For a group of 50 male workers, the mean and the standard deviation of their daily wages are ₹63 and ₹9, respectively. For a group of 40 female workers, these values are ₹54 and ₹6, respectively. What is the standard deviation of the combined group of these 90 workers?
9
Let group 1 (males): \(n_1=50,\ \bar{x}_1=63,\ \sigma_1=9\). Group 2 (females): \(n_2=40,\ \bar{x}_2=54,\ \sigma_2=6\).
Combined mean: \(\bar{x}=\dfrac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}=\dfrac{50\cdot 63+40\cdot 54}{90}=\dfrac{3150+2160}{90}=\dfrac{5310}{90}=59\).
Deviations: \(d_1=\bar{x}_1-\bar{x}=63-59=4\) and \(d_2=\bar{x}_2-\bar{x}=54-59=-5\).
Combined variance: \(\sigma^2=\dfrac{n_1(\sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)}{n_1+n_2}\)
\(=\dfrac{50(81+16)+40(36+25)}{90}=\dfrac{50\cdot 97+40\cdot 61}{90}=\dfrac{4850+2440}{90}=\dfrac{7290}{90}=81\).
Therefore \(\sigma=\sqrt{81}=9\). Hence the standard deviation of the combined group is 9.
Similar Questions
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SECOND: 2, 3, 4, 6, 8, 9, 10, 14, 16, 18
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Important Questions from Measures of dispersion
For the series of the following observations, the coefficient of range leads to which of the following conclusions?
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Which of the following is the relative measure of dispersion?
