Stat as Tics and Their Use in Hr 179

8/4/2019 Stat as Tics and Their Use in Hr 179

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1

Slide

Chapter 3 Part ADescriptive Statistics: Numerical Methods

Measures of Location

Measures of Variability

x



2

Slide

Measures of Location

Mean

Median

Mode

Percentiles

Quartiles



3

Slide

Example: Apartment Rents

Given below is a sample of monthly rent values ($)

for one-bedroom apartments. The data is a sample of 70

apartments in a particular city. The data are presented

in ascending order.

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



4

Slide

Mean

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

values. If the data are from a sample, the mean is denoted by

.

If the data are from a population, the mean is

denoted by m (mu).

xx

n

i

x

N

i

x



5

Slide


Mean

xx

n

i

34 356

70490 80

,.

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



6

Slide

Median

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.


http://slidepdf.com/reader/full/stat-as-tics-and-their-use-in-hr-179 7/277Slide

Median

The median of a data set is the value in the middle

when the data items are arranged in ascending order. For an odd number of observations, the median is the

middle value.

For an even number of observations, the median is

the average of the two middle values.




Median

Median = 50th percentile

i = ( p/100)n = (50/100)70 = 35.5Averaging the 35th and 36th data values:

Median = (475 + 475)/2 = 475

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



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.




Mode

450 occurred most frequently (7 times)

Mode = 450

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



Percentiles

A percentile provides information about how the

data are spread over the interval from the smallestvalue to the largest value.

Admission test scores for colleges and universitiesare 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 lessand at least (100 - p) percent of the items take on thisvalue or more.

•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.

Percentiles




90th Percentile

i = ( p/100)n = (90/100)70 = 63

Averaging the 63rd and 64th data values:

90th Percentile = (580 + 590)/2 = 585

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



14Slide

Quartiles

Quartiles are specific percentiles

First Quartile = 25th Percentile

Second Quartile = 50th Percentile = Median

Third Quartile = 75th Percentile



15Slide


Third Quartile

Third quartile = 75th percentile

i = ( p/100)n = (75/100)70 = 52.5 = 53

Third quartile = 525

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



16Slide


It is often desirable to consider measures of

variability (dispersion), as well as measures oflocation.

For example, in choosing supplier A or supplier B wemight consider not only the average delivery time for

each, but also the variability in delivery time for each.



17Slide


Range

Interquartile Range

Variance

Standard Deviation

Coefficient of Variation



18Slide

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 datavalues.



19Slide


Range

Range = largest value - smallest value

Range = 615 - 425 = 190

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



20Slide

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.



21Slide


Interquartile Range

3rd Quartile (Q3) = 525

1st Quartile (Q1) = 445

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

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615



22Slide

Variance

The variance is a measure of variability that utilizes

all the data. It is based on the difference between the value of

each observation (xi) and the mean (x for a sample, for a population).



23Slide

Variance

The variance is the average of the squared differences

between each data value and the mean. If the data set is a sample, the variance is denoted by

s2.

If the data set is a population, the variance is denoted

by 2.

s

xi

x

n

22

1

( )

2

2

( ) x

N

i



24Slide

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 comparable, than the variance, to themean.

If the data set is a sample, the standard deviation isdenoted s.

If the data set is a population, the standard deviation

is denoted (sigma).

s s2

2



25Slide


The coefficient of variation indicates how large the

standard deviation is in relation to the mean. If the data set is a sample, the coefficient of variation

is computed as follows:

If the data set is a population, the coefficient ofvariation is computed as follows:

s

x ( )100

( )100



26Slide


Variance

Standard Deviation


s xi x

n

2 2

12 996 16

( )

, .

s s 2 2996 47 54 7. .

s

x

100

54 74

490 80 100 1115

.

. .



27Slide

End of Chapter 3, Part A

Stat as Tics and Their Use in Hr 179

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