Chapter 6 – Variance and Standard Deviation
The variance of a distribution (that belongs to a population) is nothing but the average of
the squared deviations.
δ2 = ∑ (X- μ) 2
[X: score, μ: population mean, N: size of the population]
N
The expression above is for calculating the variance for a population. We usually do not
know every score or the distribution of scores for an entire population. Therefore we take
a representative sample and calculate the variance for that sample and use it as an
estimate for the population variance.
The expression for calculating the variance for a given sample is quite similar to the one
above.
s2 = ∑ (X- X) 2
[X: score, X: sample mean, N: sample size (number of scores)]
N–1
This expression is simply telling you to do the followings:
1. Take first score and find how much it differs from the mean.
2. Take the square of this difference.
3. Do this for each one of the rest of the scores and add the differences up.
4. Divide this sum by (sample size – 1)
Example 1: Data Set 1: 3, 5, 7, 10, 10
X
1
2
3
4
5
Totals
3
5
7
10
10
XX
( X  X )2
Example 2
Dive
Mark Myrna
1
28
27
2
22
27
3
21
28
4
26
6
5
18
27
Find the mean, median, mode, range?
What can be said about this data?
Which diver was more consistent?
Dive
Mark's
Score
X
1
2
3
4
5
28
22
21
26
18
xx
( x  x )2
xx
( x  x )2
Totals
Dive
1
2
3
4
5
Totals
Myrna's
Score
X
THE STANDARD DEVIATION
The square root of the variance is the standard deviation.
S2 
 ( X  X)
f
2
f
S
(X  X )
f
2
f
1) Find the variance and standard deviation for the following data.
Myrna’s Score
X
F
28
27
6
1
3
1
XX
( x  x )2
( x  x )2 x f
2) Find the variance and the standard deviation for the following distribution of scores.
Score (X)
Frequency (f)
7
4
6
4
5
6
4
3
3
2
(xx)
(xx)2
( x  x )2 x f