4. Describing Variability

Question 1

Measures of Variability

Answer 1

Descriptions of the amount by which scores are dispersed or scattered in a distribution.

Question 2

For a given mean difference - says, 10 points - between two groups, what degree of variability within each of these groups (small, medium, or large) makes the mean difference…

a) … most conspicuous with more statistical stability?
b) … least conspicuous with less statistical stability?

Answer 2

a) small

b) large

Question 3

Range

Answer 3

The difference between the largest and smallest scores.

Question 4

Variance

Answer 4

The mean of all squared deviation scores.

Question 5

Standard Deviation

Answer 5

A rough measure of the average (or standard) amount by which scores deviate on either side of their mean.

Question 6

Employees of a Corporation A earn annual salaries described by a mean of $90,000 and a standard variation of $10,000.

The majority of all salaries fall between what two values?

Answer 6

$80,000 to $100,000

Question 7

Employees of a Corporation A earn annual salaries described by a mean of $90,000 and a standard variation of $10,000.

A small minority of all salaries are less than what value?

Answer 7

$70,000

Question 8

Employees of a Corporation A earn annual salaries described by a mean of $90,000 and a standard variation of $10,000.

A small minority of all salaries are more than what value?

Answer 8

$110,000

Question 9

Employees of a Corporation B earn annual salaries described by a mean of $90,000 and a standard variation of $2,000.

The majority of all salaries fall between what two values?

Answer 9

$88,000 to $92,000

Question 10

Employees of a Corporation B earn annual salaries described by a mean of $90,000 and a standard variation of $2,000.

A small minority of all salaries are less than what value?

Answer 10

$86,000

Question 11

Employees of a Corporation B earn annual salaries described by a mean of $90,000 and a standard variation of $2,000.

A small minority of all salaries are more than what value?

Answer 11

$94,000

Question 12

Can the value of the standard deviation be negative?

Answer 12

No, the value of the standard deviation can never be negative.

Question 13

Assume that the distribution of IQ scores for all college students has a mean of 120, with a standard deviation of 15.

All students have an IQ of either 105 or 135 because everybody in the distribution is either one standard deviation above or below the mean. True or false?

Answer 13

False. Relatively few students will score exactly one standard deviation from the mean.

Question 14

Assume that the distribution of IQ scores for all college students has a mean of 120, with a standard deviation of 15.

All students score between 105 and 135 because everybody is within one standard deviation on either side of the mean. True or false?

Answer 14

False. Students will score both within and beyond one standard deviation from the mean.

Question 15

Assume that the distribution of IQ scores for all college students has a mean of 120, with a standard deviation of 15.

On the average, students deviate approximately 15 points on either side of the mean. True or false?

Answer 15

True

Question 16

Assume that the distribution of IQ scores for all college students has a mean of 120, with a standard deviation of 15.

Some students deviate more than one standard deviation above or below the mean. True or false?

Answer 16

True

Question 17

Assume that the distribution of IQ scores for all college students has a mean of 120, with a standard deviation of 15.

All students deviate more than one standard deviation above or below the mean. True or false?

Answer 17

False. Students will score both within and beyond one standard deviation from the mean.

Question 18

Assume that the distribution of IQ scores for all college students has a mean of 120, with a standard deviation of 15.

Scott’s IQ score of 150 deviates two standard deviations above the mean. True or false?

Answer 18

True

Question 19

Sum of Squares (SS)

Answer 19

The sum of squared deviation scores.

Question 20

Sum of Squares (SS) for Population (Definition Formula)

Answer 20

SS = ∑ (X - µ)²

Question 21

How to reconstruct the definition formula of Sum of Squares for Population?

Answer 21

1. Subtract the population mean, µ, from each original score, X, to obtian a deviation score, X - µ.
2. Square each deviation score (X - µ)², to eliminate negative signs.
3. Sum all squared deviation scores, ∑ (X - µ)².

Question 22

Sum of Suares (SS) for Population (Computation Formula)

Answer 22

SS = ∑ X² - [ (∑ X)² / N ]

1. The sum of the squared X scores, ∑ X², is obtained by first squaring each X score an then summing all squared scores.
2. The square of sum of all X scores, (∑ X)², is obtained by first adding all X scores and then squaring the sum of all X scores.
3. N is the population size.

Question 23

Give the 8 steps of the computational sequence for calculating the population standard deviation σ (Definition Formula).

Answer 23

1. Assign a value to N representing the number of X scores.
2. Sum all X scores
3. Obtain the mean of these scores.
4. Subtract the mean from each X score to obtain a deviation score.
5. Square each deviation score.
6. Sum all squared deviation scores to obtain the sum of squares.
7. Substitute numbers into the formula to obtain the population variance, σ².
8. Take the square root of σ² to obtain the population standard deviation, σ.

Question 24

Give the 8 steps of the computational sequence for calculating the population standard deviation σ (Computation Formula).

Answer 24

1. Assign a value to N representing the number of X scores.
2. Sum all X scores.
3. Square the sum of all X scores.
4. Square each X score.
5. Sum all squared X scores.
6. Substitute numbers into the formula to obtain the sum of squares, SS.
7. Substitute numbers into the formula to obtain the population variance, σ².
8. Take the square root of σ² to obtain the population standard deviation, σ.

