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Question

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$\left({x}_{i}\right)$ ${d}_{i}={x}_{i}-A\phantom{\rule{0ex}{0ex}}={x}_{i}-25$ ${d}_{i}^{2}$ Frequency $\left({f}_{i}\right)$ ${f}_{i}{d}_{i}$ ${f}_{i}{d}_{i}^{2}$ 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 = $A+\frac{\sum _{}{f}_{i}{d}_{i}}{\sum _{}{f}_{i}}=25+\left(\frac{-274.5}{70}\right)=25-3.92=21.08$ Variance = ${\sigma }^{2}=\left(\frac{1}{N}\sum _{}{f}_{i}{d}_{i}^{2}\right)-{\left(\frac{1}{N}\sum _{}{f}_{i}{d}_{i}\right)}^{2}=\frac{12182.75}{70}-{\left(\frac{-274.5}{70}\right)}^{2}=174.02-15.37=158.65$ Standard deviation = $\sigma =\sqrt{158.65}=12.6$

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