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Calculate Sample Standard Deviation for N 5

Reviewed by Calculator Editorial Team

Sample standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data. When you have a sample size of 5, the calculation follows a specific formula that accounts for the degrees of freedom in the sample.

What is Sample Standard Deviation?

Sample standard deviation measures the dispersion of individual data points from their mean in a sample. Unlike population standard deviation, which uses the population mean, sample standard deviation uses the sample mean and adjusts for degrees of freedom.

For small samples like n=5, the sample standard deviation provides a more accurate estimate of the population standard deviation. It's commonly used in quality control, scientific research, and data analysis to understand variability in measurements.

Formula

The formula for sample standard deviation (s) with n=5 is:

s = √[Σ(xi - x̄)² / (n - 1)]

Where:

  • xi = each individual data point
  • x̄ = sample mean
  • n = sample size (5 in this case)

The denominator (n - 1) accounts for the degrees of freedom in the sample, providing an unbiased estimate of the population standard deviation.

How to Calculate

  1. Collect your 5 data points
  2. Calculate the sample mean (x̄) by summing all values and dividing by 5
  3. For each data point, subtract the mean and square the result
  4. Sum all these squared differences
  5. Divide the sum by (5 - 1) = 4
  6. Take the square root of the result to get the sample standard deviation

Example Calculation

Example with n=5

Data points: 10, 12, 11, 13, 9

  1. Mean (x̄) = (10 + 12 + 11 + 13 + 9) / 5 = 55 / 5 = 11
  2. Squared differences:
    • (10-11)² = 1
    • (12-11)² = 1
    • (11-11)² = 0
    • (13-11)² = 4
    • (9-11)² = 4
  3. Sum of squared differences = 1 + 1 + 0 + 4 + 4 = 10
  4. Divide by (5-1) = 4 → 10 / 4 = 2.5
  5. Square root → √2.5 ≈ 1.581

Sample standard deviation ≈ 1.581

Interpreting Results

A sample standard deviation of 1.581 for n=5 indicates that, on average, the data points deviate about 1.581 units from the sample mean of 11. This suggests relatively low variability in the sample.

In practical terms, this means the measurements are generally close to each other, which might be expected in a controlled experiment or when measuring similar items.

FAQ

What's the difference between sample and population standard deviation?

Sample standard deviation divides by (n-1) to provide an unbiased estimate of the population standard deviation. Population standard deviation divides by n.

Why is n-1 used in the denominator?

Using n-1 accounts for degrees of freedom, providing a more accurate estimate of the population standard deviation when working with samples.

Can I use this calculator for any sample size?

This calculator is specifically designed for n=5. For other sample sizes, the formula would need to be adjusted.