How to Find Sample Variance: A Step‑by‑Step Guide
2025-08-31 - 11:37
What Is Sample Variance?
In descriptive statistics, sample variance measures how far each observation in a data set deviates from the sample mean. It is the cornerstone for assessing data spread, estimating population variance, and performing hypothesis tests.
Why Use Sample Variance Instead of Population Variance?
When you only have a subset of the entire population, you calculate sample variance (denoted s²) to obtain an unbiased estimate of the true population variance (σ²). The formula includes a correction factor—dividing by n − 1 instead of n—to compensate for the smaller sample size.
Step‑by‑Step Calculation
Follow these five simple steps to compute sample variance manually:
Step 1: Collect the Data
List all observations: x₁, x₂, …, xₙ. For example, a sample of five test scores might be 78, 85, 92, 88, and 81.
Step 2: Find the Sample Mean (σ̅)
Calculate the average: mean = (Σxᵢ) / n. Using the example: (78+85+92+88+81) / 5 = 84.8.
Step 3: Compute Each Deviation
Subtract the mean from each observation: (xᵢ − mean). For the first score: 78 − 84.8 = ‑6.8.
Step 4: Square the Deviations
Square each deviation to eliminate negative signs: (xᵢ − mean)². Continuing the example: (‑6.8)² = 46.24.
Step 5: Sum the Squared Deviations and Divide
Add all squared deviations and divide by n − 1:
s² = Σ(xᵢ − mean)² / (n − 1)
For our data: (46.24 + 2.56 + 51.84 + 10.24 + 14.44) / 4 = 31.58.
Common Mistakes to Avoid
Using n Instead of n − 1: This underestimates variance, especially with small samples.
Forgetting to Square Deviations: Adding raw deviations cancels out positive and negative values, leading to a zero result.
Mixing Units: Ensure all data points share the same unit before calculating.
Practical Tips for Quick Computation
• Use a spreadsheet: Functions like VAR.S() in Excel automate the process.
• Leverage statistical software: R’s var() or Python’s numpy.var(..., ddof=1) handle large datasets efficiently.
• Check assumptions: Sample variance assumes data are drawn from a roughly normal distribution for inferential purposes.
Conclusion
Mastering the calculation of sample variance equips you with a fundamental tool for data analysis. Whether you compute it by hand or use software, remembering the n − 1 denominator and the squaring step ensures accurate, unbiased results that can be confidently applied in research, business, and everyday decision‑making.
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