

s s



A little closer than normal toxxbut not unusualbut not unusual



Unusually close toxx



A little farther than normal fromxxbut not unusualbut not unusual

x – x – 22ss x + x + 22ss



Unusually far fromxx

x – x – 22ss x + x + 22ss

Let’s Do It!Let’s Do It! Let’s Do It! 5.13, p. 329 – Increasing Let’s Do It! 5.13, p. 329 – Increasing

Spread.Spread. Example 5.10, p. 329 – There Are Example 5.10, p. 329 – There Are

Many Measures of Variability.Many Measures of Variability.

standard deviation

Documents