Mean of Grouped Data

Grouped data is the set of data that is obtained by forming individual observations of variables into groups. Grouped data is divided into groups. A frequency distribution table is required for the grouped data, which helps showcase the frequencies of the given data. The mean of grouped data can be obtained using three methods. The methods are:

• Direct Method
• Assumed Mean Method
• Step Deviation Method

Calculating Mean Using Direct Method

The direct method is the simplest method to find the mean of grouped data. The mean of grouped data using the direct method can be calculated using the following steps: 

• Four columns are created in the table. The columns are Class interval, class marks (x_i), frequencies (f_i), the product of frequencies, and class marks (f_i x_i).
• Now, calculate the mean of grouped data using the formula

Mean Formula For Grouped Data (Using Direct Method)

The mean formula for grouped data using direct method is added below,

Example: Calculate the mean height for the following data using the direct method.

Solution:

As, 

Height (in inches)	Frequency(fi)	Midpoint (xi)	fi × xi
60 – 62	3	61	183
62 – 64	6	63	378
64 – 66	9	65	585
66 – 68	12	67	804
68 – 70	8	69	552
70 – 72	2	71	142
	∑fi = 40		∑fi xi = 2644

⇒ Mean = 2644/40 = 66.1

Thus, mean height is 66.1 inches.

Calculating Mean Using Assumed Mean Method

When the calculation of the mean for grouped data using the direct method becomes very tedious, then the mean can be calculated using the assumed mean method. To find the mean using the assumed mean method the following steps are needed:

• Five columns are created in the table i.e., class interval, class marks (x_i), corresponding deviations (d_i = x_i – A) where A is the central value from class marks as assumed mean, frequencies (f_i), and product of f_i and d_i.
• Now, the mean value can be calculated for the given data using the following formula.

Mean Formula For Grouped Data (Using Assumed Mean Method)

The mean formula for grouped data using assumed mean method is added below,

Example: Calculate the mean of the following data using the Assumed Mean Method.

Solution:

Let us assume the value of mean be A = 53,

and the required table for the given data is as follows for A = 53:

Weight (in kg)	Frequency(fi)	Midpoint (xi)	Deviation (di = xi – A)
40 – 44	2	42	-11
44 – 48	3	46	-7
48 – 52	5	50	-3
52 – 56	7	54	1
56 – 60	2	58	5
60 – 64	1	62	9

Add one more column to the table which give product of f_i and d_i :

Weight (in kg)	Frequency(fi)	Midpoint (xi)	Deviation (di = xi – A)	fi × di
40 – 44	2	42	-11	-22
44 – 48	3	46	-7	-21
48 – 52	5	50	-3	-15
52 – 56	7	54	1	7
56 – 60	2	58	5	10
60 – 64	1	62	9	9
	∑fi = 20			∑fi di = -32

Thus, Mean = 53 + (-32)/20 = 53 – 1.6 = 51.4

Thus, mean weight of the given data using assumed mean method is 51.4 Kg.

Calculating Mean Using Step Deviation Method

Step deviation method is also famously known as the scale method or shift of origin method. When finding the mean of grouped data becomes tedious, using step deviation method can be used. Following are the steps that should be followed while using the step deviation method:

• Five columns are created in the table. They are class interval, class marks (x_i, here the central value is A), deviations (d_i), u_i = d_i/h (h is class width), and product of f_i and UI_i.
• Now, the mean of the data can be calculated using the following formula

Mean Formula For Grouped Data (Using Step Deviation Method)

The mean formula for grouped data using step deviation mean method is added below,

Example: Calculate the mean of the following data using the Step Deviation method.

Solution:

Range of the data is 20 to 48, for assumption of mean, lets take average of the range values,

Assumed mean = (20 + 48) /2 = 68/2 = 34

Let’s A = 34 be the assumed mean of the data,

Now, using assumed mean value, let’s create the table for step deviation as follows:

Age (in years)	Frequency(fi)	Class Mark(xi)	Deviation(di = xi – A)	Step Deviation (ui = di/h)	fi × ui
20 – 24	3	22	-12	-3	-9
24 – 28	6	26	-8	-2	-12
28 – 32	8	30	-4	-1	-8
32 – 36	5	34	0	0	0
36 – 40	5	38	4	1	5
40 – 44	2	42	8	2	4
44 – 48	1	46	12	3	3
	∑fi = 20				∑fi ui =- 17

Thus, Mean = 34 + 4 × (-17)/20 = 34 + 4 × (-0.85) = 34 – 3.4 = 30.6 

Thus, mean age of data using step deviation method is 30.6

Mean in Mathematics is the measure of central tendency and is mostly used in Statistics. Mean is the easiest of all the measures. Data is of two types, Grouped data and ungrouped data. The method of finding the mean is also different depending on the type of data. Mean is generally the average of a given set of numbers or data. It is one of the most important measures of the central tendency of distributed data.

In statistics, the mean is the average of a data set. It is calculated by adding all the numbers in the data set and dividing by the number of values in the set. The mean is also known as the average. It is sensitive to skewed data and extreme values. For example, when the data are skewed, it can miss the mark.

In this article, we’ll explore all the things you need to know about What is Mean, Mean Definition, Mean Formula, Mean Examples, and others in detail.