Question 25

Sum of Squares (SS) for Sample (Definition Formula)

Answer 25

SS = ∑ ( ̅X - µ)²

Question 26

Sum of Squares (SS) for Sample (Computation Formula)

Answer 26

SS = ∑ X² - [ (∑ X)² / n ]

Question 27

Variance for Population (Formula)

Answer 27

σ² = SS / N

Question 28

Population Standard Deviation (σ)

Answer 28

A rough measure of the average amount by which scores in the population deviate on either side of their population mean.

Question 29

Standard Deviation for Population (Formula)

Answer 29

σ = √σ² = √(SS / N)

Question 30

Variance for Sample (Formula)

Answer 30

s² = SS / (N - 1) = SS / df

Question 31

Sample Standard Deviation (s)

Answer 31

A rough measure of the average amount by which scores in the sample deviate on either side of their sample mean.

Question 32

Standard Deviation for Sample (Formula)

Answer 32

s = √s² = √ [ (SS / N - 1) ] = √ (SS / df)

Question 33

When do we replace n with n - 1 ?

Answer 33

Only when dividing SS to obtain s² and s.

Question 34

Using the definition formula for the sum of squares, calculate the sample standard deviation for the following four scores: 1, 3, 4, 4.

Answer 34

s = √s² = √ [ (SS / N - 1) ] s=√ { [(1-3)² - (3-3)² + (4-3)² + (4-3)²] / (4 - 1) } = √ (6 / 3) = 1.41

Question 35

Give the 8 steps for calculation of sample standard deviation (s) (Definition Formula)

Answer 35

1. Assign a value to N representing the number of X scores. 2. Sum all X scores. 3. Obtain the mean of these scores. 4. Subtract the mean from each X score to obtain a deviation score. 5. Square each deviation score. 6. Sum all squared deviation scores to obtain the sum of squares. 7. Substitute numbers into the formula to obtain the sample variance, s². 8. Take the square root of s² to obtain the sample standard deviation, s.

Question 36

Give the 8 steps for calculation of sample standard deviation (s) (Computation Formula)

Answer 36

1. Assign a value to N representing the number of X scores. 2. Sum all X scores. 3. Square the sum of all X scores. 4. Square each X score. 5. Sum all squared X scores. 6. Substitute numbers into the formula to obtain the sum of squares, SS. 7. Substitute numbers into the formula to obtain the sample variance, s². 8. Take the square root of s² to obtain the sample standard deviation, s.

Question 37

Using the computation formula for the sum of squares, calculate the population standard deviation for the following scores: 1, 3, 7, 2, 0, 4, 7, 3

Answer 37

2.39

Question 38

Using the computation formula for the sum of squares, calculate the sample standard deviation for the following scores: 10, 8, 5, 0, 1, 1, 7, 9, 2

Answer 38

3.87

Question 39

Days absent from school for a sample of 10 first-grade children are: 8, 5, 7, 1, 4, 0, 5, 7, 2, 9 Before calculating the standard deviation, decide whether the definitional or computational formula would be more efficient. Why? Then use the more efficient formula to calculate the sample standard variation.

Answer 39

Computation formula since the mean is not a whole number. s = 3.05

Question 40

Degrees of Freedom (df)

Answer 40

The number of values free to vary, given one or more mathematical restrictions.

Question 41

As a first step toward modifying his study habits, Phil keeps daily records of his study time. During the first two weeks, Phil's mean study time equals 20 hours per week. If he studied 22 jours during the first week, how many hours did he study during the second week? If this information is to be used to estimate some unkown population characteristic, how many degrees of freedom are associated with it? Describe the mathematical restriction that causes a loss of degrees of freedom.

Answer 41

18 hours df = 1 When all observations are expressed as deviations from their mean, the sum of all deviations must equal zero.

Question 42

As a first step toward modifying his study habits, Phil keeps daily records of his study time. During the first four weeks, Phil's mean study time equals 21 hours. If he studied 22, 18, and 21 hours during the first, second, and third weeks, respectively, how may hours did he study during the fourth week? If this information is to be used to estimate some unkown population characteristic, how many degrees of freedom are associated with it? Describe the mathematical restriction that causes a loss of degrees of freedom.

Answer 42

23 hours df = 3 When all observations are expressed as deviations from their mean, the sum of all deviations must equal zero.

Question 43

Interquartile Range (IQR)

Answer 43

The range for the middle 50 percent of the scores.

Question 44

Give the 7 steps for the calculation of the IQR.

Answer 44

1. Order scores from least to most. 2. To determine how far to penetrate the set ofordered scores, begin at either end, then add 1 to the total number of scores and divide by 4. If necessary, round the result to the nearest whole number. 3. Beginning with the largest score, count the requisite number of steps (calculated in step 2) into the ordered scores to find the location of the third quartile. 4. The third quartile equals the value of the score at this location. 5. Beginning with the smallest score, again count the requisite number of steps into the ordered scores to find the location of the first quartile. 6. The first quartile equals the value of the score at this location. 7. The IQR equals the third quartile minus the first quartile.

Question 45

Determine the values of the range and the IQR for the following set of data: Retirement ages: 60, 63, 45, 63, 65, 70, 55, 63, 60, 65, 63

Answer 45

range = 25 IQR = 65 - 60 = 5

Question 46

Determine the values of the range and the IQR for the following set of data: Residence changes: 1, 3, 4, 1, 0, 2, 5, 8, 0, 2, 3, 4, 7, 11, 0, 2, 3, 4

Answer 46

range = 11 IQR = 4 - 1 = 3