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(a) Explain when to use the mean, median and mode as measures of central tendency. List and describe some common measures of dispersion. Why are measures of central tendency not appropriate in interpreting a distribution without being accompanied by measures of dispersion? (a) We have many different measures of central tendency because each measure of central tendency requires a minimum level of measurement. Each is applicable under specific conditions. The following table shows when each is applicable: Level of Mean Median Mode Variance / Standard Deviation Nominal x x x Ordinal x x Interval / Ratio Measurement Here are some actual examples for each application: (i) We use Mean as the appropriate measure of central tendency for interval measures such as the temperature in Celsius scale or Fahrenheit scale, and for ratio measures as in physical quantities, such as mass, length, energy, and temperature measured in K. (ii) We use Median as the appropriate measure of central tendency for ordinal data such as the results of a horse race, which say only which horses arrived first, second, third, etc. but no time intervals are mentioned, and for interval measures such as the temperature in Celsius scale or Fahrenheit scale. (iii) We use Mode as the appropriate measure of central tendency for nominal/categorical data such as gender, race, religious affiliation and birthplace. It is clear from the above discussion that not all measures of tendency are required to describe the data (nor all can be used). Depending upon the data type, we use one or the other to best describe the data. (b) A dataset cannot be described completely using either the measures of central tendency or the measures of dispersion (Range, Variance, Standard Deviation or Quartile Deviation) alone. They are always used together so that the distribution can be described in sufficient detail. Range provides information about the extreme values in a data set because Range = Maximum value - Minimum value. Standard deviation provides information about how much the data vary from the mean value, on an average. Two data sets may have the same measures of central tendency, say mean = median = mode = 6. This may make one to believe that they are both identical. But when standard deviation is computed for both the sets (assume it is 2.3 for one set and 3.0 for the other), it becomes clear that the second set has more spread than the first. Therefore, for a complete description of the data (this is very important for further statistical analysis), measures of central tendency and measures of spread are used together.