Measures of Central Tendency, Mean, Median and Mode Definition, Measure of central tendency is a single value, the value lies within the range of data, it is known as a measure of central tendency.

## What are the Properties of a good Measures of Central Tendency?

1) It should be based on all observations

2) Easily computable

3) It should be lease affected by extreme values

4) It should least fluctuation from sample to sample drawn from the same population.

## What are the major average classifications?

**Mathematical Averages**

The arithmetic mean geometric mean and harmonic mean comes under mathematical averages.

**Positional Averages**

Median, mode, quartiles, octiles, deciles, and percentiles come under positional averages.

**Commercial Averages**

Moving average, progressive average, and composite averages are coming under commercial averages.

**Usages of Average Classifications**

These represent a simple and concise picture of complex data sets, easy to compare 2 0r more data sets, and provide us to make appropriate decisions.

**Arithmetic Mean**

AM=Sum of the values divided by the number of values in the set.

X: 2, 3, 5, 8, 12

AM=(2+3+5+8+12)/5

AM=6.0

### Sample Mean Vs Population Mean

The population means are based on entire values, whereas the sample mean is based on the values of items selected in the sample from the population. The sample mean is an estimate of the population mean.

**Moving Average**

We get a series of averages out of a series of data. We take a fixed number of beginning items to find its average. To get the next average, delete the first item of the previous group and add the next item of the series to it and find its average. Continue this process till all the values are exhausted.

Moving Averages mainly used time-series data sets, to find a particular trend exists or not.

**Geometric Mean**

The geometric mean is the nth root of the product of n values of a set of observations. GM is not affected by extreme values, its difficult to calculate calculation is not possible in case of negative values or zero values.

For index numbers, geometric means is more suitable.

X: 2, 4, 8, 64

GM=8

**Harmonic Mean**

The harmonic mean is the inverse of the arithmetic mean of the reciprocals of the observations of a set.

Always AM>=GM>=HM

**Median**

Median is the value of the variable that divides the ordered set of values into two equal halves. Median is not influenced by extreme values. The second quartile and 50th percentile are the same as the median.

If N is odd Median=(N+1)/2 th item

If N is even Median= Average of Nth/2 and (N+1)/2 the items.

X: 3,18,7, 20, 11,12, 9,17, 22

Median=12

**Mode**

The maximum frequency in distribution is known as mode. The distribution which has only one mode is called unimodal.

x: 5, 7, 9, 9, 8, 5, 6, 8, 7, 7, 5, 7, 9, 2, 7

Mode=7

Median and Mode are least affected by the extreme values as a measure of central tendency.

## Summary

Based on type type of Variable the measure of central tendency changes.

When the scale is Nominal, Mode is the appropriate measure,

When the scale is Ordinal, Median is the appropriate measure,

When the scale is Interval/Ratio (not skewed), Mean is the appropriate measure

When the scale is Interval/Ratio (skewed), Median is the appropriate measure.