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Table 1. Number of Minutes 20 Clients Waited to See a Consultant
Consultant X Consultant Y
05 15 11 12
12 03 10 13
04 19 11 10
37 11 09 13
06 34 09 11
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Something to think about…
•What can you say to the
time allotted by the clients
to wait for the Consultant
X? How about for the
Consultant Y?
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Table 1. Number of Minutes 20 Clients Waited to See a Consultant
Consultant X Consultant Y
05 15 11 12
12 03 10 13
04 19 11 10
37 11 09 13
06 34 09 11
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In Consultant X:
•Sees some clients
almost immediately
•Others wait over 30
minutes
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In Consultant Y:
•Clients wait about 10
minutes
•9 minutes least wait
and 13 minutes most
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Something to think about…
•What can you say to the
time the Consultant X let
the clients wait? How
about for the Consultant
Y?
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In Consultant X:
•It is highly
inconsistent
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In Consultant Y:
•It is highly
consistent
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CHAPTER 6MEASURES OF VARIABILITY
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Measures of Variability
•It is a single number that
describes how the data
are scattered or how
much they are bunched.
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Going Back:
•In Consultant X, we can say
that the data are scattered.
While, in Consultant Y the
data are bunched.
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Furthermore,
•Measures of
Dispersion
•Measures of Spread
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Note:
•Bunched data/closely
grouped data will have
relatively small values of
Measures of Variability
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Note:
•Scattered data/more widely
distributed data will have
relatively small values of
Measures of Variability
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Measures of Variability
•It is an indicator of
consistency among
a set of data
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Measures of Variability
•It indicates how close
data are clustered
about Measures of
Central Tendency
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Measures of Variability
COMMONLY USED TYPES
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Measures of Variability•Index of Qualitative Variation (IQV)
•The Range
•The Interquartile Range (IQR)
•The Standard Deviation (𝝈)
Additional:
•The Variance (𝝈𝟐)
•The Mean Deviation (MD)
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Index of Qualitative
Variation (IQV)
MEASURES OF VARIABILITY
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Index of Qualitative Variation (IQV)
•A measure of variability for nominal
variables.
•It is based on the ratio of the total
number of differences in the
distribution to the maximum number
of possible differences within the
same distribution.
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Index of Qualitative Variation (IQV)
•Since the mode is the preferred
measure of central tendency for a
nominal variable, the measure of
dispersion for a nominal variable
would indicate the degree to which
cases fall in the non-modal
categories.
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Formula for IQV:
•𝑰𝑸𝑽 = 𝒕𝒐𝒕𝒂𝒍 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔
𝒎𝒂𝒙𝒊𝒎𝒖𝒎 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔
•𝑰𝑸𝑽 = 𝒇𝒊𝒇𝒋
𝑲(𝑲−𝟏)
𝟐
𝑵
𝑲
𝟐
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In the formula:
𝑲 = 𝒕𝒉𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒄𝒂𝒕𝒆𝒈𝒐𝒓𝒊𝒆𝒔
𝒊𝒏 𝒕𝒉𝒆 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆
𝑵 = 𝒕𝒉𝒆 𝒕𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓
𝒐𝒇 𝒄𝒂𝒔𝒆𝒔
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Understanding the IQV:
•The IQV is a single
number that expresses
the diversity of a
distribution.
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Understanding the IQV:
•The IQV ranges
from 0 to 1.
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Understanding the IQV:
•An IQV of 0 would
indicate that the
distribution has NO
diversity at all.
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Understanding the IQV:
•An IQV of 1 would
indicate that the
distribution is maximally
diverse.
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Further Our Knowledge:
•Let’s have IQV in Real-Life.
The next example will show
the ethnical and racial
diversity in the US.
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Something to think about…
•Based on the IQV of
Diversity in the US,
what can we therefore
conclude?
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The Range
MEASURES OF VARIABILITY
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The Range
•It is the least
complicated measure of
describing the dispersion
of a set of data.
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The Range
•It is the distance given by
the highest observed value
minus the lowest observed
value in the distribution.
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The Range
•It indicates how spread out
the data are
•It is dependent on two
extreme values
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The Range
•Somewhat dependent on
the number of values in the
data set (i.e., more: the
larger the range)
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Formula for Range:
•𝑹𝒂𝒏𝒈𝒆 = 𝑯𝑶𝑽 − 𝑳𝑶𝑽where:
𝑯𝑶𝑽 = 𝒉𝒊𝒈𝒉𝒆𝒔𝒕 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒗𝒂𝒍𝒖𝒆
𝑳𝑶𝑽 = 𝒍𝒐𝒘𝒆𝒔𝒕 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒗𝒂𝒍𝒖𝒆
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Table 1. Number of Minutes 20 Clients Waited to See a Consultant
Consultant X Consultant Y
05 15 11 12
12 03 10 13
04 19 11 10
37 11 09 13
06 34 09 11
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On the same example:
•The range of Consultant X is 34
minutes
•The range of Consultant Y is 4
minutes
“What can you conclude?”
