Mean - Step - Deviation Method

Step deviation method is a method of obtaining the mean of grouped data when the values are large. In statistics, there are three types of mean - arithmetic mean, geometric mean, and harmonic mean. When the values of the data are large and the deviation of the class marks have common factors, the step deviation method is used.

Definition

Step deviation method can be defined as the method used to obtain the mean of large values which is divisible by a common factor. These values of deviations are reduced to a smaller value by dividing all the values by a common factor. In other words, the step deviation method is used when the deviations of the class marks from the assumed mean are large and they all have a common factor. This method is also called a change of origin or scale method. The mean of grouped data is obtained by three different methods, direct method, assumed or short-cut method, and step deviation method. The step deviation method is considered as the extension of the assumed method as we use the deviation formula used in the assumed method.

Mean - Step - Deviation Method Formula

Step Deviation of Mean = ${\bar {x}}=a+h\left[{\frac {\sum f_{i}u_{i}}{\sum f_{i}}}\right]$

where $a=$ assumed mean

$h=$ class size

$d_{i}=x_{i}-a$

$x_{i}=$ midpoint of the class interval

$u_{i}={\frac {d_{i}}{h}}$

$f_{i}=$ frequency

$d_{i}=x_{i}-a$

Example: Find the mean of the following using the step-deviation method.

Class Interval	$f_{i}$
10 - 25	2
25 - 40	3
40 - 55	7
55 - 70	6
70 - 85	6
85 - 100	6

Solution:

Class Interval	$f_{i}$	$x_{i}$	$d_{i}=x_{i}-a$ ($a=47.5$)	$u_{i}={\frac {d_{i}}{h}}$ ($h=15$)	$f_{i}u_{i}$
10 - 25	2	17.5	-30	-2	-4
25 - 40	3	32.5	-15	-1	-3
40 - 55	7	47.5	0	0	0
55 - 70	6	62.5	15	1	6
70 - 85	6	77.5	30	2	12
85 - 100	6	92.5	45	3	18
Total	${\sum f_{i}}$=30				$\sum f_{i}u_{i}$=29

Step Deviation of Mean = ${\bar {x}}=a+h\left[{\frac {\sum f_{i}u_{i}}{\sum f_{i}}}\right]$

${\bar {x}}=47.5+15$$\left[{\frac {29}{30}}\right]$

${\bar {x}}=47.5+14.5=62$