Statistics

1. Chapter Two
The mean, the variance,
and the standard deviation
1. The Mean and other Measures of Central Tendency
The arithmetic mean is the average of a set of numbers. It can be
symbolised as M and its formula is:
N
∑X ∑X
Arithmetic Mean = M = i=1
= (2 :1)
N N
It is the most important of three measures of central tendency.
The other two are the median and the mode. The mode is defined
simply as the value which occurs most frequently. The median is
the value below which exactly fifty percent of cases fall, and above
which are exactly fifty percent of cases. It is the point which splits
the set of scores into two equal parts.
Problems
Find the mean, mode and median of the following sets of scores.
A. 1, 2, 2, 3, 4, 5, 6.
B. 10, 11, 12, 14, 15, 15.
C. 7, 7, 8, 8, 10, 10.

2. Philip Ley. Quantitative Aspects of Psychological Assessment 11
Answers
A. Mean 3.3; mode 2; median 3.
B. Mean 12.8; mode 15; median 13
(by convention half way between the two mid-most scores
when there is an even number of scores).
C. Mean 8.3; mode 8; median 8.
From this point onwards subscripts will be used only when
necessary.
Returning now to the formula for the mean it can be shown that:
∑ X = NM (2:2)
Proof
⎛∑ X ⎞
(1) NM = N ⎜ ⎟.
⎝ N ⎠
N∑ X
(2) = .
N
(3) The N’s cancel one another out leaving ∑X.
This general principle that the sum of a set of values equals N
times the mean of that set will be useful at several later points.
It is also true that:
∑ (X − M ) = 0 (2:3)
© 1972, 2007 Philip Ley Text re-typed for computer by Irene Page

3. Philip Ley. Quantitative Aspects of Psychological Assessment 12
Proof
(1) ∑ (X − M ) = ∑ X − ∑ M by Summation Rule 1.
(2) Further by Summation Rule 3, as the mean is a constant:
∑ M = NM , so (1) becomes ∑ X − NM .
(3) But we have just shown in equation (2:2) that NM = ∑ X
so we obtain ∑ X − ∑ X = 0.
The value X i − M x , the deviation of a score from the mean, is
called a deviation score and is sometimes symbolised as x i .
Similarly Yi − M y is symbolised as yi . As demonstrated in (2:3)
∑ x = 0; ∑y=0
2. The Variance
The variance is a measure of dispersion. It tells us something
about the scatter of scores around the mean. It is defined as the
mean squared deviation from the mean, and symbolised by a
small sigma squared - σ . Its formula is:
2
∑ (X − M )
2
Variance = σ x = (2:4)
2
N
or using x for X – M
∑x 2
σx =
2
(2:5)
N
It follows from this formula that:
© 1972, 2007 Philip Ley Text re-typed for computer by Irene Page

4. Philip Ley. Quantitative Aspects of Psychological Assessment 13
∑ x = ∑ (X − M ) = Nσ
2
(2:6)
2 2
x
(2:6) is obtained from (2:5) by multiplying both sides of the
equation by N.
Another variant of the formula for the variance is:
∑x 2
σx = 2
−M2 (2:7)
N
Proof
σ x2 = ∑ (X − M ) / N .
2
(1)
(2) = ∑ (X 2 + M 2 − 2XM )/ N.
(3) Using Summation Rules 1 and 3 this becomes:
(∑ X 2
+ NM 2 − 2∑ XM )/ N
(4) but we have shown in (2:2) that ∑ X = NM so we can write:
σ x2 = (∑ X 2 + NM 2 − 2NMM )/ N
(5) but 2NMM = 2NM so (4) becomes:
2
σ x2 = (∑ X 2 − NM 2 )/ N
(6) Dividing by N this gives:
∑X 2
σx =
2
−M2
N
© 1972, 2007 Philip Ley Text re-typed for computer by Irene Page

5. Philip Ley. Quantitative Aspects of Psychological Assessment 14
The numerator (top part) of equation (2:4) for the variance is the
sum of squared deviations from the mean. This sum is usually
called the sum of squares.
Sum of squares = SS = ∑ (X − M ) (2:8)
2
An alternative formula for this value is:
SS = ∑ X 2
−
(∑ X ) 2
(2:9)
N
Proof
(1) In the proof of (2:7) at (5) it has been shown that
∑ (X − M ) = ∑ X − NM
2 2 2
⎛ ∑ X ⎞⎛ ∑ X ⎞
(2) But NM = N ⎜ ⎟⎜ ⎟
2
⎝ N ⎠⎝ N ⎠
N∑ X∑ X
(3) Multiplying this becomes:
N2
(4) Dividing numerator and denominator by N gives:
(∑ X ) 2
N
(5) Substituting this in (1) we obtain:
∑ (X − M ) = ∑ X
2 2
−
(∑ X ) 2
N
© 1972, 2007 Philip Ley Text re-typed for computer by Irene Page

6. Philip Ley. Quantitative Aspects of Psychological Assessment 15
1. The Standard Deviation
The standard deviation is the square root of the variance and is
symbolised by a small Greek sigma - σ . Its formula is the square
root of any of the formulae for the variance, e.g.
∑x 2
σx = (2:10)
N
The mean, the variance and the standard deviation are important
in psychometrics because of their relationships to the normal
curve. These relationships will be discussed in the next chapter.
Problems
Given the following set of scores:
1, 2, 3, 4, 5, 6, 7.
Find:
A. The mean.
B. The variance.
C. The standard deviation.
D. ∑ (X − M ).
Answers
A. 4; B. 4; C. 2; D. 0.
© 1972, 2007 Philip Ley Text re-typed for computer by Irene Page