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Descriptive Statistics: Numerical Measures

Location and Variability

Chapter 3BA 201

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Sample Statistics, Population Parameters,and Point Estimators

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.

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LOCATION

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Measures of Location

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.

The mean provides a measure of central location.

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Mean

ixx

n ix

N

Sample Population

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Sample Mean

Apartment Rents445 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

34,356 490.80

70ix

xn

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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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Median

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

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

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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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.

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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

Apartment Rents

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PRACTICE

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Practice #1

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37411241265

• Compute the … Mean Median Mode

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Practice #1 - Median

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113312861409125915791113126513541255

37411241265

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Practice #1 - Mode

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111311241133125512591265126512861354140914791579

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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. 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

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

Apartment Rents

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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

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

Apartment Rents

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PRACTICE

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Practice #2 - Percentiles

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111311241133125512591265126512861354140914791579

80th Percentile

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VARIABILITY

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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.

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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

Apartment Rents

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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

Apartment Rents

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PRACTICERANGE

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Practice #3 - Range

311162318

7

Range

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VARIANCE

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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

22

( )xNis

xi x

n2

2

1

( )

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• Variance

Sample Variance

Apartment Rents

22 ( )

2,996.161

ix xs

n

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Detailed Example - Variance

1 1-2 = -1 -12 = 13 3-2 = 1 12 = 12 2-2 = 0 02 = 01 1-2 = -1 -12 = 13 3-2 = 1 12 = 1

2 4

ix xxi 2)( xxi

1

)( 22

n

xxs i

s2 = 4/(5-1) = 1

b

c

d

e

f

g

a

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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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• Standard Deviation

Standard Deviation

Apartment Rents

2 2996.16 54.74s s

22 ( )

2,996.161

ix xs

n

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Detailed Example – Standard Deviation

1

)( 22

n

xxs i

s2 = 4/(5-1) = 1

2ss

11s a

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PRACTICEVARIANCE AND STANDARD DEVIATION

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Practice #4 – Variance

3

7

11

16

18

23

ix xxi 2)( xxi

1

)( 22

n

xxs i

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Practice #4 – Standard Deviation

1

)( 22

n

xxs i 2ss

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The coefficient of variation is computed as follows:

Coefficient of Variation

100 %s

x

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

for asample

for apopulation

100 %

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54.74100 % 100 % 11.15%

490.80sx

2 2996.16 54.74s s

22 ( )

2,996.161

ix xs

n

the standard

deviation isabout 11%

of the mean

• Variance

• Standard Deviation

• Coefficient of Variation

Sample Variance, Standard Deviation,And Coefficient of Variation

Apartment Rents

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PRACTICECOEFFICIENT OF VARIATION

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Practice #5 – Coefficient of Variation

x

s=

%100*

x

s

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