Mean, Median, and Mode

The mean or so called as the arithmetic average is when we add all the values of the data set and dividing that by the total number of values in the given data set. It is used when the data is evenly distributed. It is represented by the symbol “μ”.

The median is the middle value of a data set when the data set is arranged in ascending or descending order. If the number of values is odd, then the median is the middle value, and if the number of values in the data set is even, then the median is the average of the two middle values. It is represented by M.

The mode is the value that appears most often in a data set. The mode can be applied to both numerical data and categorical data. The mode is very useful to identify the most common values in the given data set. The mode is represented by Z.
 

The mean is found by adding all the values in the data set and dividing it by the total number of values in the data set. The mean is also called the arithmetic average. Mean is denoted with the symbol x
The formula used to calculate mean is:

Mean x = Sum of Values / Number of Values

For example, find the mean of the data sets 10, 40, 60, 80, and 110.

Mean = 10 + 40 + 60 + 80 + 110/5

Mean = 300/5 = 60

To find the mean of a grouped data, we use three methods, which are mentioned below:

Direct Method: To calculate the mean of a grouped data, we use the formula mentioned below:

x = fixi/fi

Where fi is the sum of all the frequencies.

Assumed Mean Method: To calculate the mean using assumed mean method, we use the following formula:

x = a + fixi/fi

Where a is the assumed mean

di  stands for deviation, where di = xi - a

fi is the sum of all the frequencies.

Step Deviation Method: To calculate the mean using step deviation method, we use the following formula:

x = a + hfixifi

Where a is the assumed mean

ui is called as the reduced deviation, where ui = (xi - a)/h

h is the class size

fi is the sum of all the frequencies.

The median is a middle value of a data that is sorted. To find the median, we sort the data in ascending or descending order. The median divides a given data set into two equal halves.

The formula to calculate the median is given below:

For even number of values:

[n2th term + {n2 + 1}th term / 2]

For odd number of values:

(n + 1)2th term

For example, find the median of the data 10, 20, 30, 40, and 50

Step 1: As the given data set is already in ascending order, we shall move on to the next step

Step 2: Check the number of terms in the data set “n”, whether it is even or odd

Step 3: Here n = 5 (odd)

Median = (n + 1)2th term

Median = [(5 + 1)/2]th term

 = 30.

To find the median of a grouped data, we use the following formula:

Median = l + [(n/2 - cf)/f] x h

Where, 

l is the lower limit of the median class

n is the number of observations

f is the frequency

h is class size

cf is cumulative frequency.

Mode is the highest number of repeated values in the given data set. To find mode, the formula is given below:

Mode = Highest Frequency Term

For example, find the mode in the given data set {1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4}

As the highest number of repeated values is 3.

Mode = 3 because 3 appears most frequently.

In a grouped data, the relation between the measures of the three tendencies which are mean, median and mode is shown below:

Mode = 3 Median - 2 Mean

The relationship between these three tendencies helps us to understand how to find the other tendency. Say for example we know what is the mean and mode, we can convert the formula by solving for median. 

For example, median = 20 and mode = 45, find the median:

Substituting the values in the formula:

45 = 3 median - 2(20)

45 = 3(20) - 2 mean

2 mean = 45 + 60

2 mean = 105

Mean = 105/2 = 52.5

The differences between mean, median, and mode are given below:
 

There are a lot of ways to master mean, median, and mode. Some ways to master mean, median, and mode are mentioned below:

Understanding Concepts Clearly: We have to understand the concepts of mean, median, and mode properly to master it. Mean is the average of all the values in the dataset, median is the middle value of the dataset, and mode is the most frequently occurring value.

Handling Grouped Data (Class intervals): When calculating for grouped data for mean, we use assumed mean or step deviation method; for median, we use the median formula with cumulative frequency tables; and for mode, use the modal class and the formula:

        Mode = L + f1 - f0 / 2f1 - f0 -f2  x h

Where, 
L = lower boundary of modal class
f1 = frequency of modal class
f0 = frequency before modal class
f2 = frequency after modal class
h = class width

Practice Strategies: To master the concept of mean, median and mode, we can use real-life examples, solve previous year exam question papers and practice timed quizzes for a quick recall of the concepts.

There are a lot of real-world applications of mean, median and mode. Let us now see what kind of applications and fields we use mean, median and mode in:

• Education and Academic Performance: We use mean to calculate the average exam scores in a class, median is used to report the middle score from the very high scores to the lower scores, and mode is used to identify the most common grade that is achieved by students.

• Business and Finance: In business and finance, we use mean to determine the average sales per month, median is used for analyzing the median salary to represent typical employee earnings without being skewed by executives’ high salaries, and mode is used to find the most frequently sold product.

• Healthcare and Medicine: In healthcare and medicine, we use means to calculate the average blood pressure and cholesterol levels in a patient, we use median to find the median survival time in clinical trials, and we use mode to find the most common symptoms of a disease.