Q.3: Explain the measures of central tendency and measures of
dispersion. How these two concepts are related?
How these two concepts are related? Suggest one measure of dispersion
for each measure of central tendency with logical reasons.
-
Course: Introduction to Educational Statistics (8614)
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Level: B.Ed (1.5 Years)
Answer:
Collecting data can be easy and fun.
But sometimes it can be hard to tell other people about what you have found.
That’s why we use statistics. Two kinds of statistics are frequently used to describe
data. They are measures of central tendency and dispersion. These are often called
descriptive statistics because they can help you describe your data.
Mean, median, and mode
These are all measures of central
tendency. They help summarize a bunch of scores with a single number. Suppose
you want to describe a bunch of data that you collected to a friend for a
particular variable like the height of students in your class. One way would be to
read each height you recorded to your friend. Your friend would listen to all
of the heights and then come to a conclusion about how tall students generally
are in your class, but this would take too much time. Especially if you are in
a class of 200 or 300 students! Another way to communicate with your friend
would be to use measures of central tendency like the mean, median, and mode.
They help you summarize bunches of numbers with one or just a few numbers. They
make telling people about your data easy.
Range, variance, and standard deviation
These are all measures of dispersion.
These help you to know the spread of scores within a bunch of scores. Are the
scores really close together or are they really far apart? For example, if you
were describing the heights of students in your class to a friend, they might
want to know how much the heights vary. Are all the men about 5 feet 11 inches
within a few centimeters or so? Or is there a lot of variation where some men
are 5 feet and others are 6 foot 5 inches? Measures of dispersion like the
range, variance, and standard deviation tell you about the spread of scores in a
data set. Like central tendency, they help you summarize a bunch of numbers
with one or just a few numbers.
How these two concepts are related? Suggest one measure of dispersion for each measure of central tendency with logical reasons.
In many ways, measures of central
tendency are less useful in statistical analysis than measures of dispersion of
values around the central tendency. The
dispersion of values within variables is especially important in social and
political research because:
•
Dispersion or "variation" in observations is
what we seek to explain.
• Researchers want to know WHY some cases lie above
average and others below average for a given variable:
•
TURNOUT in voting: why do some states show higher
rates than others?
•
CRIMES in cities: why are there differences in crime
rates?
•
CIVIL STRIFE among countries: what accounts for
differing amounts?
•
Much of statistical explanation aims at explaining
DIFFERENCES in observations -- also known
as
§ VARIATION, or
the more technical term, VARIANCE.
The SPSS Guide contains only the
briefest discussion of measures of dispersion on pages 23-24.
•
It mentions the minimum and maximum values as the
extremes, and
•
it refers to the standard deviation as the "most
commonly used" measure of dispersion.
This is not enough, and we'll discuss
several statistics used to measure variation, which differ in their importance.
•
We'll proceed from the less important to the more
important, and
•
we'll relate the various measures to measurement
theory.
Easy-to-Understand Measures of dispersion for NOMINAL and ORDINAL variables
In the great scheme of things,
measuring dispersion among nominal or ordinal variables is not very
important.
• There is inconsistency in methods to measure
dispersion for these variables, especially for nominal variables.
• Measures suitable for nominal variables (discrete,
non-order able) would also apply to discrete order able or continuous variables,
order able, but better alternatives are available.
• Whenever possible, researchers try to re-conceptualize
nominal and ordinal variables and operationalize (measure) them with an
interval scale.
Variation Ratio, VR
•
VR = l - (proportion of cases in the mode)
•
The value of VR reflects the following logic:
§ The larger the
proportion of cases in the mode of a nominal variable, the less the variation
among the cases of that variable.
§ By subtracting
the proportion of cases from 1, VR reports the dispersion among cases.
o
This measure has an absolute lower value of 0,
indicating NO variation in the data (occurs when all the cases fall into one
category; hence no variation).
o
Its maximum value approaches one as the proportion of
cases inside the mode decreases.
•
Unfortunately, this measure is a "terminal
statistic":
§ VR does not
figure prominently in any subsequent procedures for statistical analysis.
§ Nevertheless,
you should learn it, for it illustrates
o
That even nominal variables can demonstrate variation
o
That the variation can be measured, even if somewhat
awkwardly.
Easy-to-understand measures of variation for CONTINUOUS variables.
RANGE: the distance between the highest and
lowest values in a distribution
•
Uses information on only the extreme values.
•
Highly unstable as a result.
SEMI-INTERQUARTILE
RANGE: distance
between scores at the 25th and the 75th percentiles.
•
Also uses information on only two values, but not ones
at the extremes.
•
More stable than the range but of limited utility.
AVERAGE DEVIATION:
where |X,
-X’| = absolute
value of the differences
Absolute values of the differences are summed, rather
than the differences themselves, for summing the positive and negative values
of differences in a distribution calculating from its mean always yields 0.
•
The average deviation is simple to calculate and
easily understood.
• But it is of limited value in statistics, for it does
not figure in subsequent statistical analysis.
• For mathematical reasons, statistical procedures are
based on measures of dispersion that use SQUARED deviations from the mean
rather than absolute deviations.
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