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CHAPTER 4
MEASURES OF CENTRAL TENDENCY
AND VARIABILITY
CHAPTER OUTLINE
I.
Measures of Central Tendency. There are three commonly used measures of
central tendency.
A. Arithmetic mean. Symbolized by " X " (read X bar), or "µ" (read mu). This is
simply the average of a set of raw scores. When referring to an entire
population, the symbol used is µ. " X " is used for sample data.
1. Equation. Equation for calculating the mean of sample raw scores
X   Xi  / N
This equation simply tells us to add all of the scores and divide by the total
number of scores.
2. Properties:
a. The mean is sensitive to the exact value of all the scores in the
distribution.
b.  ( X  X )  0 . The sum of the deviations about the mean equals zero.
c. The mean is very sensitive to extreme scores when the scores are not
balanced at both ends of the distribution.
d.  ( X  X ) 2 is a minimum. The sum of the squared deviations of all the
scores about their mean is a minimum.
e. Under most circumstances, of the measures used for central tendency,
the mean is least subject to sampling variation.
3. Overall mean equation.
X overall 
sum of all scores n1X 1  n2 X 2      nk X k

N
n1  n2      nk
Signifies that the overall mean is equal to the sum of the mean of each
group times the number of scores in the group, divided by the sum of the
number of scores in each group.
B. The median (Mdn) This is the value below which 50% of the scores fall. It is
the same thing as P50.
1. Equation for grouped scores:
Mdn = XL + (i/fi)(cum fP  cum fL)
2. Calculation with raw scores. The median is the centermost score, if the
number of scores is odd. If the number is even, the median is taken as the
average of the two centermost scores. (Note: scores are rank ordered
first.)
3. Properties of the median:
a. The median is less sensitive than the mean to unbalanced extreme
scores.
b. Under usual circumstances, the median is more subject to sampling
variability than the mean, but less subject to sampling variability than the
mode.
C. The mode. This is simply the most frequent score in the distribution. It is
calculated by inspection.
D. Symmetry. In a unimodal (one mode) symmetrical distribution, the mean,
median and mode are all equal. In a positively skewed distribution, the mean
will be larger than the median. In a negatively skewed distribution, the mean
will be lower than the median.
II.
Measures of Variability. These measures quantify the extent of dispersion in a
distribution.
A. Range. The range is the difference between the highest and lowest score in
the distribution. The range measures only the spread of the two extreme
scores.
B. Deviation scores. These tell how far away the raw score is from the mean of its
distribution. It is symbolized by ( X  X ) or ( X   ).
(X  X )
for sample data
(X  μ)
for population data
C. The standard deviation. This is a measure of an average deviation of raw
scores about the mean. It is symbolized by "s" for the sample standard
deviation, and by "" (small sigma) for the population standard deviation.
1. Conceptual formula:
s
 (X  X )

N 1
2
SS
N 1
2. Computational formula:
s
because
X
2
 X 

2
N
N 1
 ( X  X )2   X 2   X 
2
/N
3. For the population standard deviation, the formula is just a little different.
The denominator of the formula is N, instead of N  1.
a. Conceptual formula:
σ
 (X  μ)

N
2
SSpop
N
b. Computational formula:
σ
X
 X 

2
2
N
N
4. Properties of the standard deviation.
a. The standard deviation gives a measure of dispersion relative to the
mean.
b. The standard deviation is sensitive to each score in the distribution.
c. It is stable with regard to sampling fluctuations.
D. The variance. This is simply the square of the standard deviation. The
variance is symbolized as "s2" for the sample, and "2" for the population. This
is read as "the variance" or "s squared."
1. Conceptual formula:
s2 
σ2 
2
SS
 (X  X )

N 1
N 1
2
SSpop
 (X  μ)

N
N
for a sample
for a population
2. Computational formula:
s2 
σ 
2
X
X
2
2
 X 

N
N 1
 X 

N
N
2
for a sample
2
for a population