PSYC202 Chapter 4 – Variability
Variability: a quantitative measure of the degree to which scores in a distribution
are spread out or clustered together
Variability serves 2 purposes:
o Describes the distribution ▯whether scores clustered together or spread out
over a large distance. Difference between scores and difference between a
score and the mean.
o Measures how well an individual score (or group of scores) represents the
entire distribution. Also tells you how much error to expect if you are
using a sample to represent a population
3 measures of variability: range, interquartile range, standard deviation
Range and Interquartile Range
Range = difference between largest score (Xmax) and smallest (Xmin) in
distribution (*remember to include real limits)
Since range described in terms of distance, it is typically used with interval or
ratio scales (continuous variable) ** remember that the concept of real limits
applies to continuous variables
Problem with range ▯it’s limited by the 2 extremes (highest and lowest scores)
and my not reflect that actual distribution of scores
For Discrete variables, range is determined by counting the number of categories
from the one containing the smallest score to the one containing the largest score.
However, don’t forget to include the first category (ex. If counting categories
from 08, there’s actually 9 categories if you include 0).
Interquartile Range: the range covered by the middle 50% of the distribution
(ignore extreme scores).
o Finding the interquartile range ▯find first quartile or Q1 (boundary
separating lowest 25% from the rest of the distribution), then second
quartile or Q3 (boundary separating top 25% from the rest of the
distribution). The interquartile range = the distance between Q1 and Q3
o Easiest way to find it – draw a box histogram and count off lowest quarter
of boxes, and highest quarter of boxes, and you’re left with the middle.
(similar to finding a median for a continuous variable)
Semiquartile Range: half the interquartile range. Measures distance from the
middle of the quartile range to either boundary of the quartile range. This is
usually done when interquartile range is used to describe variability
o Calculated by dividing quartile range in half.
o Excludes extreme scores but also loses 50% of info
Standard Deviation and Variance for a Population
Most common & important measure of variability
Uses variability to consider the distance between each score and the mean ▯
averages the average distance of scores from the mean (tells us if scores scattered
or close together)
Deviation: distance from the mean Deviation Score = Xμ (score-mean)
Positives and negatives indicate whether a score is above (+) the mean or below
() the mean
To find standard distance from the mean, add up all deviations and divide by N.
Mean of deviations is always 0
Population Variance: equals the mean squared deviation (each Xμ value is
squared to get rid of positive/negatives). Variance is the average squared distance
from the mean, and gets rid of the problem of 0.
Since the squared distance from the mean isn’t what we want, we must square
root the variance to get the standard deviation.
Standard Deviation = √Variance
Only used with interval or ratio scales
Can guess standard deviation if looking at a histogram. Look at mean, and check
how close the closest point to the mean is vs. how far the farther point from the
mean is. Subtract these and your standard deviation should fall somewhere in that
interval (ex. If closest score is 1 point away from mean and farthest score is 5
points away from mean, SD should fall somewhere between 15, particularly in
the middle, around ~3.)
Formulas For Population Variance and Standard Deviation
Standard deviation and variance calculations differ slightly depending on whether
you’re using population or sample
Sum of Squared Deviations (SS): Sum of squared deviation scores
o Definitional formula: SS = ∑(X μ) 2
o Computational formula: SS = ∑X - (∑X)
(Avoids rounding error) N
Both produce the same answer, but definitional formula should only be used
if N is small and the Mean is a whole number.
Final Formulas and Notation
Population Variance: SS Also called σ 2
Population Standard Deviation: √(SS) Square root of populatio