Swati Lathia

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Introduction to Measure of Central Tendency

Table of Contents

Arithmetic Average or Mean

Calculating the mean from Ungrouped Data

Calculate simple mean. The observations is 42. One observation is taken as 34 instead of 43 by data entry error. Find the correct mean.

Calculate the mean from the discrete series

For the given data, x+y=24 and the mean is 3.62. Find the values of x & y.

Calculate mean from Continuous Series

The mean of the following distribution of 125 observation is 22.62. Find the missing frequency of the distribution.

Combined Mean

Sometimes we require to find the mean of entire group of data on the basis of different group of data. If we have two or more group of data, these groups are combined to get a combined or entire group of data. The mean the combine group of data is called the combined mean and is denoted by symbol x̄̄ (x double bar).

If 1, 2, 3, …,n are the mean of n group of data with number of observations n1,n2,n3,…nn respectively, then combined mean x̄̄ of the group is obtained by the formula :

Combined Mean = x̄̄ = (1n1 + 2n2 + 3n3 + … +nnn) / n1+n2+n3+…+nn

Example : The mean of 10 observations is 35 and mean of other 15 observations is 25. Find the mean of all 25 observations.

Example : A group of 60 values has a mean 10. A subgroup of this has n1=40, x̄1 = 11. Find the mean of other group.

Weighted Mean

Weighted mean is an average, whose observations are being multiplied by certain values known as “weights” and the aggregate of the multiplied results are being divided by the total sum of their “weights” instead of the sum of the items (number of observations) The formula is, Xw(sounds Xw bar) = (x1w1+x2w2+x3w3+….+xnwn)/(w1+w2+w3+..wn)

Properties of Mean

1. Sum of deviations of actual observations taken from the mean is always zero , i,e, Σ(x-) = 0

2. Mean is not independent of change of origin and scale. i.e. y = (x + or – a)/b where y is the mean of new observations, x is the old mean, a and b are constants which can change the value of old observations.


The median is the middle value of a series arranged in any order of magnitude. It is also known as “positional average”. It is just the 50th percentile value below which 50% of the value in the sample fall. Median is value diving observations into two equal parts such that the number of observations less than or equal to and the number of observations greater than or equal to this value are equal. To calculate median, observations must be in any of the order either in ascending or in descending.

Calculate median from ungrouped data

Find the median for the following data : 35, 10, 28, 8, 24, 5, 20, 19, 12, 30, 39

Calculate median from the discrete series

Calculate median from the continuous series

Calculate Missing frequency : Find missing frequencies for total 400 frequencies if median of frequency distribution is 38.


In some cases, some of the observations occur more frequently . The value of an observation that occurs most frequently in the observation data is called Mode of the data.

It is defined as the value of the variable which is repeated maximum number of times in the data set. It is denoted by Z. Sometimes, the data may have one or two modes or no mode at all. The data with one mode is called Uni-modal data and data with two modes are called bi-modal data.

Calculate Mode from ungrouped data, Calculate Mode from Discrete Series and Calculate Mode from Continuous Series

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