The data values in a data distribution is generally spread over a range of values. The central tendency of the data set when measured helps us to understand the central behavior of data values.

Statisticians are interested in knowing the average of data values, how the data values are positioned in the data set and the most typical data value of the observations. The three types of central tendency define these characteristics and thus lead to the methods of determining them.

A measure of central tendency of a data distribution is a number which convey an idea of centarlness of the data set. The centralness may refer to the average of all values, the middle value when the data values are ordered or the most typical or the most occurring value in the data set. Three measures of central tendency, the mean, median and mode of a data set are defined to measure these three central behaviors of the data set.
The three measures of central tendency are:
  1. Mean
  2. Median
  3. Mode

Mean:
The mean in Statistics refer to the arithmetic mean of a data set. When we use the word mean or average, The formula used to calculate the mean of the data set with n observations x1, x2,.......xn is
x = $\frac{x_{1}+x_{2}+......+x_{n}}{n}$ or using Sigma notation $\frac{\sum_{i=1}^{n}x_{i}}{n}$.

If the above data values are given with corresponding frequencies f1, f2,.....fn, then the mean of the frequency distribution is
x = $\frac{f_{1}x_{1}+f_{2}x_{2}+......f_{n}x_{n}}{N}$ where N = f1+ f2 +.........+ fn.
It can be shown that mean is the central value where the sum of the negative deviations and the sum of the positive deviations from the mean are equal. For this reason the mean is considered as a central value.
The Merits of Mean are

  1. It is well defined.
  2. It is based on all observations
  3. It is available for algebraic treatment.
  4. It is easy to compute.
  5. It is least affected by sampling fluctuations.

Against these merits, the disadvantages of using mean as a central value are,

  1. It is affected by extreme values.
  2. It cannot be determined if a single observation is not known or if the extreme classes in a frequency distribution are not well defined.

Median:

The median of a numerical data set is the value in the middle when the data is arranged in ascending or descending order. It is the halfway point in the data set partitioning the entire distribution into two equal halves.. 50% of the values of the distribution are less than the median and 50% are greater than the median. Because of this reason the median is considered as a measure of central tendency.

Steps to to be followed for finding the median.

  1. Order the data values, ascending or descending
  2. When the number of values is odd, median will be the middle value in the ordered array. That is the $\frac{n+1}{2}^{th}$ term in the array. When the number of values is even, the median is the average of the two values. That is the average of the $\frac{n}{2}^{th}$ and the $(\frac{n}{2}+1)^{th}$ terms.

The Median is a central value which is easy to understand and easy to compute. As against the mean it is less affected by Outliers or extreme values.
The disadvantages with median are

  1. it is not well defined.
  2. It is not based on all values and not suitable for algebraic treatment.
  3. It is affected by sampling fluctuations.

Mode:
Mode is defined to be the value that occurs most often. Because of the highest number of occurrence in the data set, Mode can be considered as the most typical value in the data set and hence a measure of central tendency.

Mode can be easily found by sheer observation counting or plotting the frequencies of each item. Mode is not affected by extreme values. The advantage of mode over the other two measures of central tendencies is it can also be found for non numerical data sets.

This against, the demerits of mode are

  1. It is not well defined and not based on all items.
  2. It is not suitable for algebraic treatment.
  3. it is affected by sampling fluctuations.


Solved Example

Question: Find the Mean, Median and Mode of the 15 data values
25, 32, 28, 34, 32, 24, 31, 36, 32, 27, 28, 31, 29, 30, 26.
Solution:
 
The sum of the data values
∑x = 25 + 32 + 28 + 34 + 32 + 24 + 31 + 36 + 32 + 27 + 28 + 31 + 29 + 30 + 26 = 445
and n =15.

Mean = $\frac{445}{15}$ = 29.67  (rounded to two decimal values).

Median:

To find the median let us arrange the data values in order. As there are 15 items (odd number) there is one middle value which is the 8th item in the array. Thus as shown below Median of the data set = 30.

Mean Mode

Mode:
It can be observed that the value 32 occurs 3 times which is the highest number of occurrences.
Hence the Mode of the data set = 32.