Sample Mean and Sample Standard Deviation: Concepts and Calculation

Describing Data Numerically

Sample Mean and Sample Standard Deviation

The sample mean and sample standard deviation are fundamental statistics used to describe the center and variability of a dataset. These measures help summarize data and are widely used in introductory statistics courses.

• Sample Mean (\( \bar{x} \)): The sample mean is a measure of the central tendency of the sample. It is calculated as the sum of all sample values divided by the number of values in the sample.

• Sample Standard Deviation (\( s \)): The sample standard deviation quantifies the spread or variability of the sample values around the mean. It is roughly the average distance of sample points from the mean.

Definitions and Formulas

• Sample Mean Formula:

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

• Sample Standard Deviation Formula:

$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$

• Sample Variance Formula: The sample variance is the square of the sample standard deviation.

$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$

Step-by-Step Calculation Recipe

To calculate the sample standard deviation, follow these steps:

1. Compute the sample mean \( \bar{x} \).

2. Find the deviation of each sample value from the mean: \( x_i - \bar{x} \).

3. Square each deviation: \( (x_i - \bar{x})^2 \).

4. Sum all squared deviations.

5. Divide the sum by \( n-1 \) to get the sample variance.

6. Take the square root of the variance to obtain the sample standard deviation.

Example Calculation

Given sample: 4, 4, 5, 6, 8, 14, 22 (\( n = 7 \))

• Step 1: Calculate the sample mean:

$\bar{x} = \frac{4 + 4 + 5 + 6 + 8 + 14 + 22}{7} = \frac{63}{7} = 9$

• Step 2: Calculate deviations from the mean:

Value (\( x_i \))	Deviation (\( x_i - \bar{x} \))	Squared Deviation (\( (x_i - \bar{x})^2 \))
4	-5	25
4	-5	25
5	-4	16
6	-3	9
8	-1	1
14	5	25
22	13	169

• Step 3: Sum the squared deviations:

$25 + 25 + 16 + 9 + 1 + 25 + 169 = 270$

• Step 4: Divide by \( n-1 = 6 \) to get the sample variance:

$s^2 = \frac{270}{6} = 45$

• Step 5: Take the square root to get the sample standard deviation:

$s = \sqrt{45} \approx 6.7082$

Interpretation

• Sample Mean (\( \bar{x} \)): Represents the "center of mass" or balancing point of the sample values.

• Sample Standard Deviation (\( s \)): Measures how much the sample values differ from the mean; it is sensitive to extreme values (outliers).

Additional Example (Table Format)

For a sample of 11 values, the calculation can be organized in a table for clarity:

Raw Data (\( x_i \))	Deviation (\( x_i - \bar{x} \))	Squared Deviation (\( (x_i - \bar{x})^2 \))
15	-1	1
357	87	7569
1225	94	8836
5	-3	9
9	-1	1
4479	49	2401
1	-1	1
49	0	0
1	-2	4
1	-4	16
57	7	49

Sum of squared deviations: 2994 Sample variance: $s^2 = \frac{2994}{10} = 299.4$ Sample standard deviation: $s = \sqrt{299.4} \approx 17.3031$

Notes and Properties

• The sample standard deviation is sensitive to extreme values; large deviations have a significant impact on the result.

• Organizing calculations in a table helps systematically compute deviations and squared deviations.

• Software tools (e.g., Excel's STDEV function) can automate these calculations.

Example Application: In real-world data analysis, the sample mean and standard deviation are used to summarize datasets, compare groups, and assess variability.

Additional info: The notes reference Excel for calculation, which is a common practice in statistics courses for handling larger datasets.