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Understanding the Range
•The range is the preferred
measure of variability for ordinal
level variables, and for interval
level variables that have a badly
skewed distribution.
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Understanding the Range
•The range can be computed
for interval level variables, but
is not an appropriate statistic
for nominal or dichotomous
variables.
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Understanding the Range
•The range is usually
described as the total
spread in the
distribution.
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Understanding the Range
•The range is based only on two
scores in the distribution, the
highest and the lowest, and it tells
us nothing about the distribution
of the majority of scores in
between.
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Understanding the Range
•The range is most useful when we
are comparing groups and can
describe one group as having a
larger or smaller range, or
spread, than the other groups.
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The
Interquartile Range (IQR)
MEASURES OF VARIABILITY
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Something to think about...
•Since many variables contain
one or more extremely large
or extremely small scores, the
range may be misleading.
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Solution:
•That problem is
avoided with IQR.
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Interquartile Range (IQR)
•It is the difference between the
3rd quartile and the 1st quartile.
The 3rd quartile is the value below
which 75% of the cases fall. The
1st quartile is the value below
which 25% of the cases fall.
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Interquartile Range (IQR)
•While less subject to the influence
of extreme cases than the range,
the interquartile range still uses
information for only two cases or
values in the distribution.
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Understanding the IQR
•It is the modified
version of the
range
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Understanding the IQR
•It is the positional
measure of the
variability
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Understanding the IQR
•It is the range of
the middle 50% of
scores or ranks.
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Understanding the IQR
•It is not sensitive
to extreme values
in a data set.
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Understanding the IQR
•It is not
sensitive to the
sample size.
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Formula:
•𝑰𝑸𝑹 = 𝑸𝟐−𝑸𝟏
𝟐
•𝑰𝑸𝑹 = 𝑷𝟕𝟓−𝑷𝟐𝟓
𝟐
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The
Standard
Deviation (𝛔)MEASURES OF VARIABILITY
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The Standard Deviation•The standard deviation
measures the deviations
between the mean of the
distribution and each of
the individual scores.
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The Standard Deviation•It is the preferred measure of
variability for interval level
variables, unless the distribution
is badly skewed. For badly skewed
distribution, the range is a
preferred measure of variability.
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The Standard Deviation
•It is most frequently
used measure of
dispersion.
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The Standard Deviation
•It is the average of the
distances of the observed
values from the mean
value for a set of data.
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The Standard Deviation
•Here, the basic rule:
more spread will
yield a larger SD.
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Formula:
•𝝈 = (𝑿−𝒙)
𝒏−𝟏
•𝝈 =𝒏 𝑿𝟐 − 𝑿 𝟐
𝒏(𝒏−𝟏)
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Interpreting the Standard Deviation
•The standard deviation does not
have any inherent or intuitive
meaning; it is a statistical
measure of the variability of cases
around the mean for an interval
level variable.
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Interpreting the Standard Deviation
•Standard deviation is commonly
presented in terms of the
proportion of cases that fall
between the mean plus/minus 1,
2, or 3 standard deviation
measures.
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The Mean Deviation
(MD)MEASURES OF VARIABILITY
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Mean Deviation
•It is the average
distance between the
mean and the scores
in the distribution.
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Mean Deviation
•This technic provides
a reasonably stable
estimate variation.
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Mean Deviation
•It is also called
Average Deviation.
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Formula for MD:
•𝑴𝑫 = 𝑿−𝒙
𝒏
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The Variance
(𝝈𝟐)MEASURES OF VARIABILITY
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The Variance•The variance is another
measure of variability
that is equal to the
square the standard
deviation.
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The Variance
•The variance is the
average of the squared
deviations from the
mean.
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The Variance
•It is expected value of
the square of the
deviation from the
mean.
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The Variance•In describing distributions, the
standard deviation is the more
commonly cited statistics. Variance is
used primarily in inferential statistics
such as the analysis of variance and
correlation, which you will study later
in Advanced Statistics.
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Formula:
•𝝈𝟐 = (𝑿−𝒙)
𝒏−𝟏
•𝝈𝟐 =𝒏 𝑿𝟐 − 𝑿 𝟐
𝒏(𝒏−𝟏)