Understanding Combined Statistics
When you have data from two separate groups and want to analyze them as one large dataset, you cannot simply average their means or standard deviations. Instead, we use specific formulas to account for the size of each group.
The Given Data
- Sample 1: Size $n_1 = 50$, Mean $\bar{x}_1 = 54.1$, Standard Deviation $\sigma_1 = 8$
- Sample 2: Size $n_2 = 100$, Mean $\bar{x}_2 = 50.3$, Standard Deviation $\sigma_2 = 7$
1. Calculating Combined Mean ($\bar{x}_{12}$)
The combined mean is the weighted average of the individual means: $$\bar{x}_{12} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$$ $$\bar{x}_{12} = \frac{50(54.1) + 100(50.3)}{50 + 100} = \frac{2705 + 5030}{150} = \frac{7735}{150} \approx 51.57$$
2. Calculating Combined Standard Deviation ($\sigma_{12}$)
To find the combined standard deviation, we first define $d_1 = \bar{x}_1 - \bar{x}_{12}$ and $d_2 = \bar{x}_2 - \bar{x}_{12}$.
- $d_1 = 54.1 - 51.57 = 2.53$
- $d_2 = 50.3 - 51.57 = -1.27$
The formula is: $$\sigma_{12} = \sqrt{\frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}}$$
Substitute the values:
- $\sigma_1^2 = 64$, $\sigma_2^2 = 49$
- $d_1^2 = 6.4009$, $d_2^2 = 1.6129$
$$\sigma_{12} = \sqrt{\frac{50(64 + 6.4009) + 100(49 + 1.6129)}{150}}$$ $$\sigma_{12} = \sqrt{\frac{50(70.4009) + 100(50.6129)}{150}}$$ $$\sigma_{12} = \sqrt{\frac{3520.045 + 5061.29}{150}} = \sqrt{\frac{8581.335}{150}} = \sqrt{57.2089} \approx 7.56$$
Thus, the combined mean is 51.57 and the combined standard deviation is approximately 7.56.