Variability

Variability: provides an objective, quantitative measure of the degree to which scores in a distribution are spread out or clustered together.

Uses:
> Describes the distribution: clustered/spread; how much distance is typical between scores, or between an individual score and the mean.
> Measures how well an individual score (or group of scores) represents the entire distribution (important for inferential statistics).
> Helps determine cutoffs for abnormal behavior, extreme scores (e.g., genius on an IQ test) (SD in particular does this).
Range
Definition:
Difference between the largest
score and the smallest score (taking
into account real limits).
(Semi-)Interquartile Range
Distance between the 1st quartile (25th
percentile) and the 3rd quartile (75th
percentile). Or half that amount.
In other
words:
Formulas:
(sample formulas for
variance and SD)
Xmax – Xmin
Highest X – Lowest X
(know concept of real limits here,
but okay to disregard in calculating)
IQR:
Q3 – Q1
SIQR: Q3 – Q1
2
Variance
Standard Deviation
Mean squared deviation
(distance from the mean).
Square root of the mean
squared deviation.
Mean of the squared deviation
scores. Square of the standard
deviation.
I.e., square root of the variance.
Conceptually: Typical distance
of scores from the mean.
Def:
Comp:
s² = SS = (X - M)²
n–1
n–1
.
s = / SS
√ n–1
Deviation score: X – M
SS: X²– (X)² (computational)
Sum of Squares (SS): (X - M)²
n
Other useful
formulas:
Quick and easy way to assess
variance.
Relatively unaffected by extreme scores
and sample size. Stable under sampling
(p. 129). Only measure of variability for
open-ended distributions (p. 129).
SD is most common measure of variability, and goes hand-in-hand
with the mean. Var and SD take each score into account, so
representative of every score in the sample. ~ Consider distance
between scores, so give information on how scattered or clustered
scores are (not just overall range). ~ Stable under sampling (p. 129).
Does not take entire population/sample
into account.
Drawbacks:
Most affected by extreme scores.
Only uses most extreme score on
either end; Does not take other
scores into account. Affected by
sample size and unstable under
sampling (p. 128-129).
Influenced by extreme scores. Deviation scores are squared, so if
one score has a large deviation from the mean  very large
(disproportionate) contribution to variance and SD.
Other notes:
For the sake of simplicity, you can
disregard real limits when computing
the range. I.e.:
Range = Xhighest - Xlowest
Generally goes together with the median
(see end of p. 129)
Samples tend to have less variability than the population they come
from, so sample formulas adjust for this bias by using (n-1) rather
than n (the population formulas use N).
Uses,
Advantages:
Psyc 281 / Conley