Download Psychology 210 Psychometric Methods

Chapter 3
Central Tendency
and Variability
Characterizing Distributions Central Tendency
Most people know these as
“averages”
 scores near the center of the
distribution - the score towards which
the distribution “tends”

– Mean
– Median
– Mode
Arithmetic Mean (Mean)

Mean (μ; M or X ) - the numerical
average; the sum of the scores (Σ)
divided by the number of scores (N or n)
X
å
m=
N
X
å
M=
n
Σ - The Summation Operator

Sum the scores
– In general, “Add up all the scores”
– Sum all the values specified
n
åX
i=1
i
= X1 + X2 +
+ Xn
Central
Tendency (CT)
Median (Md) - the score which divides
the distribution in half; the score at
which 50% of the scores are below it;
the 50%tile
 Order the scores, and count “from the
outside, in”

Central
Tendency (CT)



Mode (Mo) - the
most frequent
score
To find the mode
from a freq. dist.,
look for the highest
frequency
For this
distribution, the
mode is the interval
24 - 26, or the
midpoint 25
Mode
Interval
15 - 17
18 - 20
21 - 23
24 – 26
27 – 29
30 – 32
33 – 35
36 – 38
39 – 41
42 – 44
45 – 47
48 – 50
Total
f
1
2
3
6
2
2
2
0
1
0
1
1
21
cf
1
3
6
12
14
16
18
18
19
19
20
21
Characterizing Distributions Variability
Variability is a measure of the extent to
which measurements in a distribution
differ from one another
 Three measures:

– Range
– Variance
– Standard Deviation
Variability

Range - the highest score minus the
lowest score
Variability

Variance (σ2) - the average of the
squared deviations of each score from
their mean (SS(X)), also known as the
Mean Square (MS)
n
1
2
s = å (Xi - m X )
N i=1
1
2
s x = SS(X)
N
2
x
Variance

the average of the squared deviations
of each score from their mean
1. Deviation of a score from the mean
 2. Squared
 3. All added up
Average
 4. Divide by N

Computing Variance
Score (X)
μ
X-μ
(X – μ)2
2
4
-2
4
3
4
-1
1
4
4
0
0
4
4
0
0
5
4
1
1
6
4
2
2
Σ(X – μ)=0*
Σ(X – μ)2=
SS(X) = 10
*When computing the sum of the deviations
of a set of scores from their mean, you will
always get 0. This is one of the special
mathematical properties of the mean.
s
2
(X - m )
å
=
N
10
s =
6
2
s = 1.67
2
2
Variability

Sample Variance (s2) – (sort of) the
average of the squared deviations of
each score from their mean (SS(X))
n
1
2
s =
(Xi - M X )
å
n - 1 i=1
1
2
sx =
SS(X)
n -1
2
x
Unbiased Estimates

M for μ (M is an unbiased estimate of μ)
X
å
m=
N

X
å
M=
n
The average M (of all the Ms) from all
random samples of size n is guaranteed to
equal μ
Samples systematically
underestimate the variability
in the population
If we were to use the formula for
population variance to compute sample
variance
 We would systematically
underestimate population variance by
a factor of 1 in the denominator

Therefore:

Sample Variance (s2) – (sort of) the
average of the squared deviations of
each score from their mean; the
unbiased estimate of σ2
n
1
2
s =
(Xi - M X )
å
n - 1 i=1
1
2
sx =
SS(X)
n -1
2
x
Squared the Units?

Score (X)
μ
X-μ
(X – μ)2
2
4
-2
4
3
4
-1
1
4
4
0
0
4
4
0
0
5
4
1
1
6
4
2
2
Σ(X – μ)=0*
Σ(X – μ)2=
SS(X) = 10



Let’s say that these
scores represent
cigarettes smoked per
day
In the first column, for
example, “2” represents
the quantity “2
cigarettes”
The third column
represents 2 fewer
cigarettes than the mean
The fourth column
represents “-2cigarettessqured” or 4 cigarettessquared
Variability

Standard Deviation (σ) - the square
root of the variance (σ2)
sX =
1
SS(X)
N
sX = s
2
X
Variability in samples

Sample Standard Deviation (s) - the
square root of the variance (s2)
1
sX =
SS(X)
n -1
sX = s
2
X