Module 3aDescriptive Statistics: Numerical Methods Measures of Location Percentiles and Quartiles Measures of Variability

Learning GoalsUnderstand the purpose of measures of location.Be able to compute the mean, median, mode, quartiles, and various percentiles.Understand the purpose of measures of variability.Be able to compute the range, interquartile range, variance, standard deviation, and coefficient of variation.

Measures of locationThe table on the right contains excerpt from a data set that contains salaries for 474 employees at a Midwestern bank.

We want to use measures of location to describe this data set.

Sheet1

ObservationSalary

16300

26360

36480

46480

56480

66540

76600

86660

96720

106780

116780

126780

136840

146840

156900

166960

176960

187080

197260

207260

217380

227500

237680

247680

257860

267860

277860

287860

297860

307860

317920

327980

337980

348040

358040

368160

378160

388160

398220

408280

418280

428340

438340

448340

458340

468340

478400

488400

498460

508460

518520

528520

538520

548520

558580

568580

578580

588640

598640

608640

618700

628700

638700

648760

658760

668760

678760

688760

698820

708820

718820

728820

738880

748880

758880

768940

778940

788940

798940

808940

818940

829000

839000

849000

859000

869000

879000

889000

899060

909060

919120

929120

939180

949180

959180

969180

979240

989240

999240

1009240

1019300

1029300

1039360

1049360

1059360

1069360

1079420

1089420

1099480

1109480

1119540

1129540

1139600

1149600

1159600

1169600

1179600

1189600

1199600

1209600

1219660

1229660

1239660

1249660

1259720

1269720

1279720

1289720

1299780

1309780

1319780

1329780

1339780

1349780

1359780

1369780

1379840

1389840

1399900

1409900

1419900

1429900

1439960

14410020

14510020

14610020

14710020

14810080

14910080

15010080

15110080

15210080

15310140

15410140

15510200

15610200

15710200

15810200

15910260

16010260

16110320

16210320

16310380

16410380

16510380

16610380

16710440

16810440

16910500

17010500

17110500

17210500

17310500

17410500

17510500

17610560

17710560

17810560

17910560

18010620

18110620

18210620

18310620

18410620

18510680

18610680

18710680

18810680

18910680

19010680

19110680

19210740

19310740

19410800

19510800

19610800

19710860

19810860

19910920

20010920

20110920

20210920

20310920

20410980

20510980

20610980

20710980

20810980

20911040

21011040

21111100

21211100

21311100

21411100

21511100

21611100

21711100

21811160

21911160

22011160

22111160

22211220

22311220

22411220

22511280

22611340

22711340

22811340

22911400

23011400

23111400

23211400

23311400

23411400

23511460

23611520

23711520

23811580

23911640

24011640

24111640

24211640

24311640

24411664

24511700

24611700

24711736

24811760

24911760

25011760

25111760

25211760

25311820

25411880

25511940

25611940

25711940

25811940

25912000

26012000

26112000

26212000

26312000

26412060

26512060

26612108

26712120

26812120

26912120

27012120

27112180

27212240

27312240

27412240

27512300

27612300

27712300

27812300

27912300

28012300

28112300

28212300

28312300

28412300

28512300

28612300

28712300

28812360

28912360

29012360

29112420

29212480

29312480

29412480

29512540

29612540

29712540

29812600

29912600

30012600

30112660

30212660

30312660

30412660

30512780

30612780

30712780

30812780

30912840

31012960

31113020

31213020

31313020

31413140

31513200

31613260

31713320

31813320

31913320

32013380

32113416

32213500

32313560

32413560

32513560

32613560

32713560

32813560

32913764

33013800

33113800

33213800

33313800

33413800

33513848

33613920

33713920

33813980

33914040

34014040

34114100

34214100

34314100

34414100

34514220

34614220

34714280

34814280

34914280

35014400

35114400

35214400

35314400

35414460

35514640

35614820

35715000

35815060

35915120

36015120

36115120

36215360

36315420

36415480

36515540

36615540

36715660

36815720

36915840

37015960

37116020

37216080

37316080

37416080

37516080

37616140

37716140

37816320

37916320

38016440

38116620

38216800

38316920

38416920

38517200

38617364

38717400

38817460

38917580

39017950

39118000

39218060

39318100

39418250

39518400

39618400

39718750

39818900

39919020

40019200

40119500

40219600

40320000

40420220

40520400

40620500

40720580

40820850

40921060

41021250

41121600

41221750

41321950

41421960

41522000

41622000

41722000

41822200

41922300

42022600

42122620

42222700

42322800

42423250

42523500

42623750

42723760

42824000

42924000

43024150

43124250

43224500

43324750

43424750

43525000

43626000

43726000

43826000

43926400

44026500

44126700

44226750

44326750

44427000

44527250

44627250

44727500

44827500

44927700

45028000

45128000

45228350

45329000

45429400

45529500

45630000

45730000

45831250

45931300

46031400

46132000

46232500

46333000

46433500

46534500

46636250

46736500

46836800

46938800

47040000

47141400

47241500

47344250

47454000

Sheet2

ObservationSalary

16300

26360

36480

46480

56480

66540

76600

86660

96720

106780

46433500

46534500

46636250

46736500

46836800

46938800

47040000

47141400

47241500

47344250

Sheet3

Measures of LocationThe following are measures of location:MeanMedianModePercentilesQuartiles

MeanThe mean of a data set is the average of all the data values.If the data are from a sample, the mean is denoted by If the data are from a population, the mean is denoted by m (mu).

