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MA 6.21
Measures of Variability (with exercises and answers)
Basic Statistics
Many courses require that students in undergraduate degrees have a basic
understanding of descriptive statistics. Descriptive statistics are statistics that
collect, summarize, classify and present data. This handout gives an overview of
one type of descriptive statistics, measures of variability.
Measures of Variability
Measures that allow you to: determine the degree of variation within a population
or sample; determine how representative a particular score is of a data set; and
determine the scope and validity of any generalizations you wish to make based
on your observations in research are measures of variability. The measures of
variability discussed in this handout are:
 Range
 Variance
 Standard Deviation
Range
The range is the difference between the highest and lowest scores in a
distribution. It is calculated by subtracting the lowest score from the
highest score.
EXAMPLE:
14 40 42 47 49 51 71 81
I)
Find nouns that are “Specific”
(81 – 14) = 67
The range for this distribution would be (81 – 14) = 67. However, as you
can see, 14 is an outlier and skews this distribution because if it were not
in the distribution, the range would be (81 – 40) = 41.
When is the range useful?
The range gives you a rough guide to the variability in a data set, as it tells
you how a particular score compares to the highest and lowest scores
within a data set. The range gives t you only a limited amount of
information, as data sets which are skewed towards a low score can have
the same range as data sets which are skewed towards a high score, or
those which cluster around some central score.
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Learning Centre
MA 6.21
Variability
Variance is the degree to which scores vary from their mean. The variance
takes into account every score in the data set. The variance is calculated
by getting the average of the squared deviations from the mean.
To calculate the variance for a set of quiz scores:
1. Find the mean (M).
2. Find the deviation of each raw score from the mean (D).
3. To do this, subtract the mean from each raw score. To check your
calculations sum the deviation scores. This sum should be equal to
zero. **Note that deviation scores that are below the mean will be
negative
4. Square the deviation scores (SS). We do this because by squaring
the scores, negative scores are made positive and extreme scores
are given relatively more weight.
5. Find the sum of the squared deviation scores.
6. Divide the sum by the number of scores. This yields the average of
the squared deviations from the mean, or the variance Range.
EXAMPLE:
Data Set: 5 6 7 11 12 12 13 14 18 19 21 21 22 34 35 35 46 50
1.
5+6+7+11+12+12+13+14+18+19+21+21+22+24+35+35+35+50 = 360
Mean = 360/N = 360/18 = 20
2. D = N-M
6 – 20 = (-14)
7 – 20 = (-13)
11 – 20 = (-9)
12 – 20 = (-8)
12 – 20 = (-8)
13 – 20 = (-7)
14 – 20 = (-6)
18 – 20 = (-2)
19 – 20 = (-1)
21 – 20 = 1
21 – 20 = 1
22 – 20 = 2
24 – 20 = 4
35 – 20 = 15
35 – 20 = 15
35 – 20 = 15
SS
(-15)2 =225
(-14)2 =196
(-13)2 =169
(-9)2 =81
(-8)2 =64
(-8)2 =64
(-7)2 =49
(-6)2 =36
(-2)2 =4
(-1)2 =1
12 =1
12 =1
22 =4
42 =16
152 =225
152 =225
152 =225
50 – 20 = 30
302 =900
Standard
5 –Deviation
20 = (-15)
Sum: [(-83) + 83] = 0
2486/18=138.11
Variance = 138.11
H. Williams, 2015
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Learning Centre
MA 6.21
EXERCISE: Using the steps above, find the variance for the following
data set.
a. 8 11 12 14 17 17 18 19 22 29 35 38
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Standard Deviation(SD)
The standard deviation is the square root of the variance. Unlike the variance,
the standard deviation is in the same units as the raw scores themselves,
therefore one cannot just use the variance. This is what makes the standard
deviation more meaningful. For example, it would make more sense to discuss
the variability of a set of IQ scores in IQ points than in squared IQ points because
they would not be congruent with the scores meaning.
Sum of Squares
The sum of squares is a measure of variance or deviation from the mean. It is
calculated by summing of the squares of each score’s difference from the mean.
Total sum of squares is another sum of squares; it considers not only the sum of
squares from the factors, but also from randomness or error because the
squaring of each score rids the equation of negative numbers. As you can see in
the above example showing how to obtain the variance, step 5 requires you to
find the sum of squares (SS).
Useful definitions
Measures of Variability: Measures that allow you to: determine the degree of
variation within a population or sample; determine how representative a particular
score is of a data set; and determine the scope and validity of any
generalizations you wish to make based on your observations in research are
measures of variability.
Range: The range is the difference between the highest and lowest scores in a
distribution
Variability: degree to which the scores vary from their mean.
Standard Deviation: Square root of the variance.
Sum of Squares: The sum of squares is a measure of variance or deviation from
the mean. It is calculated by summing of the squares of each score’s difference
from the mean. It is the sum of squared deviations.
Other Exercises and Resources
For more information about or practice with measures of variability, see:
H. Williams, 2015
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Learning Centre
MA 6.21
http://study.com/academy/lesson/skewed-distribution-examples-definitionquiz.html
http://www.learningcommons.uoguelph.ca/guides/university_learning/handouts/hi
ghlighting.pdf
http://www.stats.ox.ac.uk/~marchini/teaching/L1/L1.slides.pdf
http://davidmlane.com/hyperstat/intro.html
http://study.com/academy/lesson/outlier-in-statistics-definition-lesson-quiz.html
http://www.statisticshowto.com/skewed-distribution/
http://support.minitab.com/en-us/minitab/17/topic-library/modelingstatistics/anova/anova-statistics/understanding-sums-of-squares/
MEASURES OF VARIABILITY – ANSWER KEY
1. 8 11 12 14 17 17 18 19 22 29 35 38
M = (240/N)
240/12 = 20
Variance
(8 – 20) = -12
(11 – 20) = -9
(12 – 20) = -8
(14 – 20) = -6
(17 – 20) = -3
(17 – 20) = -3
(18 – 20) = -2
(19 – 20) = -1
(22 – 20) = 2
(29 – 20) = 9
(35 – 20) = 15
(38 – 20) = 18
(-44) + 44 = 0
Variance = 81.83
SS
(-12)2 = 144
(-9)2 = 81
(-8)2 = 64
(-6)2 = 36
(-3)2 = 9
(-3)2 = 9
(-2)2 = 4
(-1)2 = 1
22 = 4
92 = 81
152 = 225
182 = 324
982/12 =81.83
H. Williams, 2015
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