Variability

Statistics means never having to say you're certain.

Statistics - Chapter 4 2

VariabilityThe amount by which scores are dispersed or

scattered in a distribution.Page 74 graphs

3Statistics - Chapter 4

RangeDifference between the largest and smallest

scores.

Problem: large groups may have large range

Variance and Standard DeviationStandard Deviation - The square root of the

variance.OrThe square root of the mean of all the

squared deviations from the mean!!???

The value of a standard deviation can NOT be negative.

Standard DeviationA rough measure of the average (or standard)

amount by which scores deviate on either side of their mean.

Progress Check 4.1 Page 80Employees of Corporation A earn annual salaries

described by a mean of $90,000 and a standard deviation of $10,000.a. The majority of all salaries fall between what two

values?b. A small minority of salaries are less than what

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

value?c. Answer parts (a), (b), and (c) for Corporation B’s

employees, who earn annual salaries described by a mean of $90,000 and a standard deviation of $2,000.

Standard DeviationDeviations from the mean.The sum of all the deviations equals the

variance.

To calculate the varianceThe sum of squares equals the sum of all

squared deviation scores (p. 83)

Sum of SquaresCalculation example of sample sum of

squares (SS) using the computation formula (p. 83)

SS=ΣX2 – (ΣX)2

n

Standard DeviationCalculation example of sample standard

deviation using the computation formula (p. 86)

s = √s2 = √SS2

n-1

Why n-1? (p88)This applies the sample estimate to the

variance rather then the population estimate.If we use the population estimate we would

underestimate the variability.

In other words, this allows a more conservative and accurate estimate of the variance within the sample.

Degrees of freedomDegrees of freedom (df) refers to the number

of values that are free to vary, given one or more mathematical restrictions, in a sample being used to estimate a population characteristic. (p. 90)

The value of the population mean – mu (μ)Most of the time the population mean is

unknown so we use the value of the sample mean and the degrees of freedom (df) = n-1.

Standard Deviation calculation (p88)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 in the formula to obtain the

sample variance, s2.8. Take the square root of s2 to obtain the sample

standard deviation, s.

Qualitative data and varianceNo measures of variability exist for

qualitative data!

However, if the data can be ordered, then the variability can be described by identifying extreme scores (ranks).

Progress CheckCalculate the mean, median, mode, and

standard deviation for the following height of students in inches.

64, 61, 73, 70, 71, 75, 69, 60, 63, 71, 65, 62

Statistics - Chapter 4 16