slides by john loucks st . edward’s university

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Slides by

JohnLoucks

St. Edward’sUniversity



Chapter 3, Part A Descriptive Statistics: Numerical

Measures Measures of Location Measures of Variability



Measures of Location

If the measures are computed for data from a sample,

they are called sample statistics.

If the measures are computed for data from a population,

they are called population parameters.

A sample statistic is referred toas the point estimator of the

corresponding population parameter.

Mean Median Mode Percentiles Quartiles



Mean

The mean of a data set is the average of all the data values.

x The sample mean is the point estimator of the population mean m.

Perhaps the most important measure of location is the mean.

The mean provides a measure of central location.



Sample Mean x

Number ofobservationsin the sample

Sum of the valuesof the n observations

ixx

n



Population Mean m

Number ofobservations inthe population

Sum of the valuesof the N observations

ix

Nm



Seventy efficiency apartments were randomly

sampled in a small college town. The monthly rent

prices for these apartments are listed below.

Sample Mean

Example: Apartment Rents

445 615 430 590 435 600 460 600 440 615440 440 440 525 425 445 575 445 450 450465 450 525 450 450 460 435 460 465 480450 470 490 472 475 475 500 480 570 465600 485 580 470 490 500 549 500 500 480570 515 450 445 525 535 475 550 480 510510 575 490 435 600 435 445 435 430 440



Sample Mean

34,356 490.8070ix

xn

445 615 430 590 435 600 460 600 440 615440 440 440 525 425 445 575 445 450 450465 450 525 450 450 460 435 460 465 480450 470 490 472 475 475 500 480 570 465600 485 580 470 490 500 549 500 500 480570 515 450 445 525 535 475 550 480 510510 575 490 435 600 435 445 435 430 440




Median

Whenever a data set has extreme values, the median is the preferred measure of central location.

A few extremely large incomes or property values can inflate the mean.

The median is the measure of location most often reported for annual income and property value data.

The median of a data set is the value in the middle when the data items are arranged in ascending order.



Median

12 14 19 26 2718 27

For an odd number of observations:

in ascending order

26 18 27 12 14 27 19 7 observations

the median is the middle value.

Median = 19



12 14 19 26 2718 27

Median

For an even number of observations:

in ascending order

26 18 27 12 14 27 30 8 observations

the median is the average of the middle two values.

Median = (19 + 26)/2 = 22.5

19

30



Median

Averaging the 35th and 36th data values:Median = (475 + 475)/2 = 475

Note: Data is in ascending order.

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615




Trimmed Mean

It is obtained by deleting a percentage of the smallest and largest values from a data set and then computing the mean of the remaining values. For example, the 5% trimmed mean is obtained by removing the smallest 5% and the largest 5% of the data values and then computing the mean of the remaining values.

Another measure, sometimes used when extreme values are present, is the trimmed mean.



Mode The 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. Caution: If the data are bimodal or multimodal, Excel’s MODE function will incorrectly identify a single mode.



Mode

450 occurred most frequently (7 times)Mode = 450


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615




Excel Formula Worksheet

Note: Rows 7-71 are not shown.

Using Excel to Computethe Mean, Median, and Mode

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean =AVERAGE(B2:B71)3 2 440 Median =MEDIAN(B2:B71)4 3 450 Mode =MODE.SNGL(B2:B71)5 4 6156 5 480



Excel Value Worksheet


Using Excel to Computethe Mean, Median, and Mode

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean 490.803 2 440 Median 475.004 3 450 Mode 450.005 4 6156 5 480



Percentiles A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. Admission test scores for colleges and universities are frequently reported in terms of percentiles. The 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.



Percentiles

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 p th percentile is the value in the i th position.

If i is an integer, the p th percentile is the average of the values in positions i and i +1.



80th Percentile

i = (p/100)n = (80/100)70 = 56Averaging the 56th and 57th data values:80th Percentile = (535 + 549)/2 = 542


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615




80th Percentile

“At least 80% of the items take on a

value of 542 or less.”

“At least 20% of theitems take on a

value of 542 or more.”56/70 = .8 or 80% 14/70 = .2 or 20%

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615




Using Excel’s Percentile FunctionThe formula Excel uses to compute the location (Lp)of the pth percentile is

Lp = (p/100)n + (1 – p/100)

Excel would compute the location of the 80th percentile for the apartment rent data as follows:L80 = (80/100)70 + (1 – 80/100) = 56 + .2 = 56.2The 80th percentile would be

535 + .2(549 - 535) = 535 + 2.8 = 537.8

Using Excel’s Rank and Percentile Toolto Compute Percentiles and Quartiles

Excel interpolates over the interval from 0 to n.






A B C D

1Apart-ment

Monthly Rent ($) 80th Percentile

2 1 525 =PERCENTILE.INC(B2:B71,.8) 3 2 440 4 3 450 5 4 6156 5 480

80th percentile

It is not necessaryto put the data in ascending

order.






