CHAPTER 8: Measures of Dispersion

Question 1

What are measures of dispersion?

Answer 1

Descriptive summary measure of how varied are observations from each other

Question 2

What does a small measure of dispersion indicate?

Answer 2

Observations do not vary much and are concentrated about the center of the distribution

Question 3

What does a large measure of dispersion indicate?

Answer 3

Observations greately vary and are spread out from the center of the distribution

Question 4

What indicates an absence of variation?

Answer 4

Value of 0

Measure of dispersion is never negative

Question 5

Differentiate measures of absolute dispersion and measures of relative dispersion

Answer 5

Measures of absolute dispersion:
has the same unit of the observations

Measures of relative dispersion:
has no unit, can be used to compare data sets

Question 6

Give examples of measures of absolute dispersion

Answer 6

range, interquartile range, standard deviation

*variance is not counted because it is a squared value

Question 7

Give examples of measures of relative dispersion

Answer 7

coefficient of variation, z score

Question 8

Define the range

Answer 8

Distance between minimum and maximum values in a data set

Range = Maximum - Minimum

Sometimes presented by stating largest and smallest value

Question 9

How is the range approximated from the FDT?

Answer 9

Subtracting the LCL of the lowest class interval and the HCL of the highest class interval

Range = UCLhci - LCLlci

Question 10

Describe the characteristics of the range

Answer 10

An easy-to-use measure of dispersion, not mathematically tractable

Fails to examine the clustering of observations in the middle of the data set

Greately affected by outliers, cannot be approximated if there are open class intervals

Small values = small range, large values = large range

Question 11

Define the interquartile range (IQR)

Answer 11

IQR = Q3-Q1

Reflects the range of the middle 50% of a data set

Seen as a data set trimmed 25% (tig top and bottom)

The value is dependent on how the quartiles were determined

Question 12

What does the IQR address?

Answer 12

Addresses sensitivity of the range to outliers, but does not address possible variations in the outer 25% portions

Question 13

Define population variance

Answer 13

Mean of squared deviations between each observed value and the mean

Squared deviation: (Observation - mean)^2

Question 14

Define sample variance

Answer 14

Summation of squared deviations between each observed value and the mean, divided by the total number of observations minus 1

Divided by (n-1) to prevent underestimation of the actual variance

Question 15

Define standard deviation

Answer 15

Positive square root of the variance

Question 16

What is the purpose of computational formulas?

Answer 16

Removes computation errors if the mean used for it is rounded off

Question 17

If each observation is transformed by addition or subtraction of a constant, what is the effect on the standard deviation?

Answer 17

According to Property 1, the standard deviation will not change

Question 18

If each observation is transformed by multiplication or division by a constant, what is the effect on the standard deviation?

Answer 18

According to Property 2, the standard deviation will be multiplied or divided by the same constant

Question 19

What are the characteristics of the standard deviation?

Answer 19

Uses every observation in the data set

Distorted by outliers

Never negative

Subject to algebraic treatment

Applies to at least interval level data

Question 20

What does the Bienayme-Chebyshev Rule determine?

Answer 20

How values are described based on the standard deviation

Percentage of observations that are within k standard deviations below and above the mean must be at least [1 - (1/k^2)] * 100
(k > 1)

Question 21

What percent of observed values are 2 standard deviations below and above the mean?

Answer 21

75%

[1 - (1/2^2)] * 100 = 75%

Question 22

What percent of observed values are 3 standard deviations below and above the mean?

Answer 22

88.89%

[1 - (1/3^2)] * 100 = 88.89%

Question 23

What percent of observed values are 3 standard deviations below and above the mean?

Answer 23

99.75%

[1 - (1/4^2)] * 100 = 99.75%

Question 24

Define the coefficient of variation (CV)

Answer 24

The ratio of the standard deviation to the mean in percentage

Can be used to compare variability of two or more data sets even with different means and units

(Standard deviation/mean)*100

Question 25

What will the coefficient of variation be when the mean is zero?

Answer 25

Undefined

Question 26

What will the coefficient of variation be when the mean is negative?

Answer 26

Meaningless

Question 27

Define the standard score or the Z-score

Answer 27

Measures by how many standard deviations an observation is above or below the mean (relative position of the observation in the data set)

(Observation-mean)/standard deviation

Can be used to compare data sets that differ in unit, mean, standard deviation, or both mean and standard deviation

Question 28

Based on the z-score, when is an observation an outlier?

Answer 28

When the absolute value of the computed z-score is greater than or equal to 3

Question 29

How is the z-score interpreted?

Answer 29

If positive: number of standard deviations the observation is above the mean

If negative: number of standard deviations the observation is below the mean

If zero: the observations are equal to the mean