Measures of Central Tendency: Mean, Median, and Mode

Mean, median and mode are some of the measures of central tendency. These are three different properties of data sets that can give us useful, easy to understand information about a data set to see the big picture and understand what the data means about the world in which we live.

Mean

“Mean” and “average” are just two different terms for the same property of a data set. It is also known as the arithmetic mean. The mean or average is beneficial to property and one of the most significant, easy and most used calculations out of all the three central tendencies. The mean is basically the summation of all the values in the set of data after it is divided by the total number of values in the set of the data. 

There are three methods of taking out averages – or mean in this case – and they are: direct method, assumed mean approach and step deviation method.

The above definition is of Arithmetic Mean, one of the many types of Mean. In detail, the types of mean are explained although most of them are out of scope for elementary Statistics

1. Arithmetic Mean 

Arithmetic Mean is the average of all the observations. Generally, if the mean is mentioned without any adjective, it is assumed to be Arithmetic Mean.

Example- We have a set of observations-x=1,3,5,7,91,3,5,7,91,3,5,7,9. The Arithmetic Mean is computed as (x/n) where n is the number of observations which is equal to 5 in this case. Thus x=25 in this case and n=5  so the mean comes out to be 5

2. Weighted Mean

Weighted mean is almost the same as Arithmetic Mean, the difference being that in weighted Mean, some values contribute more than the others. 2 Cases arise while calculating Weighted Mean. The weighted mean is useful in situations when one observation is more important than others.

Case 1- When the sum of weights is 1- Simply multiply each weight by its corresponding value and sum it all up.

Example- In the previous example, let us assume that w=0.2 for all the observations, then the weighted mean is- W_mean= (0.2*1)+(0.2*3)+(0.2*5)+(0.2*7)+(0.2*9)=5 which is the same as Arithmetic Mean but if we change the weights then the mean also changes.

Case 2- When the sum of weights is not equal to 1- In this case it is beneficial to make a table that shows each weight against each observation. Then calculate the product of each observation and its corresponding weight.

3. Harmonic Mean 

Harmonic Mean is calculated by dividing the total number of observations by the reciprocal of each observation. It is quite useful in Physics and has many other applications

(example- average speed when the duration of several trips is known). 

It is given by the formula- \[H.M= \frac{ n}{(1/x1)+(1/x2)+(1/x3)+.....(1/xn)}\]

4. Geometric Mean

The Geometric Mean indicates the central tendency using the product of the observations rather than their sum(which is used in calculating Arithmetic Mean). It is used in the field of finance and social sciences. In finance, it is used to calculate the average growth rates. The Geometric Mean is most useful when the observations are dependent on each other or they have large fluctuations. It is given by(INSERT EQUATION) 

Solved Example of Mean 

1. Find the mean for the following frequency table:

Solution :

Arithmetic mean  =  \[\Sigma \frac{fx}{N}\]  

Arithmetic mean  =  \[\Sigma \frac{fx}{N}\]  =  1165 / 40

  =  29.125

Hence the required arithmetic mean for the given data is 29.125.

Median

As the name suggests, the median is nothing but the middle – or “mid” – of all the values presented in the data set. This shows what the middle of the data is. For example: in a data set of 5, 10, 15, 20, 25, 15 is the median. 

There are two different methods of finding out the mean. They are the odd number of values and even numbers of values. 

Solved Example of Median

1. Find the median for the following frequency table:

Solution:

Here, the total frequency, N = \[\Sigma f\] = 40

N/2  =  40 / 2  =  20

The median is (N/2)th value = 20th value.

Now, the 20th value happens in the cumulative frequency 22, whose corresponding x value is 25.

Hence, the median = 25.

Mode

is defined as the value that is found mostly in a data set. When the frequencies in the data keep repeating, the mode takes place. This is mainly used for taking out most of the averages. For example, if you want to calculate the average of how many students scored the most, you might want to use the mode. 

Solved Example of Mode

1. Find the mode for the following frequency table:

By observing the given data set, the number 40 occurs more often. That is 10 times.

Hence the mode is 40.

Mean  =  29.125

Mode  =  25 and

Mode  =  40.