Question

e. A set of observations has a mean of 9.0 and standard deviation 4.0. Another set of 20 observations has a mean of 10.0. Given the standard deviation of the combined set of 35 observations is 3.0, find the variance of the second set.

Answer 1

From the combined standard deviation, with the mean, and standard

deviation of the data sets, the required variance can be found.

• $\underline{\mathrm{The \ variance \ of \ the \ second \ set \ is} \ 4\dfrac{5}{28}}$

Reasons:

The given data information are;

Mean of the first set of observation, $\overline x_1$ = 9.0

Standard deviation of the the first set, σ₁ = 4.0

Sample size of second observation, n₂ = 20

Mean of second sample, $\overline x_2$ = 10

Standard deviation of the combined set, σ₁₂ = 3.0

Sample size of combined set of observation, n₂ = 35

Solution:

The number of elements in the second sample, n₁ = 35 - 20 = 15

The standard deviation of the combined set is given by the formula;

${\sigma_{12}} = \mathbf{\sqrt{\dfrac{n_1 \cdot \left ( \sigma^2_{1}-d_1^2 \right ) + n_2 \cdot \left ( \sigma^2_{2}-d_2^2 \right )}{n_{1}+n_{2}}}}$

Where;

$\overline x_{12} =\mathbf{\dfrac{n_1 \cdot \overline x_1 +n_2 \cdot \overline x_2 }{n_1 + n_2}}$

Therefore;

$\overline x_{12} =\dfrac{15 \times 9.0 +20 \times10.0 }{15 + 20} = \dfrac{67}{7}$

d₁ = $\mathbf{\overline x_{12}}$ - $\mathbf{\overline x_1}$

$d_1 = \dfrac{67}{7} - 9.0 = \dfrac{4}{7}$

d₂ = $\mathbf{\overline x_{12}}$ - $\mathbf{\overline x_2}$

$d_2 = \dfrac{67}{7} - 10.0 = -\dfrac{3}{7}$

Therefore;

$3=\sqrt{\dfrac{15 \times \left ( 4.0^2-\left(\dfrac{4}{7}\right) ^2 \right ) + 20 \cdot \left ( \sigma^2_{2}-\left(-\dfrac{3}{7}\right) ^2 \right )}{15+20}}$

Solving gives;

$Variance, \ \sigma^2_2 = \mathbf{\dfrac{{117} }{28}}$

$\mathrm{The \ variance \ of \ the \ second \ set, \ \sigma_2^2} = \dfrac{{117} }{28} = 4\dfrac{5}{28}$

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