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

Prepp Team · Answer

5

Prepp Team · Answer

1.5