Statistics Example

Easy DifficultyFE Mathematics and Statistics

Compute the sample mean and sample standard deviation for a small data set. These are building blocks for confidence intervals and regression.

Concept

For a sample x_{1}, x_{2}, \dots, x_{n}, the sample mean is the average. The sample variance uses divisor n - 1 (degrees of freedom); the sample standard deviation is the square root of the variance. \overset{x}{ˉ} = \frac{1}{n} i = 1 \sum n x_{i} s^{2} = \frac{1}{n - 1} i = 1 \sum n (x_{i} - \overset{x}{ˉ})^{2} s = s^{2}

A sample of five measurements is: 12, 15, 18, 22, 23. Find: Sample mean \overset{x}{ˉ} Sample standard deviation s

Data: 12, 15, 18, 22, 23 n = 5

\overset{x}{ˉ} = \frac{1}{n} i = 1 \sum n x_{i} \sum x_{i} = 12 + 15 + 18 + 22 + 23 = 90, n = 5 \overset{x}{ˉ} = \frac{90}{5} = 18

Variance uses n - 1 = 4 in the denominator (sample): (12 - 18)^{2} + (15 - 18)^{2} + (18 - 18)^{2} + (22 - 18)^{2} + (23 - 18)^{2} = 36 + 9 + 0 + 16 + 25 = 86 s^{2} = \frac{86}{4} = 21.5 s = 21.5 \approx 4.64

(1) Sample mean: \overset{x}{ˉ} = 18 (2) Sample standard deviation: s \approx 4.64

\overset{x}{ˉ} = \frac{1}{n} \sum x_{i}, s = \frac{1}{n - 1} \sum (x_{i} - \overset{x}{ˉ})^{2} Confidence interval for the mean (when population is approximately normal): \overset{x}{ˉ} \pm t_{α /2, n - 1} \frac{s}{n}, where t_{α /2, n - 1} is the critical value from the t -distribution.