A B C D

1Apart-ment

Monthly Rent ($) 80th Percentile

2 1 525 537.8 3 2 440 4 3 450 5 4 6156 5 480



Quartiles

Quartiles are specific percentiles. First Quartile = 25th Percentile

Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile



Third Quartile

Third quartile = 75th percentilei = (p/100)n = (75/100)70 = 52.5 = 53

Third quartile = 525


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615




Using Excel’s QUARTILE.INC FunctionExcel computes the locations of the 1st, 2nd, and 3rd

quartiles by first converting the quartiles topercentiles and then using the following formula tocompute the location (Lp) of the pth percentile:

Lp = (p/100)n + (1 – p/100)Excel would compute the location of the 3rd quartile(75th percentile) for the rent data as follows:

L75 = (75/100)70 + (1 – 75/100) = 52.5 + .25 = 52.75The 3rd quartile would be

515 + .75(525 - 515) = 515 + 7.5 = 522.5

Third Quartile





Third Quartile

A B C D

1Apart-ment

Monthly Rent ($) Third Quartile

2 1 525 =QUARTILE.INC(B2:B71,3) 3 2 440 4 3 450 5 4 6156 5 480

3rd quartile

It is not necessaryto put the data in ascending

order.




Third Quartile


A B C D

1Apart-ment

Monthly Rent ($) Third Quartile

2 1 525 522.5 3 2 440 4 3 450 5 4 6156 5 480



Using Excel’s QUARTILE.INC Function

If the value of 1 in the QUARTILE.INC function is changed to 0, Excel computes the minimum value in the data set. If the value of 1 is changed to 4, Excel computes the maximum value in the data set.



Excel’s Rank and Percentile Tool

Step 1 Click the Data tab on the RibbonStep 2 In the Analysis group, click Data AnalysisStep 3 Choose Rank and Percentile from the list of Analysis ToolsStep 4 When the Rank and Percentile dialog box appears (see details on next slide)




Step 4 Complete the Rank and Percentile dialog box as follows:






B C D E F G1 Rent Point Rent Rank Percent2 525 4 615 1 98.50%3 440 63 615 1 98.50%4 450 35 600 3 92.70%5 615 42 600 3 92.70%6 480 49 600 3 92.70%7 510 56 600 3 92.70%8 575 28 590 7 91.30%9 430 21 580 8 89.80%10 440 7 575 9 86.90%



Measures of Variability

It is often desirable to consider measures of variability (dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.



Measures of Variability

Range Interquartile Range Variance Standard Deviation Coefficient of Variation



Range

The 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.



Range

Range = largest value - smallest valueRange = 615 - 425 = 190


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615




Interquartile Range

The 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.



425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

Interquartile Range

3rd Quartile (Q3) = 5251st Quartile (Q1) = 445

Interquartile Range = Q3 - Q1 = 525 - 445 = 80





The variance is a measure of variability that utilizes all the data.

Variance

It is based on the difference between the value of each observation (xi) and the mean ( for a sample, m for a population).

x

The variance is useful in comparing the variability of two or more variables.



Variance

The variance is computed as follows:

The variance is the average of the squared differences between each data value and the mean.

for asample

for apopulation

m22

( )xNis

xi xn

22

1

( )



Standard Deviation

The 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 interpreted than the variance.



The standard deviation is computed as follows:

for asample

for apopulation

Standard Deviation

s s 2 2



The coefficient of variation is computed as follows:

Coefficient of Variation

100 %sx

The coefficient of variation indicates how large the standard deviation is in relation to the mean.

for asample

for apopulation

100 %m



54.74100 % 100 % 11.15%490.80sx

22 ( ) 2,996.161

ix xs

n

2 2996.16 54.74s s

the standard

deviation isabout 11%

of the mean

• Variance

• Standard Deviation

• Coefficient of Variation

Sample Variance, Standard Deviation,And Coefficient of Variation




Using Excel to Compute the Sample Variance, Standard Deviation, and

Coefficient of Variation Formula Worksheet


A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean =AVERAGE(B2:B71)3 2 440 Median =MEDIAN(B2:B71)4 3 450 Mode=MODE.SNGL(B2:B71)5 4 615 Variance=VAR.S(B2:B71)6 5 480 Std. Dev.=STDEV.S(B2:B71)7 6 510 C.V. =E6/E2*100



Value Worksheet

Using Excel to Compute the Sample Variance, Standard Deviation, and

Coefficient of Variation


A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean 490.803 2 440 Median 475.004 3 450 Mode 450.005 4 615 Variance 2996.166 5 480 Std. Dev. 54.747 6 510 C.V. 11.15



Using Excel’sDescriptive Statistics Tool

Step 1 Click the Data tab on the RibbonStep 2 In the Analysis group, click Data AnalysisStep 3 Choose Descriptive Statistics from the list of Analysis ToolsStep 4 When the Descriptive Statistics dialog box appears: (see details on next slide)




Excel’s Descriptive Statistics Dialog Box



Excel Value Worksheet (Partial)



A B C D E

1Apart-ment

Monthly Rent ($) Monthly Rent ($)

2 1 5253 2 440 Mean 490.84 3 450 Standard Error 6.5423481145 4 615 Median 4756 5 480 Mode 4507 6 510 Standard Deviation 54.737211468 7 575 Sample Variance 2996.162319



Excel Value Worksheet (Partial)


Note: Rows 1-8 and 17-71 are not shown.

A B C D E9 8 430 Kurtosis -0.33409329810 9 440 Skewness 0.92433047311 10 450 Range 19012 11 470 Minimum 42513 12 485 Maximum 61514 13 515 Sum 3435615 14 575 Count 7016 15 430



End of Chapter 3, Part A

slides by john loucks st . edward’s university

Documents

cengage learning

accessible website

data values

data set

data items

property value data

sample mean example

sample statistics