bia 2610 – statistical methods chapter 3 – descriptive statistics: numerical measures

BIA 2610 – Statistical MethodsChapter 3 – Descriptive Statistics:

Numerical Measures

Chapter 3 Descriptive Statistics: Numerical Measures

Measures of Variability

Measures of Location

Measures of Location

If the measures are computed for data from a sample, theyare called sample statistics.

If the measures are computed fordata from a population, they arecalled population parameters.

A sample statistic is referred to asthe point estimator of thecorresponding population parameter.

Mean Median Mode

Percentiles Quartiles

Weighted Mean Geometric Mean

Mean

The sample mean is the point estimator of the population mean µ (mu).

Perhaps the most important measure of location is the mean.

The mean provides a measure of central location. The mean of a data set is the average of all the data

values.

Sample Mean

Number ofobservationsin the sample

Sum of the valuesof the n observations

𝑥=∑ 𝑥𝑖𝑛

Population Mean m

Number ofobservations inthe population

Sum of the valuesof the N observations

𝜇=∑ 𝑥𝑖𝑁

Sample Mean

Seventy efficiency apartments were randomly

sampled in a college town. The monthly rents for

these apartments are listed below.

Example: Apartment Rents

545 715 530 690 535 700 560 700 540 715

540 540 540 625 525 545 675 545 550 550565 550 625 550 550 560 535 560 565 580

550 570 590 572 575 575 600 580 670 565700 585 680 570 590 600 649 600 600 580670 615 550 545 625 635 575 650 580 610

610 675 590 535 700 535 545 535 530 540

Central Location Measures

Averaging the 35th and 36th data values:

Median = (575 + 575)/2 = 575


525 530 530 535 535 535 535 535 540 540

540 540 540 545 545 545 545 545 550 550550 550 550 550 550 560 560 560 565 565

565 570 570 572 575 575 575 580 580 580580 585 590 590 590 600 600 600 600 610610 615 625 625 625 635 649 650 670 670

675 675 680 690 700 700 700 700 715 715Note: Data is in ascending order.

= = 590.80Mode = 550

550 occurred most frequently (7 times)

Excel’s Mean, Median, and Mode Functions

Excel’s Mean function

=AVERAGE(data cell range)

Excel’s Median function

=MEDIAN(data cell range)

Excel’s Mode function

=MODE.SNGL(data cell range)

Percentiles

Arrange the data in ascending order.

Compute Lp, the location of the pth percentile.

Lp = (p/100)(n + 1)

For example, in a sample of n = 70 values, the location of the 80th percentile (p = 80) would be calculated as:Lp = (p/100)(n + 1) = (80/100)(70 + 1) = 56.8

80th Percentile

Lp = (p/100)(n + 1) = (80/100)(70 + 1) = 56.8

(the 56th value plus .8 times the difference between the 57th and 56th values)

80th Percentile = 635 + .8(649 – 635) = 646.2


525 530 530 535 535 535 535 535 540 540

540 540 540 545 545 545 545 545 550 550550 550 550 550 550 560 560 560 565 565

565 570 570 572 575 575 575 580 580 580580 585 590 590 590 600 600 600 600 610610 615 625 625 625 635 649 650 670 670


80th Percentile

525 530 530 535 535 535 535 535 540 540

540 540 540 545 545 545 545 545 550 550550 550 550 550 550 560 560 560 565 565

565 570 570 572 575 575 575 580 580 580580 585 590 590 590 600 600 600 600 610610 615 625 625 625 635 649 650 670 670

675 675 680 690 700 700 700 700 715 715

“At least 80% of the items take on a value of 646.2 or less.”

“At least 20% of theitems take on a value of 646.2 or more.”

56/70 = .8 or 80% 14/70 = .2 or 20%


Quartiles Quartiles are specific percentiles.

First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile

Third Quartile (75th Percentile)

Lp = (p/100)(n + 1) = (75/100)(70 + 1) = 53.25

Third quartile = 625 + .25(625 – 625) = 625


525 530 530 535 535 535 535 535 540 540

540 540 540 545 545 545 545 545 550 550550 550 550 550 550 560 560 560 565 565

565 570 570 572 575 575 575 580 580 580580 585 590 590 590 600 600 600 600 610610 615 625 625 625 635 649 650 670 670


(the 53rd value plus .25 times the difference between the 54th and 53rd values)

Measures of Variability Range

Variance

Standard Deviation

Coefficient of Variation

Range

525 530 530 535 535 535 535 535 540 540

540 540 540 545 545 545 545 545 550 550550 550 550 550 550 560 560 560 565 565

565 570 570 572 575 575 575 580 580 580580 585 590 590 590 600 600 600 600 610610 615 625 625 625 635 649 650 670 670

675 675 680 690 700 700 700 700 715 715

Range = largest value - smallest valueRange = 715 - 525 = 190

Note: Data is in ascending order.


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 ( for a sample,

m for a population).

The variance is useful in comparing the variability of two or more variables.

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

𝑠2=∑ (𝑥𝑖−𝑥 )2

𝑛−1𝜎 2=

∑ (𝑥 𝑖−𝜇 )2𝑁

Standard Deviation

The standard deviation is computed as follows:

for asample

for apopulation

s = s =

Excel’s Variance and Standard Deviation Functions

Excel’s Sample Variance function

=VAR.S(data cell range)

Excel’s Sample Standard Deviation function

=STDEV.S(data cell range)

Coefficient of Variation

The coefficient of variation is computed as follows:

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

for asample

for apopulation

% %

Sample Variance, Standard Deviation,and Coefficient of Variation

Standarddeviation isabout 9% of the mean

• Variance

• Standard Deviation

• Coefficient of Variation


s2 = = 2,996.16

s =

% =

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 1 545 Mean =AVERAGE(B2:B71)3 2 715 Median =MEDIAN(B2:B71)4 3 530 Mode=MODE.SNGL(B2:B71)5 4 690 Variance=VAR.S(B2:B71)6 5 535 Std. Dev.=STDEV.S(B2:B71)7 6 700 C.V. =E6/E2*100

Using Excel to Compute the Sample Variance, Standard Deviation, and

Coefficient of Variation Value Worksheet

Note: Rows 8-71 are not shown.

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 545 Mean 590.803 2 715 Median 575.004 3 530 Mode 550.005 4 690 Variance 2996.166 5 535 Std. Dev. 54.747 6 700 C.V. 9.27

End of Chapter 3

bia 2610 – statistical methods chapter 3 – descriptive statistics: numerical measures

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