Measures of Variability

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

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?

a) small

b) large

Range

The difference between the largest and smallest scores.

Variance

The mean of all squared deviation scores.

Standard Deviation

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

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?

$80,000 to $100,000

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?

$70,000

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?

$110,000

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?

$88,000 to $92,000

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?

$86,000

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?

$94,000

Can the value of the standard deviation be negative?

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

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?

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

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?

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

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?

True

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

True

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

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

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

True

Sum of Squares (SS)

The sum of squared deviation scores.

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

SS = ∑ (X - µ)²

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

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

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

SS = ∑ X² - [ (∑ X)² / N ]

- The sum of the squared X scores, ∑ X², is obtained by first squaring each X score an then summing all squared scores.
- 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.
- N is the population size.

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

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

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

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

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

SS = ∑ ( ̅X - µ)²

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

SS = ∑ X² - [ (∑ X)² / n ]

Variance for Population (Formula)

σ² = SS / N

Population Standard Deviation (σ)

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

Standard Deviation for Population (Formula)

σ = √σ² = √(SS / N)

Variance for Sample (Formula)

s² = SS / (N - 1) = SS / df

Sample Standard Deviation (s)

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

Standard Deviation for Sample (Formula)

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

When do we replace n with n - 1 ?

Only when dividing SS to obtain s² and s.

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

s = √s² = √ [ (SS / N - 1) ]

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

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

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

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

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

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

2.39

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

3.87

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.

Computation formula since the mean is not a whole number.

s = 3.05

Degrees of Freedom (df)

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

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.

18 hours

df = 1

When all observations are expressed as deviations from their mean, the sum of all deviations must equal zero.

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.

23 hours

df = 3

When all observations are expressed as deviations from their mean, the sum of all deviations must equal zero.

Interquartile Range (IQR)

The range for the middle 50 percent of the scores.

Give the 7 steps for the calculation of the IQR.

- Order scores from least to most.
- 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.
- 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.
- The third quartile equals the value of the score at this location.
- Beginning with the smallest score, again count the requisite number of steps into the ordered scores to find the location of the first quartile.
- The first quartile equals the value of the score at this location.
- The IQR equals the third quartile minus the first quartile.

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

range = 25

IQR = 65 - 60 = 5

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

range = 11

IQR = 4 - 1 = 3