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# Descriptive Statistics- Measures of Central tendencies

Descriptive Statistics is the discipline of quantitatively describing the main features of a given data set. It provides simple summary measures about the sample about the observations that has been made in the set. These summary measures may form the basis of the initial description of a data as a part of a more extensive statistical analysis or they may suffice in themselves for some particular statistical investigations.

The most commonly used descriptive statistics in statistical analysis are:

• Measures of Central Tendency, which yield a representative value for a set of observations.
• Measures of Dispersion, which show how different are the observations in a given data set different from the central value on an average.
• Measures examining the shape of a given data distribution.
• Measures aimed at examining the most unusual observations of a data set.

In this module we’ll learn about the different measures of central tendency.

Measures of Central Tendency

Central tendency refers to the propensity of quantitative data to cluster around a particular value. The particular value around which the observations in the data set fluctuate is called the central value. It is a representative value of the set of given observations. The objective of the analyst is to find out functional forms based on the observations of the data set which would give a ‘good’ representative central value. Such functional forms are known as measures of central tendency. The most widely used measures of central tendency are: Mean, Median and Mode.

For Example;

The vice president of marketing of a fast – food chain is studying the sales performance of the 100 stores in the eastern part of the country. He would be looking at the distribution with an eye toward getting information about the central tendency to compare the eastern part with other parts of country. Central tendency is basically the central most value of a distribution. Now how do we know which one is the central most value?

There are precisely three ways to find the central value: Mean, Median and Mode.

Mean

As a measure of central tendency, Mean gives the average value of a set of observations. The idea of average is a familiar one. Suppose, we say ‘Germans live longer than Indians’. This does not mean that every Germans live longer than every Indians. All we mean is that the longevity of a typical German is more than the average longevity of a typical Indian.

Properties

• The sum of the deviations of the given values of a variable from its mean is necessarily zero
• The Arithmetic Mean of a sample of observations depends both on the change of scale and origin.
• The combined Arithmetic Mean for two groups of with mean x1 and x2 and with n1 and n2 observations respectively is defined by

Median

Median of a set of statistical observations is the middlemost value of a data set when they are arranged in the increasing order of the magnitude. Median is that value of the variable which divides the group into two equal parts, one comprising all the values greater and the other, all values less than median.

Properties

Based on the construction and the nature of operation Median, as a measure of central tendency exhibits the following important properties:

• Median obeys linearity, i.e. it depends simultaneously on the change of scale and origin.
• The combined median of the two groups lies in between the median of the two individual groups.

Mode

Mode, for a given set of observations is that value of the variable, where the variable occurs with considered to represent the true characteristics of a frequency distribution and it is referred as the most typical or the fashionable value of the variate.