Slides by John Loucks St . Edward’s University
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Transcript of Slides by John Loucks St . Edward’s University
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Slides by
JohnLoucks
St. Edward’sUniversity
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Chapter 3, Part A Descriptive Statistics: Numerical
Measures Measures of Location Measures of Variability
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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
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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.
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Sample Mean x
Number ofobservationsin the sample
Sum of the valuesof the n observations
ixx
n
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Population Mean m
Number ofobservations inthe population
Sum of the valuesof the N observations
ix
Nm
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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
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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
Example: Apartment Rents
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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.
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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
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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
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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
Example: Apartment Rents
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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.
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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.
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Mode
450 occurred most frequently (7 times)Mode = 450
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
Example: Apartment Rents
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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 ($)
2 1 525 Mean =AVERAGE(B2:B71)3 2 440 Median =MEDIAN(B2:B71)4 3 450 Mode =MODE.SNGL(B2:B71)5 4 6156 5 480
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Value Worksheet
Note: Rows 7-71 are not shown.
Using Excel to Computethe Mean, Median, and Mode
A B C D E
1Apart-ment
Monthly Rent ($)
2 1 525 Mean 490.803 2 440 Median 475.004 3 450 Mode 450.005 4 6156 5 480
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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.
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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.
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80th Percentile
i = (p/100)n = (80/100)70 = 56Averaging the 56th and 57th data values:80th Percentile = (535 + 549)/2 = 542
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
Example: Apartment Rents
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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
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
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Excel Formula Worksheet
Note: Rows 7-71 are not shown.
Using Excel’s Rank and Percentile Toolto Compute Percentiles and Quartiles
A B C D
1Apart-ment
Monthly Rent ($) 80th Percentile
2 1 525 =PERCENTILE.INC(B2:B71,.8) 3 2 440 4 3 450 5 4 6156 5 480
80th percentile
It is not necessaryto put the data in ascending
order.
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24 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Value Worksheet
Note: Rows 7-71 are not shown.
Using Excel’s Rank and Percentile Toolto Compute Percentiles and Quartiles
A B C D
1Apart-ment
Monthly Rent ($) 80th Percentile
2 1 525 537.8 3 2 440 4 3 450 5 4 6156 5 480
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Quartiles
Quartiles are specific percentiles. First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile
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Third Quartile
Third quartile = 75th percentilei = (p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525
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
Example: Apartment Rents
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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
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28 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
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Excel Formula Worksheet
Note: Rows 7-71 are not shown.
Third Quartile
A B C D
1Apart-ment
Monthly Rent ($) Third Quartile
2 1 525 =QUARTILE.INC(B2:B71,3) 3 2 440 4 3 450 5 4 6156 5 480
3rd quartile
It is not necessaryto put the data in ascending
order.
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29 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Value Worksheet
Third Quartile
Note: Rows 7-71 are not shown.
A B C D
1Apart-ment
Monthly Rent ($) Third Quartile
2 1 525 522.5 3 2 440 4 3 450 5 4 6156 5 480
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30 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
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31 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
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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)
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32 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
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Excel’s Rank and Percentile Tool
Step 4 Complete the Rank and Percentile dialog box as follows:
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Value Worksheet
Note: Rows 11-71 are not shown.
Excel’s Rank and Percentile Tool
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%
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34 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
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Measures of Variability
Range Interquartile Range Variance Standard Deviation Coefficient of Variation
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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.
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Range
Range = largest value - smallest valueRange = 615 - 425 = 190
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
Example: Apartment Rents
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38 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
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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
Note: Data is in ascending order.
Example: Apartment Rents
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40 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
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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.
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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
( )
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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.
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The standard deviation is computed as follows:
for asample
for apopulation
Standard Deviation
s s 2 2
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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
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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
Example: Apartment Rents
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Using Excel to Compute the Sample Variance, Standard Deviation, and
Coefficient of Variation Formula Worksheet
Note: Rows 8-71 are not shown.
A B C D E
1Apart-ment
Monthly Rent ($)
2 1 525 Mean =AVERAGE(B2:B71)3 2 440 Median =MEDIAN(B2:B71)4 3 450 Mode=MODE.SNGL(B2:B71)5 4 615 Variance=VAR.S(B2:B71)6 5 480 Std. Dev.=STDEV.S(B2:B71)7 6 510 C.V. =E6/E2*100
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Value Worksheet
Using Excel to Compute the Sample Variance, Standard Deviation, and
Coefficient of Variation
Note: Rows 8-71 are not shown.
A B C D E
1Apart-ment
Monthly Rent ($)
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
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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)
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Using Excel’sDescriptive Statistics Tool
Excel’s Descriptive Statistics Dialog Box
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Excel Value Worksheet (Partial)
Using Excel’sDescriptive Statistics Tool
Note: Rows 9-71 are not shown.
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
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Excel Value Worksheet (Partial)
Using Excel’sDescriptive Statistics Tool
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
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End of Chapter 3, Part A