Measures of Variability Range Interquartile range Variance Standard deviation Coefficient of variation.

Measures of Variability

• Range

• Interquartile range

• Variance

• Standard deviation

• Coefficient of variation

Consider the sample of starting salaries of business

grads. We would be interested in knowing if there was a low or high degree of variability or dispersion in starting salaries received.

Range

•Range is simply the difference between the largest and smallest values in the sample

•Range is the simplest measure of variability.

•Note that range is highly sensitive to the largest and smallest values.

Example: Apartment RentsExample: Apartment Rents

Seventy studio apartmentsSeventy studio apartments

were randomly sampled inwere randomly sampled in

a small college town. Thea small college town. The

monthly rent prices formonthly rent prices for

these apartments are listedthese apartments are listedin ascending order on the next slide. in ascending order on the next slide.

Range

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

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

Range = largest value - smallest valueRange = largest value - smallest value

Range = 615 - 425 = 190Range = 615 - 425 = 190

Interquartile Range

The The interquartile rangeinterquartile range of a data set is the difference of a data set is the difference between the third quartile and the first quartile.between the third quartile and the first quartile.

It is the range for the It is the range for the middle 50%middle 50% of the data. of the data.

It overcomes the sensitivity to extreme data valuesIt overcomes the sensitivity to extreme data values..

Interquartile Range

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

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

3rd Quartile (3rd Quartile (QQ3) = 5253) = 5251st Quartile (1st Quartile (QQ1) = 4451) = 445

Interquartile Range = Interquartile Range = QQ3 - 3 - QQ1 = 525 - 445 = 801 = 525 - 445 = 80

Variance

•The variance is a measure of variability that uses all the data•The variance is based on the difference between each observation (xi) and the mean ( for the sample and μ for the population).

x

The variance is the average of the squared differences between the observations and the mean value

For the population:N

xi2

2 )(

For the sample: 1

)( 22

n

xxs i

Standard Deviation

• The Standard Deviation of a data set is the square root of the variance.

• The standard deviation is measured in the same units as the data, making it easy to interpret.

Computing a standard deviation

1

)( 2

n

xxs i

For the population:

For the sample:

N

xi2)(

Coefficient of Variation

100

Just divide the standard deviation by the mean and multiply times 100

Computing the coefficient of variation:

For the sample100x

s

For the population

The heights (in inches) of 25 individuals were recorded and the following statistics were calculated mean = 70range = 20mode = 73variance = 784median = 74 The coefficient of variation equals

1 2 3 4

0% 0%0%0%

1. 11.2%

2. 1120%

3. 0.4%

4. 40%

10

0

0

5

If index i (which is used to determine the location of the pth percentile) is not an integer, its value should be

1 2 3 4

0% 0%0%0%

1. squared

2. divided by (n - 1)

3. rounded down

4. rounded up

10

0

0

5

Which of the following symbols represents the variance of the population?

1 2 3

0% 0%0%

1. 2

2. 3.

10

0

0

5

Which of the following symbols represents the size of the sample

1 2 3 4

0% 0%0%0%

1. 2

2. 3. N

4. n

10

0

0

5

The symbol s is used to represent

1 2 3 4

0% 0%0%0%

1. the variance of the population

2. the standard deviation of the sample

3. the standard deviation of the population

4. the variance of the sample

10

0

0

5

The numerical value of the variance

1 2 3 4

0% 0%0%0%

1. is always larger than the numerical value of the standard deviation

2. is always smaller than the numerical value of the standard deviation

3. is negative if the mean is negative

4. can be larger or smaller than the numerical value of the standard deviation

10

0

0

5

If the coefficient of variation is 40% and the mean is 70, then the variance is

1 2 3 4

0% 0%0%0%

1. 28

2. 2800

3. 1.75

4. 784

10

0

0

5

Problem 22, page 94

Broker-Assisted 100 Shares at $50 per Share

Range 45.05

Interquartile Range 23.98

Variance 190.67

Standard Deviation 13.8

Coefficient of Variation 38.02

25th percentile 6

75th percentile 18

interquart 25 24.995

interquart 75 48.975

Mean 36.32

Online 500 Shares at $50 per Share

Range 57.50

Interquartile Range 11.475

Variance 140.633

Standard Deviation 11.859

Coefficient of Variation 57.949

25th percentile  

75th percentile  

interquart 25 13.475

interquart 75 24.95

Mean 20.46

The variability of commissions is greater

for broker-assisted trades

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

Coefficient of VariationCoefficient of Variation Formula WorksheetFormula Worksheet

Note: Rows 8-71 are not shown.Note: Rows 8-71 are not shown.

