Two sets of data consisting of 10 and 20 observations have same mean 8 with standard deviations of 1 and 2, respectively. If the two data sets are combined, then the variance is

Question

Prepp Team · Accepted Answer

Problem Setup

We are given two datasets:

Dataset 1: Number of observations, $n_1 = 10$; Mean, $\bar{x}_1 = 8$; Standard Deviation, $s_1 = 1$.
Dataset 2: Number of observations, $n_2 = 20$; Mean, $\bar{x}_2 = 8$; Standard Deviation, $s_2 = 2$.

We need to find the variance of the combined dataset.

Variance Calculation

First, calculate the variance for each dataset:

Variance of Dataset 1: $s_1^2 = (s_1)^2 = 1^2 = 1$.
Variance of Dataset 2: $s_2^2 = (s_2)^2 = 2^2 = 4$.

Combined Mean

Calculate the mean of the combined dataset ($\bar{x}_{comb}$):

$ \bar{x}_{comb} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} $ $ \bar{x}_{comb} = \frac{(10 \times 8) + (20 \times 8)}{10 + 20} = \frac{80 + 160}{30} = \frac{240}{30} = 8 $

Since both original means are 8, the combined mean is also 8.

Combined Variance Formula

The formula for the combined variance ($s_{comb}^2$) of two datasets is:

$ s_{comb}^2 = \frac{n_1 s_1^2 + n_2 s_2^2 + n_1 (\bar{x}_1 - \bar{x}_{comb})^2 + n_2 (\bar{x}_2 - \bar{x}_{comb})^2}{n_1 + n_2} $

Since $\bar{x}_1 = \bar{x}_{comb} = 8$ and $\bar{x}_2 = \bar{x}_{comb} = 8$, the terms involving the difference between means are zero:

$ (\bar{x}_1 - \bar{x}_{comb})^2 = (8 - 8)^2 = 0 $ $ (\bar{x}_2 - \bar{x}_{comb})^2 = (8 - 8)^2 = 0 $

The formula simplifies to:

$ s_{comb}^2 = \frac{n_1 s_1^2 + n_2 s_2^2}{n_1 + n_2} $

Final Calculation

Substitute the known values into the simplified formula:

$ s_{comb}^2 = \frac{(10 \times 1) + (20 \times 4)}{10 + 20} $ $ s_{comb}^2 = \frac{10 + 80}{30} $ $ s_{comb}^2 = \frac{90}{30} $ $ s_{comb}^2 = 3 $

Result

The variance of the combined dataset is 3.

Prepp Team · Answer

2

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5

Prepp Team · Answer

1.5

List - I	List - II
(A) Coefficient of Dispersion	(I) Standard deviation as a percentage of mean
(B) Standard Deviation	(II) Mean of squared deviations of individual scores from mean
(C) Coefficient of variation	(III) Variance in terms of mean
(D) Variance	(IV) Positive square root of variance