Calculate the mean, variance and standard deviation of the following frequency distribution.

Class: 1–10 10–20 20–30 30–40 40–50 50–60

Frequency: 11 29 18 4 5 3

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Solution

Let the assumed mean A = 25.

Class Mid-Values $(x_{i})$ $d_{i} = x_{i} - A = x_{i} - 25$ $d_{i}^{2}$ Frequency $(f_{i})$ $f_{i} d_{i}$ $f_{i} d_{i}^{2}$

1–10 5.5 −19.5 380.25 11 −214.5 4182.75

10–20 15 −10 100 29 −290 2900

20–30 25 0 0 18 0 0

30–40 35 10 100 4 40 400

40–50 45 20 400 5 100 2000

50–60 55 30 900 3 90 2700

N = $\sum_{} f_{i}$ = 70 $\sum_{} f_{i} d_{i}$ = −274.5 $\sum_{} f_{i} d_{i}^{2} =$ 12182.75

Mean = $A + \frac{\sum_{} f_{i} d_{i}}{\sum_{} f_{i}} = 25 + (\frac{- 274.5}{70}) = 25 - 3.92 = 21.08$

Variance = $σ^{2} = (\frac{1}{N} \sum_{} f_{i} d_{i}^{2}) - {(\frac{1}{N} \sum_{} f_{i} d_{i})}^{2} = \frac{12182.75}{70} - {(\frac{- 274.5}{70})}^{2} = 174.02 - 15.37 = 158.65$

Standard deviation = $σ = \sqrt{158.65} = 12.6$

Suggest Corrections

Classes	0-10	10-20	20-30	30-40	40-50
Frequency	5	8	15	16	6

Class	:	0-5	5-10	10-15	15-20	20-25	25-30	30-35	35-40
Frequency	:	2	5	7	13	21	16	8	3

Class interval:	0−10	10−20	20−30	30−40	40−50	50−60	Total
Frequency:	5	f₁	20	15	f₂	5	60

Class	0-10	10-20	20-30	30-40	40-50	50-60
Frequence	6	7	15	16	4	2

Class:	1–10	10–20	20–30	30–40	40–50	50–60
Frequency:	11	29	18	4	5	3