Variability
What Do We Mean by Variability?
 Variability
provides a quantitative
measure of the degree to which scores in
a distribution are spread out or clustered
together.
What Does Variability Tell Us?
 It
describes the distribution
 It represents how well an individual score
is likely to represent the entire population

The more variability, the less alike scores are
from individual to individual
What Does the Range Tell Us?
 The
range tells us how spread the scores
are
 Problems with using the range

Ignores the middle scores
• Outliers can throw off the range

Gives us no idea of the scale
What Is the Interquartile Range?

The interquartile range ignores extreme scores
and measures the range covered by the middle
50% of the distribution
 How does SPSS find quartiles



How do we find the interquartile range?


(1/4)(n+1) – Gives us the number of the lower bound
element
(3/4)(n+1) – Gives us the number of the upper bound
element
Q3 – Q1
Semi-Interquartile range (divided in half)
What Are the Weaknesses of These
Methods?
 Don’t
take into account the spread of the
other scores
 Don’t take into account scaling issues
What Is a Standard Deviation?
 This
is the way we generally measure
variability
 This measure is used to determine what
the typical distance from the mean is for
any given score in our data
How Do We Calculate a Standard
Deviation?

First you must find the deviation of each score,
or the deviation


Next we attempt to find the mean of the
deviations



X-μ
What do we notice about the mean of the deviations?
Why is this happening?
How else might we find the mean of the
deviations?
How Else Might We Find the Mean
of the Deviations?
 Absolute

values?
Some use these, though they are harder to
use mathematically for inferential statistics
since the normal curve is based on S.D.
 Squaring
the numbers gives us a positive
value.

It also weights deviants differentially.
Taking the Mean of Squared
Deviations Gives Us Variance
 What


This is also a measure of variation
Helpful, but not quite as valuable in describing
the average deviation from the mean
 How


is variance?
do we solve this?
Take the square root
This yields the Standard Deviation
Samples Versus Populations
s vs. σ
 s2 vs. σ2
 As always M vs. μ
 n-1 versus N




This increases the size of the average deviant and
makes it a more accurate, unbiased estimator of the
population score
This is in essence a penalty for sampling
Another way to think about it is because of the
degrees of freedom
Degrees of Freedom
For a sample of n scores, the degrees of
freedom or df for the sample variance are
defined as df = n – 1. The degrees of
freedom determine the number of scores
in the sample that are independent and
free to vary.
Formulae? Formulas?
 Sum

of Squared Error
SS = Σ(X-μ)2
 Variance


σ2 = SS/N
s2 = SS/(n-1) = SS/df
 Standard


Deviation
σ = √(SS/n)
s = √(SS/n-1) = √(SS/df)