Combined Standard Deviation Calculation Explained

This problem requires calculating the standard deviation for a combined data set, given the characteristics (size, mean, standard deviation) of two individual data sets. We aim to find the standard deviation for the total group comprising 15 observations.

Data Set Details Provided

The information provided for the two distinct data sets is as follows:

• Data Set 1: Represents $n_1 = 9$ observations, with an arithmetic mean $\bar{x}_1 = 3$ and a standard deviation $\sigma_1 = 3$.
• Data Set 2: Represents $n_2 = 6$ observations, with an arithmetic mean $\bar{x}_2 = 3$ and a standard deviation $\sigma_2 = 4$.

Variance and Standard Deviation Formulas

The fundamental relationship connecting variance ($\sigma^2$), standard deviation ($\sigma$), mean ($\bar{x}$), and the sum of squares ($\sum x^2$) for a data set of size $n$ is:

$$ \sigma^2 = \frac{\sum x^2}{n} - (\bar{x})^2 $$

To proceed with the combined calculation, we first rearrange this formula to find the sum of squares ($\sum x^2$):

$$ \frac{\sum x^2}{n} = \sigma^2 + (\bar{x})^2 $$

$$ \sum x^2 = n(\sigma^2 + (\bar{x})^2) $$

The strategy involves calculating the sum of squares for each individual data set, then summing them to get the total sum of squares for the combined data set. We also need the combined mean.

Sum of Squares Calculation

For Data Set 1:

• First, calculate the variance for this set: $\sigma_1^2 = 3^2 = 9$.
• Next, calculate the sum of squares using the formula $\sum x^2 = n(\sigma^2 + (\bar{x})^2)$:
  $\sum x_1^2 = n_1(\sigma_1^2 + \bar{x}_1^2) = 9(9 + 3^2) = 9(9 + 9) = 9(18) = 162$.

For Data Set 2:

• Calculate the variance for this set: $\sigma_2^2 = 4^2 = 16$.
• Calculate the sum of squares:
  $\sum x_2^2 = n_2(\sigma_2^2 + \bar{x}_2^2) = 6(16 + 3^2) = 6(16 + 9) = 6(25) = 150$.

Combined Mean Calculation

The combined arithmetic mean ($\bar{X}$) for two data sets is calculated using the formula:

$$ \bar{X} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2} $$

Plugging in the values:

$$ \bar{X} = \frac{(9 \times 3) + (6 \times 3)}{9 + 6} = \frac{27 + 18}{15} = \frac{45}{15} = 3 $$

Therefore, the combined mean ($\bar{X}$) is 3.

Combined Variance Calculation

The total sum of squares for the combined data set ($N = n_1 + n_2 = 15$) is obtained by adding the sums of squares from the individual sets:

$$ \sum x^2_{combined} = \sum x_1^2 + \sum x_2^2 = 162 + 150 = 312 $$

Now, we can compute the combined variance ($\sigma^2_{combined}$) using the standard formula $\sigma^2 = \frac{\sum x^2}{N} - (\bar{X})^2$:

$$ \sigma^2_{combined} = \frac{\sum x^2_{combined}}{N} - (\bar{X})^2 $$

$$ \sigma^2_{combined} = \frac{312}{15} - (3)^2 $$

$$ \sigma^2_{combined} = \frac{312}{15} - 9 $$

To perform the subtraction, we find a common denominator:

$$ \sigma^2_{combined} = \frac{312 - (9 \times 15)}{15} = \frac{312 - 135}{15} = \frac{177}{15} $$

Combined Standard Deviation Calculation

Finally, the combined standard deviation ($\sigma_{combined}$) is the square root of the calculated combined variance:

$$ \sigma_{combined} = \sqrt{\sigma^2_{combined}} = \sqrt{\frac{177}{15}} $$

Final Result Verification

The calculated standard deviation for the combined data set is $\sqrt{\frac{177}{15}}$. This result corresponds exactly to the first option provided.