1 Slide

Descriptive Statistics: Numerical Measures

Location and Variability

Chapter 3BA 201

2 Slide

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.

3 Slide

LOCATION

4 Slide

Measures of Location Mean Median Mode Percentiles Quartiles

5 Slide

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.

6 Slide

Mean

ixx

n ix

Nm

Sample Population

7 Slide

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

34,356 490.8070ix

xn

8 Slide

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.

9 Slide

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.

10 Slide

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

11 Slide

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

12 Slide

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

13 Slide

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.

14 Slide

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

15 Slide

PRACTICE

16 Slide

Practice #1

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11241265

• Compute the … Mean Median Mode

18 Slide

Practice #1 - Median

1479983

113312861409125915791113126513541255

37411241265

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111311241133125512591265126512861354140914791579

19 Slide

Practice #1 - Mode

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111311241133125512591265126512861354140914791579

20 Slide

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.

21 Slide

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.

22 Slide

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

23 Slide

Quartiles

Quartiles are specific percentiles. First Quartile = 25th Percentile

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

24 Slide

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

25 Slide

PRACTICE

26 Slide

Practice #2 - Percentiles

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80th Percentile

27 Slide

VARIABILITY

28 Slide

Measures of Variability

Range Interquartile Range Variance Standard Deviation Coefficient of Variation

29 Slide

Range

The range of a data set is the difference between the largest and smallest data values.

30 Slide

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

31 Slide

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.

32 Slide

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

33 Slide

PRACTICERANGE

34 Slide

Practice #3 - Range

311162318

7

Range

35 Slide

VARIANCE

36 Slide

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.

37 Slide

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

( )

38 Slide

• Variance

Sample VarianceApartment Rents

22 ( ) 2,996.161

ix xs

n

39 Slide

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)( 2

2

n

xxs i

s2 = 4/(5-1) = 1

b

c

d

e

f

g

a

40 Slide

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.

41 Slide

The standard deviation is computed as follows:

for asample

for apopulation

Standard Deviation

s s 2 2

42 Slide

• Standard Deviation

Standard DeviationApartment Rents

2 2996.16 54.74s s

22 ( ) 2,996.161

ix xs

n

43 Slide

Detailed Example – Standard Deviation

1)( 2

2

n

xxs i

s2 = 4/(5-1) = 1

2ss

11s a

44 Slide

PRACTICEVARIANCE AND STANDARD DEVIATION

45 Slide

Practice #4 – Variance

37

11161823

ix xxi 2)( xxi

1)( 2

2

n

xxs i

46 Slide

Practice #4 – Standard Deviation

1)( 2

2

n

xxs i 2ss

47 Slide

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

48 Slide

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

49 Slide

PRACTICECOEFFICIENT OF VARIATION

50 Slide

Practice #5 – Coefficient of Variation

x

s=%100*

xs

51 Slide