Chapter 4

I. Variability

A. Definition - quantitative measure of the degree to which the scores in a distribution spread out or cluster together. Usually defined in terms of distance from a measure of central tendency.

II. Ranges

A. Range - largest score minus smallest score plus one unit to encompass upper and lower real limits. Need at least ordinal data.

Range = URL X_max - LRL X_min

B. Interquartile range - distance between scores that encompass the middle 50% of a distribution. Find score at Q₃ (75th percentile) and at Q₁ (25th percentile) and subtract. Need at least ordinal data.

Interquartile range = Q₃ - Q₁

C. Semi-interquartile range - distance between scores that encompass the middle 50% of a distribution. Find score at Q₃ (75th percentile) and at Q₁ (25th percentile) and subtract. Need at least ordinal data.

Semi-interquartile range = (Q₃ - Q₁) / 2

 

III. Variance and Standard Deviation - need at least interval data

A. Deviation - difference between X and the mean.

1. Can’t use deviations as our measure of distance because they sum to zero.

2. Make all deviations positive by squaring.

B. Sum of squares (SS) - sum of squared deviations

C. Variance - mean squared deviations.

D. Standard deviation - square root of variance

E. Degrees of freedom - terms that are free to vary

n-1 for denominator

F. Transformations of scale

1. Adding a constant to X - does not affect s

2. Multiplying each X by a constant - s is multiplied by the constant (so s² is multiplied by constant squared).

G. Properties of standard deviation

1. Relative distance from mean (description of single X compared to distrubtion)

2. Relative distance between means (inference about one mean compared to another)

IV. Comparing variability statistics

A. Level of measurement

1. Ordinal - range, IQR, semi-IQR

2. Interval and ratio - variance and standard deviation

B. Extreme scores -

C. Sample size -

D. Stability across samples -

E. Open-ended distributions -