vmbo havo vwo
Standard deviation calculator
Quick answer
Standard deviation (Dutch: standaardafwijking) measures spread. Population: σ = √(Σ(x − x̄)² / n). Sample (Dutch: steekproef): same sum, divide by n − 1.
For 2, 5, 5, 8, 11 the mean is 6.2. σ ≈ 3.0594; s ≈ 3.4205. Center measures first on the mean calculator.
Calculator
Choose population or sample. Paste or type numbers, separated by a comma or a space. Comma or point is fine. With one number, s is impossible (divide by zero).
What is standard deviation?
Standard deviation is spread: how far the numbers sit from their mean. It is not the mean, not the median, and not the mode. Those three live on the mean calculator. Here you compute only σ or s — the mean is a prerequisite, not the answer.
| Symbol | Divide by | When in class |
|---|---|---|
| σ (sigma) | n | Population: all observations (whole class, every measurement) |
| s | n − 1 | Sample: you estimate the spread of a larger population |
On vmbo (more practical track), havo and vwo (middle and university-preparatory tracks) the task usually says whether you have the whole group or a sample. Nothing explicit: look for “population” or “sample”, or use the mode that matches your book. Both modes stay on this one page.
Variance is the number under the root (sum of squares divided by n or n − 1). Range is simply max − min. Quartiles (boxplot, IQR) neither: Q2 is the median.
Examples by level
Scores 2, 5, 5, 8, 11. How far from the mean?
- Mean31 ÷ 5 = 6.2
- Deviations2 is 4.2 below 6.2; 11 is 4.8 above
- SpreadNot all equally close to the center
The square-root formula comes next.
Same list. What is σ (divide by n)?
- Mean6.2
- Sum of squares46.8
- Divide by n46.8 ÷ 5 = 9.36
- Square rootσ ≈ 3.0594
Same list, now s (divide by n − 1).
- n − 15 − 1 = 4
- Divide46.8 ÷ 4 = 11.7
- Square roots ≈ 3.4205
One observation: n − 1 = 0, divide by zero is impossible. No extra rule.
Common mistakes
Wrong
Treating standard deviation as sum ÷ count.
Right
That is the mean. Open the mean calculator for the center; here only spread.
Wrong
Swapping σ and s: dividing a sample by n (or a population by n − 1) and still writing the other symbol.
Right
Choose first: population → σ with n; sample → s with n − 1. Same list 2, 5, 5, 8, 11: σ ≈ 3.0594 and s ≈ 3.4205 are not interchangeable.
Wrong
Computing s from a single observation.
Right
n − 1 = 0: divide by zero is impossible. For s you need at least two numbers. With one number only σ (divide by n = 1) is mathematically possible — and it is 0.
Practice
vmbo 2, 5, 5, 8, 11. What is the mean before you do σ?
6.2
31 ÷ 5 = 6.2. Same list as on the mean page.
havo Same list, population. What is σ? (4 decimals)
3.0594
Sum of squares 46.8; 46.8 ÷ 5 = 9.36; √9.36 ≈ 3.0594.
havo 5, 5, 5. What is σ?
0
Everything sits on the mean; the deviations are 0.
vwo 2, 5, 5, 8, 11, sample. What is s? (4 decimals)
3.4205
46.8 ÷ 4 = 11.7; √11.7 ≈ 3.4205.
vwo One number, sample. What happens?
Divide by zero is impossible
n − 1 = 0. You need at least two numbers for s.
In the Netherlands
CBS uses mean and median for the center of a distribution, for example income. Standard deviation is the spread around that center. Definitions sit at CBS. σ and s are in this one calculator: it is the same question (how wide is the list?); only the divisor differs.
Next step
- How does this work? The explanation and examples are further up this page.
- Test yourself Mean practice, with an explanation after every answer.
Learning path
- Learn first: Mean calculator Arithmetic mean of a list of numbers.
- Next: Normal distribution calculator The three rules of thumb: 68%, 95% and almost 100%.
Frequently asked questions
How do you calculate the population standard deviation?
First the mean x̄, then the sum of (x − x̄)², divide by n, take the square root. Formula: σ = √(Σ(x − x̄)² / n). For 2, 5, 5, 8, 11, x̄ = 6.2, sum of squares 46.8, 46.8 ÷ 5 = 9.36, σ ≈ 3.0594.
What is the difference between population and sample?
Population (whole group): divide by n, symbol σ. Sample (Dutch: steekproef — part of a larger group): divide by n − 1, symbol s. Same list 2, 5, 5, 8, 11: σ ≈ 3.0594 and s ≈ 3.4205. With one number, s is impossible (n − 1 = 0).
When do you divide by n − 1?
When the task calls it a sample, or when you estimate the spread of a larger population. On vmbo/havo/vwo that is often in the wording; if it says “all pupils” or “the whole measurement”, choose n and σ. Unsure: pick the mode that matches your textbook — both are on this page.
Is standard deviation the same as the mean?
No. The mean is sum ÷ count (the centre). Standard deviation is how far the numbers sit from that centre. Median and mode: mean calculator. Without x̄ you cannot fill in σ or s.
What if every number is the same?
Then σ = 0 (and s = 0 with at least two equal numbers): every deviation is zero. Example: 5, 5, 5.
Where do you calculate quartiles or the range?
Not here. Range is max − min. Q2 is the median.