What is Standard Deviation?
Standard deviation measures how spread out the values in a data set are around the mean. It is the square root of variance, expressed in the same units as the original data.
Standard deviation (σ) is the square root of the average squared deviation from the mean: σ = √(Σ(x−μ)²/n). A larger σ means more spread-out data.
- •Data points cluster close to the mean
- •Narrow, tall bell curve
- •Example: 48, 50, 49, 51, 50
- •Data points are spread far from the mean
- •Wide, flat bell curve
- •Example: 10, 90, 45, 5, 95
Try it: interactive calculator
Step-by-step worked examples
Find the standard deviation of: 2, 4, 4, 4, 5, 5, 7, 9.
Mean = (2+4+4+4+5+5+7+9)/8 = 5 Squared deviations: 9,1,1,1,0,0,4,16 (sum = 32) Variance = 32/8 = 4 SD = √4 = 2
Find the standard deviation of test scores: 60, 70, 80, 90, 100.
Mean = 80 Squared deviations: 400,100,0,100,400 (sum = 1000) Variance = 1000/5 = 200 SD = √200 ≈ 14.14
A data set has variance 25. What is its standard deviation?
Variance = 25 SD = √25 = 5
Flashcards
Quick quiz
Q1.A data set has variance 16. What is its standard deviation?
Q2.What is the standard deviation of 5, 5, 5, 5?
Q3.Which best describes standard deviation?
Q4.As data spread increases, standard deviation…
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Common mistakes
Confusing standard deviation with the mean. — Correct: SD measures spread, not central location — it's the square root of variance.
Forgetting to square deviations before averaging. — Correct: Deviations must be squared first (so negatives don't cancel out), then averaged, then square-rooted.
Thinking SD can be negative for 'low' data. — Correct: SD is always ≥ 0 — it's a square root, so it can never be negative.
Using variance and SD interchangeably. — Correct: They're related but different: variance is in squared units, SD is in original units.
FAQ
What is standard deviation?
Standard deviation is a statistic that shows how spread out data values are from the mean.
What is the formula for standard deviation?
σ = √(Σ(x−μ)²/n) — the square root of the average squared deviation from the mean.
What are examples of standard deviation?
The data set 2,4,4,4,5,5,7,9 has SD = 2; the data set 60,70,80,90,100 has SD ≈ 14.14.
How do you calculate standard deviation?
Find the mean, square each value's deviation from the mean, average those squares (variance), then take the square root.