What is the standard deviation of the following data?
Measurement 0 - 10 10 - 20 20 - 30 30 - 40
Frequency 1 3 4 2

Moderate

Solution

CIass Frequency $y_{i}$ $\begin{aligned} u_{i} = \frac{y_{i} - A}{10}, \\ A = 25 \end{aligned}$ $f_{i} u_{i}$ $f_{i} u_{i}^{2}$
0 - 10 1 5 -2 -2 4
10 - 20 3 15 -1 -3 3
20 - 30 4 25 0 0 0
30 - 40 2 35 1 2 2
Total 10 -3 9
$\begin{array}{l} σ^{2} = c^{2} [\frac{\sum f_{i} u_{i}^{2}}{{Σf}_{i}} - {(\frac{\sum f_{i} u_{i}}{\sum f_{i}})}^{2}] \\ = 10^{2} [\frac{9}{10} - {(\frac{- 3}{10})}^{2}] = 90 - 9 = 81 \\ \Rightarrow σ = 9 \end{array}$

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CIass	Frequency	$y_{i}$	$\begin{aligned} u_{i} = \frac{y_{i} - A}{10}, \\ A = 25 \end{aligned}$	$f_{i} u_{i}$	$f_{i} u_{i}^{2}$
0 - 10	1	5	-2	-2	4
10 - 20	3	15	-1	-3	3
20 - 30	4	25	0	0	0
30 - 40	2	35	1	2	2
Total	10			-3	9