Question

Two samples of size hundred and 150 e respectively have means 15 and 16 and standard deviations 3 and 4 respectively. Find the combined mean and standard deviation of size 250.​

Answer 1

Given

$\tt\longmapsto{N_1 = 100, \bar{X}_1 = 15, σ_1 = 3}$

$\tt\longmapsto{N_2 = 150, \bar{X}_2 = 16, σ_2 = 4}$

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To find

• Combined mean.
• Combined standard deviation.

Solution

★ We know that

$\large{\underline{\boxed{\tt{\orange{\bar{X}_{12} = \dfrac{N_1 \bar{X}_1 + N_2 \bar{X}_2}{N_1 + N_2}}}}}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\bar{X}_{12} = \dfrac{100 \times 15 + 150 \times 16}{100 + 150}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\bar{X}_{12} = \dfrac{1500 + 2400}{250}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\bar{X}_{12} = \dfrac{3900}{250}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\bar{X}_{12} = 15.6}$

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★ Now

⇝ Now, we will calculate $\sf{d_1}$ and $\sf{d_2}$

$\sf\longrightarrow{d_1 = \bar{X}_1 - \bar{X}_{12}}$

$\sf\longrightarrow{d_1 = 15 - 15.6}$

$\sf\longrightarrow{d_1 = -0.6}$

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$\sf\longrightarrow{d_2 = \bar{X}_2 - \bar{X}_{12}}$

$\sf\longrightarrow{d_2 = 16 - 15.6}$

$\sf\longrightarrow{d_2 = 0.4}$

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★ Using the formula

$\small{\underline{\boxed{\tt{\orange{σ_{12} = \sqrt{\dfrac{N_1σ_1^2 + N_2σ_2^2 + N_1d_1^2 + N_2d_2^2}{N_1 + N_2}}}}}}}$

⠀

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\tt {\pink{Here}}\begin{cases} \sf{\green{N_1 = 100\: and\: N_2 = 150}}\\ \sf{\blue{σ_1 = 3\: and\: σ_2 = 4}}\\ \sf{\orange{d_1 = -0.6\: and\: d_2 = 0.4}}\end{cases}\end{lgathered} \:\end{lgathered}\end{lgathered}$

⠀

$\tiny\tt:\implies{σ_{12} = \sqrt{\dfrac{100 \times (3)^2 + 150 \times (4)^2 + 100 \times (-0.6)^2 + 150 \times (0.4)^2}{100 + 150}}}$

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$\tiny\tt:\implies{σ_{12} = \sqrt{\dfrac{100 \times 9 + 150 \times 16 + 100 \times 0.36 + 150 \times 0.16}{250}}}$

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$\small\tt:\implies{σ_{12} = \sqrt{\dfrac{900 + 2400 + 36 + 24}{250}}}$

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$\small\tt:\implies{σ_{12} = \sqrt{\dfrac{3360}{250}}}$

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$\small\tt:\implies{σ_{12} = \sqrt{13.44}}$

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$\small\tt:\implies{σ_{12} = 3.67}$

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Hence, the combined mean is 15.6 and combined standard deviation is 3.67.

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