Two groups of items have the following statistics. Group 1: $40$ items, mean $30$, standard deviation $4$. Group 2: $60$ items, mean $35$, standard deviation $5$. Find the combined mean and the combined standard deviation of all $100$ items.

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Accepted Answer

Here $n_1 = 40$, $\bar{x}_1 = 30$, $\sigma_1 = 4$; and $n_2 = 60$, $\bar{x}_2 = 35$, $\sigma_2 = 5$.

**Combined mean:**
$$\bar{x}_{12} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2} = \frac{40(30) + 60(35)}{40 + 60} = \frac{1200 + 2100}{100} = \frac{3300}{100} = 33.$$

**Deviations of the group means from the combined mean:**
$$d_1 = \bar{x}_1 - \bar{x}_{12} = 30 - 33 = -3, \qquad d_2 = \bar{x}_2 - \bar{x}_{12} = 35 - 33 = 2.$$

**Combined standard deviation:**
$$\sigma_{12} = \sqrt{\frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}} = \sqrt{\frac{40(4^2 + (-3)^2) + 60(5^2 + 2^2)}{100}}.$$
Evaluate inside: $40(16 + 9) + 60(25 + 4) = 40(25) + 60(29) = 1000 + 1740 = 2740$. So
$$\sigma_{12} = \sqrt{\frac{2740}{100}} = \sqrt{27.4} \approx 5.23.$$ …

Q.Two groups of items have the following statistics. Group 1: $40$ items, mean $30$ , standard deviation $4$ . Group 2: $60$ items, mean $35$ , standard deviation $5$ . Find the combined mean and the combined standard deviation of all $100$ items.

Concept understanding — Coefficient of Variation and Combined Standard Deviation

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Q.Two groups of items have the following statistics. Group 1: $40$ items, mean $30$ , standard deviation $4$ . Group 2: $60$ items, mean $35$ , standard deviation $5$ . Find the combined mean and the combined standard deviation of all $100$ items.

Unlock everything free for 14 days