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Standard Deviation Calculator
🎓 Student0
Standard Deviation
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📐 Formula
Sample Standard Deviation (s): √[Σ(xᵢ − x̄)² ÷ (n−1)]
Population Standard Deviation (σ): √[Σ(xᵢ − x̄)² ÷ n]
When to use each:
Sample (n−1): Use when your data is a sample drawn from a larger population (e.g., a survey of 30 students from a class of 200). The n−1 denominator (Bessel's correction) corrects for the bias in estimating the population variance from a sample.
Population (n): Use when you have data for the entire population (e.g., exam marks for every student in your class).
❓ Frequently Asked Questions
What does standard deviation tell me about my grades?
Standard deviation measures how spread out your scores are from the average. A low standard deviation (e.g., 5) means your marks are consistent. A high standard deviation (e.g., 20) means some subjects are significantly better or worse than others, signalling where to focus improvement.
What is the difference between variance and standard deviation?
Variance is the average of squared differences from the mean. Standard deviation is the square root of variance, bringing the result back to the same units as the original data. Standard deviation is easier to interpret: for exam marks out of 100, an SD of 12 means most marks fall within ±12 of the mean.
Why does sample standard deviation use n−1 instead of n?
When estimating population variance from a sample, using n would systematically underestimate the true variance (because the sample mean is closer to the sample data than the true population mean is). Subtracting 1 from n — known as Bessel's correction — corrects this bias. For large samples, the difference is negligible.
How is standard deviation used in statistics assignments?
Standard deviation is fundamental in hypothesis testing, confidence intervals, z-scores, and normal distribution problems. In research methods courses, it is used to describe data spread. In finance and economics, it measures risk. Understanding when to use sample vs population SD is a common exam question.