Standard Deviation Calculator

Calculating Sample Standard Deviation: A Step-by-Step Example

Consider the following scores from a final exam in a statistics course: 75, 61, 83, 65, 96

Step 1: Calculate the Mean

The mean, \( \bar{x} \), is the sum of all data points divided by the number of data points. In this case, we have:

$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = \frac{75 + 61 + 83 + 65 + 96}{5} = \frac{380}{5} = 76 $$

Therefore, the mean final exam score in this statistics course is \( \bar{x} = 76 \).

Step 2: Calculate the Sample Variance

The sample variance, \( s^2 \), measures the average squared deviation of each data point from the sample mean. It is calculated as follows:

$$ s^2 = \frac{1}{n - 1} \sum_{i=1}^{n}(x_i - \bar{x})^2 = \frac{1}{n - 1} \left[(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \cdots + (x_n - \bar{x})^2\right] $$

$$ = \frac{1}{5-1} \left[(75 - 76)^2 + (61 - 76)^2 + (83 - 76)^2 + (65 - 76)^2 + (96 - 76)^2\right] $$

$$ = \frac{1}{4} \left[1 + 225 + 49 + 121 + 400\right] = \frac{796}{4} = 199 $$

Therefore, the sample variance of the final exam scores in this statistics course is \( s^2 = 199 \).

Step 3: Calculate the Sample Standard Deviation

The sample standard deviation, \( s \), is defined as the square root of the sample variance:

$$ s = \sqrt{199} \approx 14.11 $$

Therefore, the sample standard deviation of the final exam scores in this statistics course is approximately \( 14.11 \). This value indicates that, on average, the final exam scores deviate by about 14.11 units from the sample mean.