Statistics : Basics - Mean , Median and Mode

Arithmetic mean/Mean

It is the most common measure of central tendency. It serves as a balance point in a set of data. It is the only common measure in which all the values play an equal role. Mean is computed as sum of all the values divided by number of values in the data set. It is denoted by X-bar.

Example:

Age of students in class:

12 13 12 11 14 15 12 13 14

Mean=  ( 12 + 13 + 12 + 11 + 14 + 15 + 12 + 13 +14 ) / 9 = 116 / 9 = 12.88

Therefore, the average age is 12.8 ~ 13years

Median

Median is the middle value in an ordered array of the data that has been ranked from smallest to the largest. Half of the values are smaller or equal to the median, and half of the values are larger or equal to the median. In case of extreme values, median can be used as it is not affected.

Median has different calculation in case of odd and even data numbers.

Odd number of values

Median is the middle value of the sorted data

Example:

Age of students in class: 12 13 12 11 14 15 12 13 14

First sort the data : 11 12 12 12 13 13 14 14 15

Therefore, Median is 13

Even number of values

Median is the average of two middle values of the sorted data

Example:

Age of students in class: 12 13 12 11 14 15 12 13 14  12

First sort the data : 11 12 12 12 12 13 13 14 14 15 

Median = (12 + 13) /2 =12.5

Mode

Mode is the value that appears most frequently. Like the median and unlike the mean, it is not affected by the extreme values. For a particular variable, there can be several modes or no modes at all.

Example:

Age of students in class: 12 13 12 11 14 15 12 13 14 13

Since 12 and 13 has occurred 3 times, mode :12 & 13.

Comparison of Mean , Median and Mode

The determination of which average exactly suits for a specific variable depends on many factors. Certainly depends on the data level. Below are the valid averages for each level of data.

Nominal data => Mode

                   Ordinal data   => Mode and Median

                               Interval data  => Mean, Median and Mode 

                              Ratio data      => Mean, Median and Mode

For a symmetrical distribution, all the three measures  mean, median and mode are exactly same in value.

i.e, Mean = Median = Mode

Geometric Mean

When you want to measure the rate of change of the variable over time, you need to use the geometric mean instead of arithmetic mean. It is computed as nth root of product of n values.

Harmonic Mean

It is also a mathematical average. It is the reciprocal of the average of  the reciprocal of the values i.e number of values divided by sum of its reciprocals.

Example:

The time taken by 3 teams for designing a caption are 6,3 and 8min. For computing the average rate of designing the caption is

XH= 3 / (1/6 + 1/3 + 1/8) = 3/(15/24) = 24/5 = 4.8min

Arithmetic mean would be X=6+3+8/3 = 5.6min

Harmonic mean will always be less than arithmetic mean.