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean =AVERAGE(B2:B71)3 2 440 Median =MEDIAN(B2:B71)4 3 450 Mode =MODE(B2:B71)5 4 615 Variance =VAR(B2:B71)6 5 480 Std. Dev. =STDEV(B2:B71)7 6 510 C.V. =E6/E2*100

Value WorksheetValue Worksheet

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

Coefficient of VariationCoefficient of Variation

Note: Rows 8-71 are not shown.Note: Rows 8-71 are not shown.

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean 490.803 2 440 Median 475.004 3 450 Mode 450.005 4 615 Variance 2996.166 5 480 Std. Dev. 54.747 6 510 C.V. 11.15

Using Excel’sUsing Excel’sDescriptive Statistics ToolDescriptive Statistics Tool

Step 4Step 4 When the When the Descriptive StatisticsDescriptive Statistics dialog box dialog box appears:appears:

Enter B1:B71 in the Enter B1:B71 in the Input RangeInput Range box box

Select Select Grouped By ColumnsGrouped By Columns

Select Select Labels in First RowLabels in First Row

Select Select Output RangeOutput Range

Enter D1 in the Enter D1 in the Output RangeOutput Range box box

Select Select Summary StatisticsSummary Statistics

Click Click OKOK

• Descriptive Statistics Dialog Box

Using Excel’s Descriptive Statistics ToolUsing Excel’s Descriptive Statistics Tool

Value Worksheet (Partial)Value Worksheet (Partial)

Using Excel’sUsing Excel’sDescriptive Statistics ToolDescriptive Statistics Tool

A B C D E

1Apart-ment

Monthly Rent ($) Monthly Rent ($)

2 1 5253 2 440 Mean 490.84 3 450 Standard Error 6.5423481145 4 615 Median 4756 5 480 Mode 4507 6 510 Standard Deviation 54.737211468 7 575 Sample Variance 2996.162319

Note: Rows 9-71 are not shown.Note: Rows 9-71 are not shown.

Value Worksheet (Partial)Value Worksheet (Partial)

Using Excel’sUsing Excel’sDescriptive Statistics ToolDescriptive Statistics Tool

A B C D E9 8 430 Kurtosis -0.33409329810 9 440 Skewness 0.92433047311 10 450 Range 19012 11 470 Minimum 42513 12 485 Maximum 61514 13 515 Sum 3435615 14 575 Count 7016 15 430

Note: Rows 1-8 and 17-71 are not shownNote: Rows 1-8 and 17-71 are not shown..

Measures of Relative Location and Detecting Outliers

• z-scores• Chebyshev’s Theorem• Detecting Outliers

By using the mean and standard

deviation together, we can learn more about the relative

location of observations in a

data set

z-score

s

xxz ii

Here we compare the deviation from

the mean of a single observation to the standard deviation

The z-score is compute for each xi :

Where

zi is the z-score for xi

is the sample mean

s is the sample standard deviation

x

The z-score can be interpreted as the

number of standard deviations xi is from the sample mean

Z-scores for the starting salary data

Graduate Starting Salary xi - x z-score

1 2850 -90 -0.543

2 2950 10 0.060

3 3050 110 0.664

4 2880 -60 -0.362

5 2755 -185 -1.117

6 2710 -230 -1.388

7 2890 -50 -0.302

8 3130 190 1.147

9 2940 0 0.000

10 3325 385 2.324

11 2920 -20 -0.121

12 2880 -60 -0.362

Chebyshev’s Theorem

This theorem enables us to make statements about the proportion of data values

that must be within a specified number of

standard deviations from the mean

At least (1-1/z2) of the data values must be within z standard deviations of the mean, where z is greater than 1.

Implications of Chebychev’s Theorem

• At least .75, or 75 percent of the data values must be within 2 ( z = 2) standard deviations of the mean.

• At least .89, or 89 percent, of the data values must be within 3 (z = 3) standard deviations of the mean.

• At least .94, or 94percent, of the data values must be within 4 (z = 4) standard deviations from the mean.

Note: z must be greater than one but need not be an integer.

For example:

Chebyshev’s Theorem

Let Let zz = 1.5 with = 490.80 and = 1.5 with = 490.80 and ss = 54.74 = 54.74xx

At least (1 At least (1 1/(1.5) 1/(1.5)22) = 1 ) = 1 0.44 = 0.56 or 56% 0.44 = 0.56 or 56%

of the rent values must be betweenof the rent values must be between

xx - - zz((ss) = 490.80 ) = 490.80 1.5(54.74) = 409 1.5(54.74) = 409

andandxx + + zz((ss) = 490.80 + 1.5(54.74) = 573) = 490.80 + 1.5(54.74) = 573

(Actually, 86% of the rent values(Actually, 86% of the rent values are between 409 and 573.)are between 409 and 573.)

Detecting Outliers

You can use z-scores to detect extreme values in the data set, or “outliers.” In the case of very high z-scores

(absolute values) it is a good idea to recheck the data for

accuracy.