Question

Find the mean of the following frequency distribution using step- deviation method:

Class	84 – 90	90 – 96	96 – 102	102 – 108	108 – 114	114 – 120
Frequency	15	22	20	18	20	25

Answer 1

1. Identify given values

We determine the class width (h) by subtracting the lower limit from the upper limit of any class interval:

h = 90 – 84 = 6

2. Determine class marks

The class mark (x_i) for each interval is the midpoint, calculated using the formula `x_i = ("Lower Limit" + "Upper Limit")/2`.

We choose an assumed mean (A) from near the middle of our x_i values. Let A = 99.

3. Compute step-deviations

The step-deviation (u_i) for each class interval is determined by:

`u_i = (x_i - A)/h = (x_i - 99)/6`

We tabulate the given data and calculate the summation values:

Class Interval	Frequency (fi)	Class Mark (xi)	`bb(u_i = (x_i - 99)/6)`	fi × ui
84 – 90	15	87	–2	–30
90 – 96	22	93	–1	–22
96 – 102	20	99 (A)	0	0
102 – 108	18	105	1	18
108 – 114	20	111	2	40
114 – 120	25	117	3	75
Total	Σfi = 120			Σfiui = 81

4. Apply step-deviation formula

The formula to calculate the arithmetic mean using the step-deviation method is:

Mean `(barx) = A + ((sumf_iu_i)/(sumf_i)) xx h`

Substitute the totals from our table into the formula:

`barx = 99 + (81/120) xx 6`

`barx = 99 + 486/120`

`barx = 99 + 4.05`

`barx = 103.05`

The mean of the given frequency distribution using the step-deviation method is exactly 103.05.