Step deviation Method for Finding the Mean with Examples

Statistics is a discipline of mathematics that uses quantified models and representations to gather, review, analyze, and draw conclusions from data. The most commonly used statistical measures are mean, median, and mode. Variance and standard deviation are measures of dispersion in statistics and various measures of concentration including quartiles, quintiles, deciles, and percentiles. Statistics is a way more beyond the topics mentioned, but here we stop for the “Mean” by Step Deviation method. In general, there are 3 types of mean:

1. Arithmetic mean
2. Geometric mean
3. Harmonic mean

This article is about the Arithmetic mean by Step Deviation method. The arithmetic mean, also called the average or average value is the quantity obtained by summing two or more numbers or variables and then dividing by the number of numbers or variables. The arithmetic mean is important in statistics. For example, Let’s say there are only two quantities involved, the arithmetic mean is obtained simply by adding the quantities and dividing by 2. Mean or Arithmetic Mean is the average of the numbers i.e. a calculated central value for a set of numbers. General Formulae for Mean is,

Mean = Sum of observation /  Number Of Observation

Example:

The marks obtained by 5 students in a class test are 7, 9, 6, 4, 2 out of 10. Find the mean marks for the class?

According to the formula mean marks of the class are: 

Average marks = Sum of observation / Number Of Observation

Here average marks = (7 + 9 + 6 + 4 + 2) / 5 = 28 / 5 = 5.6        

Hence the mean marks for the class is 5.6            

Derivation of Formula for Mean by Step Deviation Method

The general formula for mean in statistics is:

Mean = Σf_ix_i / Σf_i

Where,

Σf_ix_i: the weighted sum of elements and 

Σf_i: the number of elements

In the case of grouped data, assume that the frequency in each class is centered at its class-mark. If there are n classes and f_i denotes the frequency and y_i denotes the class-mark of the i_th class the mean is given by,

Mean = Σf_iy_i / Σf_i

When the number of classes is large or the value of f_i and y_i is large, an approximate (assumed) mean is taken near the middle, represented by A and deviation (d_i) is taken into consideration. Then mean is given by,

Mean = A + Σf_id_i / Σf_i 

In the problems where the width of all classes is the same, then further simplify the calculations of the mean by computing the coded mean, i.e. the mean of u₁, u₂, u₃, …..u_n where,

u_i = (y_i – A) / c

Then the mean is given by the formula,

Mean = A + c x (Σf_iu_i / Σf_i)

This method of finding the mean is called the Step Deviation Method.

Examples

Question 1: Find the mean for the following frequency distribution?

Class Intervals	84-90	90-96	96-102	102-108	108-114
Frequency	8	12	15	10	5

Solution:

Applying the Standard Deviation Method,

We take the assumed mean to A = 99, and here the width of each class(c) = 6

Classes	Class-mark(yi)	ui = (yi – A) / c	frequency(fi)	fiui
84-90	87	-2	8	-16
90-96	93	-1	12	-12
96-102	99	0	15	0
102-108	105	1	10	10
108-114	111	2	5	10
Total			50	-8

Mean = A + c x (Σf_iu_i / Σf_i)

          = 99 + 6 x (-8/50)

          = 99 – 0.96

          = 98.04

Question 2: Find the mean for the following frequency distribution?

Class Intervals	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	10	6	8	12	5	9

Solution:

Applying the Standard Deviation Method,

Construct the table as under, taking assumed mean A = 45, and width of each class(c) = 10.

Classes	Class-mark(yi)	ui = (yi – A) / c	frequency(fi)	fiui
20-30	25	-2	10	-20
30-40	35	-1	6	-6
40-50	45	0	8	0
50-60	55	1	12	12
60-70	65	2	5	10
70-80	75	3	9	27
Total			50	23

Mean = A + c x (Σf_iu_i / Σf_i)

          = 45 + 10 x (23/50)

          = 45 + 4.6

          = 49.6

Question 3: The weight of 50 apples was recorded as given below

Weight in grams	80-85	85-90	90-95	95-100	100-105	105-110	110-115
Number of apples	5	8	10	12	8	4	3

Calculate the mean weight, to the nearest gram?

Solution:

Construct the table as under, taking assumed mean A = 97.5. Here width of each class(c) = 5

Classes	Class-mark(yi)	ui = (yi – A) / c	frequency(fi)	fiui
80-85	82.5	-3	5	-15
85-90	87.5	-2	8	-16
90-95	92.5	-1	10	-10
95-100	97.5	0	12	0
100-105	102.5	1	8	8
105-110	107.5	2	4	8
110-115	112.5	3	3	9
Total			50	-16

Mean = A + c x (Σf_iu_i / Σf_i)

          = 97.5 + 5 x (-16/50)

          = 97.5 – 1.6

          = 95.9

Hence the mean weight to the nearest gram is 96 grams.

Question 4: The following table gives marks scored by students in an examination:

Marks	0-5	5-10	10-15	15-20	20-25	25-30	30-35	35-40
Number of students	3	7	15	24	16	8	5	2

Calculate the mean marks correct to 2 decimal places?

Solution:

Construct the table as under, taking assumed mean A = 17.5. Here width of each class(c) = 5

Classes	Class-mark(yi)	ui = (yi – A) / c	frequency(fi)	fiui
0-5	2.5	-3	3	-9
5-10	7.5	-2	7	-14
10-15	12.5	-1	15	-15
15-20	17.5	0	24	0
20-25	22.5	1	16	16
25-30	27.5	2	8	16
30-35	32.5	3	5	15
35-40	37.5	4	2	8
Total			80	17

Mean = A + c x (Σf_iu_i / Σf_i)

          = 17.5 + 5 x (17/80)

          = 17.5 + 1.06

          = 18.56