Measures of Central Tendency and Dispersion Their Relation. Suggest one MO dispersion for each MO central tendency with Logical Reasons

 

Explain the measures of central tendency and measures of dispersion. How these two concepts are related? Suggest one measure of dispersion for each measure of central tendency with logical reasons

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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 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:

1. Dispersion or "variation" in observations is what we seek to explain.

2. 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?

3. 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 norminal or oridinal 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-orderable) would also apply to discrete orderable or continuous variables, orderable, but better alternatives are available.
• Whenever possible, researchers try to reconceptualize nominal and ordinal variables and operationalize (measure) them with an interval scale.

Variation Ratio, VR

1.  VR = l - (proportion of cases in the mode)

2.  The value of VR reflects the following logic:

a. The larger the proportion of cases in the mode of a nominal variable, the less the variation among the cases of that variable.

b. By subtracting the proportion of cases from 1, VR reports the dispersion among cases.

• 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).
• Its maximum value approaches one as the proportion of cases inside the mode decreases.

3. Unfortunately, this measure is a "terminal statistic":

a VR does not figure prominently in any subsequent procedures for statistical analysis.

b Nevertheless, you should learn it, for it illustrates

• that even nominal variables can demonstrate variation
• 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

= 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.
• or 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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