sample standard deviation example

2 min read 13-03-2025

Understanding standard deviation is crucial in statistics. It measures the spread or dispersion of a dataset around its mean (average). While population standard deviation describes the entire population, sample standard deviation estimates this spread using only a subset of the population. This article will illustrate sample standard deviation with clear examples.

What is Sample Standard Deviation?

The sample standard deviation (represented by 's') quantifies how much individual data points deviate from the sample mean. A larger sample standard deviation indicates greater variability, while a smaller value suggests data points cluster closer to the mean. It's a vital tool for understanding data reliability and making inferences about a larger population. We use sample standard deviation because obtaining data for an entire population is often impractical or impossible.

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

Let's learn how to calculate sample standard deviation with a simple example. Suppose we have a sample of exam scores: {70, 75, 80, 85, 90}.

Step 1: Calculate the mean (average).

Add all the scores together and divide by the number of scores: (70 + 75 + 80 + 85 + 90) / 5 = 80. The mean is 80.

Step 2: Find the deviations from the mean.

Subtract the mean from each score:

70 - 80 = -10
75 - 80 = -5
80 - 80 = 0
85 - 80 = 5
90 - 80 = 10

Step 3: Square the deviations.

Squaring eliminates negative signs, ensuring all values contribute positively to the overall variance:

(-10)² = 100
(-5)² = 25
0² = 0
5² = 25
10² = 100

Step 4: Calculate the sample variance.

Sum the squared deviations and divide by (n-1), where 'n' is the sample size (5 in this case). We use (n-1) because it provides a better, unbiased estimate of the population variance.

(100 + 25 + 0 + 25 + 100) / (5 - 1) = 62.5. This is the sample variance.

Step 5: Calculate the sample standard deviation.

Take the square root of the sample variance: √62.5 ≈ 7.9. This is the sample standard deviation.

Example 2: A Larger Dataset

Let's consider a slightly more complex example. Imagine we're analyzing the daily rainfall (in mm) for a week: {10, 12, 15, 8, 11, 13, 9}.

Following the same steps:

Mean: (10 + 12 + 15 + 8 + 11 + 13 + 9) / 7 ≈ 11.14 mm
Deviations: Calculate the difference between each rainfall value and the mean.
Squared Deviations: Square each deviation.
Sample Variance: Sum the squared deviations and divide by (7-1) = 6.
Sample Standard Deviation: Take the square root of the sample variance. You'll find a value representing the typical deviation from the average daily rainfall.

Interpreting Sample Standard Deviation

In both examples, the sample standard deviation provides a measure of the data's spread. A higher standard deviation means the data points are more spread out; a lower standard deviation indicates they are clustered closer to the mean. This is useful for comparing the variability of different datasets.

Applications of Sample Standard Deviation

Sample standard deviation has wide-ranging applications across various fields:

Quality Control: Monitoring the consistency of manufactured products.
Finance: Assessing the risk associated with investments.
Healthcare: Analyzing the variability in patient outcomes.
Environmental Science: Studying the fluctuations in environmental variables.
Research: Determining the reliability of experimental results.

Conclusion

Sample standard deviation is a fundamental statistical concept. Understanding how to calculate and interpret it is essential for analyzing data and making informed decisions across numerous disciplines. By following the steps outlined above, you can confidently analyze your own datasets and gain valuable insights. Remember, using (n-1) in the variance calculation provides a more accurate estimate of the population standard deviation, particularly important when dealing with smaller sample sizes.