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Course: ORMS 5310, Fall 2008School: Texas A&M...
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Prepared Slides by JOHN S. LOUCKS St. Edward' s University 2006 Thomson/SouthWestern Slide 1 Chapter 3 Descriptive Statistics: Numerical Measures Part A s s Measures of Location Measures of Variability 2006 Thomson/SouthWestern Slide 2 Measures of Location s s s s s Mean Median Mode Percentiles Quartiles If the measures are computed for data from a sample, they are...
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Prepared Slides by
JOHN S. LOUCKS
St. Edward' s University
2006 Thomson/SouthWestern
Slide 1
Chapter 3 Descriptive Statistics: Numerical Measures Part A
s s
Measures of Location Measures of Variability
2006 Thomson/SouthWestern
Slide 2
Measures of Location
s s s s s
Mean Median Mode Percentiles Quartiles
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 to as the point estimator of the corresponding population parameter.
2006 Thomson/SouthWestern
Slide 3
Mean
s s
The mean of a data set is the average of all the data values. The sample mean is the point estimator of the population mean . x
2006 Thomson/SouthWestern
Slide 4
Sample Mean x
Sum of the values of the n observations
i
x=
x
n
Number of observations in the sample
2006 Thomson/SouthWestern
Slide 5
Population Mean
Sum of the values of the N observations
i
=
x
N
Number of observations in the population
2006 Thomson/SouthWestern
Slide 6
Sample Mean
Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed in ascending order on the next slide.
2006 Thomson/SouthWestern
Slide 7
Sample Mean
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615
2006 Thomson/SouthWestern
Slide 8
Sample Mean
x x=
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590
34, 356 = = 490.80 n 70
i
435 445 450 475 490 525 600
435 445 460 475 500 535 600
435 445 460 475 500 549 600
435 445 460 480 500 550 600
440 450 465 480 500 570 615
440 450 465 480 510 570 615
2006 Thomson/SouthWestern
Slide 9
Median
The median of a data set is the value in the middle when the data items are arranged in ascending order. Whenever a data set has extreme values, the median is the preferred measure of central location. The 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.
2006 Thomson/SouthWestern
Slide 10
Median
For an odd number of observations: 26 18 27 12 14 27 19 12 14 18 19 26 27 27 the median is the middle value. Median = 19
7 observations in ascending order
2006 Thomson/SouthWestern
Slide 11
Median
For an even number of observations: 26 18 27 12 14 27 30 19 12 14 18 19 26 27 27 30
8 observations in ascending order
the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5
2006 Thomson/SouthWestern
Slide 12
Median
Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615
2006 Thomson/SouthWestern
Slide 13
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.
2006 Thomson/SouthWestern
Slide 14
Mode
450 occurred most frequently (7 times) Mode = 450
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615
2006 Thomson/SouthWestern
Slide 15
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.
2006 Thomson/SouthWestern
Slide 16
Percentiles
s
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.
2006 Thomson/SouthWestern
Slide 17
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.
2006 Thomson/SouthWestern
Slide 18
90th Percentile
i = (p/100)n = (90/100)70 = 63 Averaging 63rd the and 64th data values: 90th Percentile = (580 + 590)/2 = 585
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615
Slide 19
2006 Thomson/SouthWestern
90 Percentile
th
"At least 90% of the items take on a value of 585 or less." 63/70 = .9 or 90%
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600
"At least 10% of the items take on a value of 585 or more." 7/70 = .1 or 10%
435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615
Slide 20
2006 Thomson/SouthWestern
Quartiles
Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile
2006 Thomson/SouthWestern
Slide 21
Third Quartile
Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615
Slide 22
2006 Thomson/SouthWestern
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.
2006 Thomson/SouthWestern
Slide 23
Measures of Variability
s s s s s
Range Interquartile Range Variance Standard Deviation Coefficient of Variation
2006 Thomson/SouthWestern
Slide 24
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.
2006 Thomson/SouthWestern
Slide 25
Range
Range = largest value smallest value Range = 615 425 = 190
425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615
2006 Thomson/SouthWestern
Slide 26
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.
2006 Thomson/SouthWestern
Slide 27
Interquartile Range
3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interqua...
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