Calculate Sample Standard Deviation

Enter your data points, separated by commas.

Understanding Sample Standard Deviation

The sample standard deviation is a crucial statistical measure that quantifies the amount of variation or dispersion in a set of data points. In simpler terms, it tells us how spread out the individual data points are from the average (mean) of the sample. A low standard deviation indicates that the data points tend to be close to the mean, suggesting uniformity. Conversely, a high standard deviation implies that the data points are spread out over a wider range of values.

It's important to distinguish between sample standard deviation and population standard deviation. The sample standard deviation is calculated from a sample of a larger population, and it uses a denominator of n-1 (where n is the number of data points) to provide a less biased estimate of the population standard deviation. The population standard deviation is calculated when you have data for the entire population and uses n as the denominator.

How to Calculate Sample Standard Deviation

The formula for sample standard deviation (often denoted by s) is:

s = sqrt( Σ(xᵢ - x̄)² / (n - 1) )

Where:

• s is the sample standard deviation.
• Σ (Sigma) is the summation symbol, meaning "sum of".
• xᵢ represents each individual data point in your sample.
• x̄ (x-bar) is the mean (average) of the sample data points.
• n is the number of data points in the sample.
• sqrt(...) denotes the square root.

The calculation involves the following steps:

1. Calculate the Mean (x̄): Sum all the data points and divide by the number of data points (n).
2. Calculate Deviations: Subtract the mean (x̄) from each individual data point (xᵢ – x̄).
3. Square Deviations: Square each of the deviations calculated in the previous step (xᵢ – x̄)².
4. Sum Squared Deviations: Add up all the squared deviations. This is the sum of squares (Σ(xᵢ – x̄)²).
5. Calculate Variance: Divide the sum of squared deviations by (n – 1). This gives you the sample variance.
6. Take the Square Root: Calculate the square root of the sample variance to obtain the sample standard deviation (s).

When is Sample Standard Deviation Used?

Sample standard deviation is widely used in various fields for its ability to estimate population variability from a subset of data. Common use cases include:

• Quality Control: Monitoring the consistency of manufactured products.
• Scientific Research: Analyzing experimental data to understand variability within a study group.
• Financial Analysis: Assessing the risk or volatility of an investment portfolio.
• Social Sciences: Studying variations in survey responses or demographic data.
• General Data Analysis: Understanding the spread of any dataset when you only have a sample.

Example Calculation

Let's calculate the sample standard deviation for the following data points: 5, 8, 11, 14, 17.

• n = 5
• Mean (x̄) = (5 + 8 + 11 + 14 + 17) / 5 = 55 / 5 = 11
• Deviations (xᵢ – x̄):
  • 5 – 11 = -6
  • 8 – 11 = -3
  • 11 – 11 = 0
  • 14 – 11 = 3
  • 17 – 11 = 6
• Squared Deviations (xᵢ – x̄)²:
  • (-6)² = 36
  • (-3)² = 9
  • 0² = 0
  • 3² = 9
  • 6² = 36
• Sum of Squared Deviations: 36 + 9 + 0 + 9 + 36 = 90
• Sample Variance: 90 / (5 – 1) = 90 / 4 = 22.5
• Sample Standard Deviation (s): sqrt(22.5) ≈ 4.74

So, the sample standard deviation for this dataset is approximately 4.74. This indicates a moderate spread of the data points around the mean of 11.

Requires at least 2 data points

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