Mean and standard deviation of two distributions of 100 and 150 items are 50, 5 and 40, 6 respectively. Find the standard deviation of all the 250 items taken together.

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Solution

$Combined Mean = \frac{N_{1}_{1} + N_{2}_{2}}{N_{1} + N_{2}} = \frac{100 \times 50 + 150 \times 40}{100 + 150} = \frac{11000}{250} = 44 Calculation of combined standard deviation σ_{1, 2} = \sqrt{\frac{N_{1} σ_{1}^{2} + N_{2} σ_{2}^{2} + N_{1} d_{1}^{2} + N_{1} d_{2}^{2}}{N_{2} + N_{2}}} d_{1} =_{1} - {¯ ¯¯¯ ¯ X}_{1, 2} and d_{2} =_{2} - {¯ ¯¯¯ ¯ X}_{1, 2} d_{1} = 50 - 44 = 6 and d_{2} = 40 - 44 = - 4 σ_{1, 2} = \sqrt{\frac{100 (5)^{2} + 150 (6)^{2} + 100 (6)^{2} + 150 (- 4)}{100 + 150}} σ_{1, 2} = \sqrt{\frac{2500 + 5400 + 3600 + 2400}{100 + 150}} = \sqrt{55.6} = 7.456$

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