Mean - Assumed Mean Method

In Statistics, the assumed mean method is used to calculate mean of a grouped data. If the given data is large, then this method is recommended rather than a direct method for calculating mean. This method helps in reducing the calculations and results in small numerical values. This method depends on estimating the mean and rounding to an easy value to calculate with. Again this value is subtracted from all the sample values. When the samples are converted into equal size ranges or class intervals, a central class is chosen and the computations are performed.

Assumed Mean Method Formula

Let $x_{1},x_{2},x_{3}.....x_{n}$ are mid-points or class marks of $n$ class intervals and $f_{1},f_{2},f_{3}.....f_{n}$ are the respective frequencies. The formula of the assumed mean method is

${\bar {x}}=a+{\frac {\textstyle \sum _{i=1}^{n}\displaystyle f_{i}d_{i}}{\textstyle \sum _{i=1}^{n}\displaystyle f_{i}}}$

Here,

$a$ = assumed mean

$f_{i}$ = frequency of ith class

$d_{i}$ = $x_{i}-a$= deviation of $i$^th class

$\sum f_{i}$ = Total number of observations

$x_{i}$= class mark = (upper class limit + lower class limit) / 2

Example: The following table gives information about the marks obtained by 110 students in an examination.

Class	Frequency
0 - 10	12
10 - 20	28
20 - 30	32
30 - 40	25
40 - 50	13

Find the mean marks of the students using the assumed mean method.

Solution:

Class	Frequency ($f_{i}$)	Class mark ($x_{i}$)	$d_{i}=x_{i}-a$	$f_{i}d_{i}$
0 - 10	12	5	5 - 25 = -20	-240
10 - 20	28	15	15 - 25 = -10	-280
20 - 30	32	25 = $a$	25 - 25 = 0	0
30 - 40	25	35	35 - 25 = 10	250
40 - 50	13	45	45 - 25 = 20	260
Total	$\sum f_{i}=110$			$\sum f_{i}d_{i}=-10$