Mean

MedianThe median is the measure of location most often reported for annual income and property value data.A few extremely large incomes or property values can inflate the mean.

MedianThe median of a data set is the value in the middle when the data items are arranged in ascending order.For an odd number of observations, the median is the middle value.For an even number of observations, the median is the average of the two middle values.

MedianMedian Median = 50th percentilei = (p/100)n = (50/100)474 = 237Because N is even, we average the 237th and 238th data values:

Median = (11,520 + 11,580)/2 = 11,550

Mean and Median ComparedBoth the mean and median are supposed to be measures of central location for the data. In the case of this data set notice that the mean is $2,217.83 more than the median (13,767.80 11,550).

Why is there such a large discrepancy?

Looking at the frequency distribution of current salaries helps to explain why this discrepancy exists.

When there are data values in a distribution that are much smaller or larger than the others such that the distribution is skewed, the mean may not be a good measure of central tendency.The histogram on the left shows the distribution of current salary. Notice two vertical lines that run from top to bottom with numbers attached. The line on the left is the median (11,550) and the line on the right is the mean (13,768).When the distribution, as in this example, has a long tail that extends to larger values (skewed right) then the mean will be larger than the median. If the distribution has a long tail that extends to smaller values (skewed left), then the mean will be smaller than the median. When the data is symmetric (not skewed) then the mean and median will be equal.

ModeThe mode of a data set is the value that occurs with greatest frequency.The greatest frequency can occur at two or more different values.If the data have exactly two modes, the data are bimodal.If the data have more than two modes, the data are multimodal.

ModeExample:SalaryIn the salary example, the modal salary was $12,300. This was the current salary of 14 of the 474 employees included in this sample.

PercentilesRecall how the median divided the sample into 2 equal parts half the observations are less than the median and half are greater than the median.There are other ways to split the sample on a percentage basis: such as finding the value where 10 percent of the observations are less than that value and 90 percent are greater.Admission test scores for colleges and universities are frequently reported in terms of percentiles.

PercentilesThe pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more.Arrange the data in ascending order.Compute index i, the position of the pth percentile. i = (p/100)n

If i is not an integer, round up. The pth percentile is the value in the ith position.If i is an integer, the pth percentile is the average of the values in positions i and i +1.Note: There is no universally accepted method to calculate percentiles. The method used in the book is not the same used in SPSS. For further information is available at http://cnx.rice.edu/content/m10805/latest

PercentilesExample: Salary (Book Method)10th Percentilei = (p/100)n = (10/100)474 = 47.4 = 48the 48th data value:10th Percentile = 8,400

QuartilesQuartiles are specific percentilesFirst Quartile = 25th PercentileSecond Quartile = 50th Percentile = MedianThird Quartile = 75th Percentile

QuartilesExample: Salaries (Book Method)Third QuartileThird quartile = 75th percent i = (p/100)n = (75/100)474 = 355.5 = 356Third quartile = 14,820

Using SPSSNotice how the value for the 75th percentile calculated using SPSS is different.

Measures of VariabilityMeasures of location do not give us an idea of how observations differ from each other.

Measures of variability quantify the spread or dispersion of observations.

Choosing suppliers is an example of why this is important In business. When choosing between suppliers we might consider not only the average delivery time for each, but also the variability in delivery time for each.

Measures of VariabilityRangeInterquartile RangeVarianceStandard DeviationCoefficient of Variation

Measures of Variability: the RangeThe range of a data set is the difference between the largest and smallest data values.It is the simplest measure of variability.It is very sensitive to the smallest and largest data values.The value of the range does not tell us anything about the variability of the values between the largest and smallest values.

Measures of Variability: the Interquartile RangeThe interquartile range of a data set is the difference between the third quartile and the first quartile.It is the range for the middle 50% of the data.It overcomes the sensitivity to extreme data values.

Measures of Variability: the Interquartile RangeExample: Salaries (Book Method)Interquartile Range3rd Quartile (Q3) = 14,8201st Quartile (Q1) = 9,600Interquartile Range = Q3 - Q1 = 14,820 9,600 = 5,220

Using SPSSInterquartile Range = Q3 - Q1 = 14,865 9,600 = 5,265

Measures of Variability: the VarianceThe variance is a measure of variability that utilizes all the data.It is based on the difference between the value of each observation (xi) and the mean ( for a sample, m for a population).

Measures of Variability: the VarianceThe variance is the average of the squared differences between each data value and the mean.If the data set is a sample, the variance is denoted by s2.

If the data set is a population, the variance is denoted by 2.

Measures of Variability: the Standard DeviationThe standard deviation of a data set is the positive square root of the variance.It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean.

If the data set is a sample, the standard deviation is denoted s.

If the data set is a population, the standard deviation is denoted (sigma).

Measures of Variability: the Coefficient of VariationThe coefficient of variation indicates how large the standard deviation is in relation to the mean. This enables the comparison of the variability of different variables.

If the data set is a sample, the coefficient of variation is computed as follows:

If the data set is a population, the coefficient of variation is computed as follows:

Example: SalaryVariance

Standard Deviation

Coefficient of Variation