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Chapter 4

DESCRIPTIVE STATISTICS:
MEASURES OF CENTRAL TENDENCY AND VARIABILITY
Understanding Statistics for International Social
Work and Other Behavioral Sciences
Serge Lee, Maria C Silveira Nunes Dinis, Lois Lowe, Kelly
Anders (2015. Oxford University Press
MEASURES OF CENTRAL TENDENCY
2
It is used to
determine how
values,
particularly
interval and ratio
data, in a given
distribution of
scores are
clustered and
identify the key
locations in the
set. For example,
X = 5, 1, 4, 5, 3,
2, and 5.
Central locations can be found by using the
mean, median, and mode
The mean X . The mean can be computed
by:
• X=
X
n
• X = Sample mean
• X = Sum of the variable (x)
Sample mean is the arithmetic
average. It is also called simple
statistic.
• X=
5+1+4+5+4+2
7
• X = 3.57
MEASURES OF CENTRAL TENDENCY CONTINUED
3
In addition to the sample mean, trimmed mean can also be computed. Typically,
15-20% is trimmed from the data set. For example, X = 1, 4, 5, 3, 5, 8, 4, 5 and 9.
Assume 20% is trimmed. Compute the trimmed mean by:
Multiply the selection proportion (in decimal) by the sample size
(.20 x 9 = 1.80)
Arrange the scores into an array, 1, 3, 4, 4, 5, 5, 5, 8, 9
Trim first two (1, 3) and last two (8, 9) from the set
Take the average of the remaining scores, = 4.6. Without trimmed
mean, the average score would have been 4.89
Trimming the mean reduced the effects of outlier bias in this
hypothetical sample by .29 (4.89- 4.6 = .29)
MEASURES OF CENTRAL TENDENCY CONTINUED
4
The Median
(Mdn). The
median is the
second quartile
or 50th
percentile.
Statistics
Rules:
• Arrange the scores into an array.
• If numbers of scores (n) are odd, the
median is the middle score
• If the numbers of scores (n) are even,
then average the two middle scores
The Mode. It is
the value or value
category that
appears most
frequently within
a data set. A data
set could have no
mode or multiple
modes. Common
terms for the
mode:
•
•
•
•
No mode (Zero mode)
Single mode
Bimodal (two modes)
Multiple modes
MEASURES OF VARIATION OR DISPERSION
5
Data spread is
called
variability or
dispersion.
Common
measures of
variation are the
range, quartile,
mean deviation,
variance, and
standard
deviation.
• Range = Maximum – minimum
• Quartile is used to locate the 25th, 50th, and
75th percentile of the distribution set
• Mean deviation (MD) is a measure of
dispersion, which is equal to the mean of the
absolute values of the deviation scores.
MD =
• Variance, also called sum of the squared
deviation from the mean, provides an
understanding of the spread of scores about
𝑋−𝑋 2
the mean.
Variance = 𝑛−1
• Standard deviation (SD) is the most
informative measure of variability
PROPERTIES OF THE MEAN DEVIATION AND VARIANCE
6
X= 5, 1, 4, 5, 3, 2, AND 5
X = Self-esteem
Mean deviation
Variance
5
1
4
5
3
2
5
ΣX = 25
= 3.57
5-3.57= +1.43
1-3.57 = -2.57
4-3.57 = +0.43
5-3.57 = +1.43
3-3.57 = -0.57
2-3.57 = -1.57
5-3.57 = +1.43
= 2.04
= 6.60
= .18
= 2.04
= .32
= 2.46
= 2.04
= 15.68/6
= 2.61
Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press
COEFFICIENT OF VARIATION (CV)
7
The CV helps
researchers to
understand
the relative
variability of
a variable
when only the
sample mean
( ) and its
corresponding
standard
deviation
(SD) are
present
• CV =
SD
X
x 100%
Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press