Standard Deviation Calculator

Paste your data set, click Calculate. The standard deviation calculator form has one box, pre-filled with 2, 4, 4, 4, 5, 5, 7, 9 — and it returns four rows at once.

• Data set (comma-separated numbers) — every value in your list, any order. The standard deviation calculator does the sorting and summing internally.
• Four rows come back: Standard deviation (sample, n−1) ≈ 2.1381, Variance (sample, n−1) ≈ 4.5714, Mean = 5, Count = 8.

Type your own list over the default — temperatures, test scores, reaction times. Decimals and negatives are welcome; keep the values separated by commas. Both spread rows use the sample formulas with $n-1$ in the denominator, the standard choice when your data is a sample rather than a complete population (why, below).

This page is more than a static standard deviation calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. “How spread out is 4, 8, 15, 16, 23, 42?” The AI computes the standard deviation step by step.
• Take a photo of your homework. The AI reads the data set from the page and shows every step.
• Ask about the ideas, not just the number. “What does a standard deviation of 2 actually mean?” The AI explains — and you can check every number against the standard deviation calculator above, so nothing is guessed.
• Work backwards. Ask which of two data sets is more consistent, and the AI compares their spreads step by step.
• Keep asking follow-ups. Ask “why?” as often as you need — the standard deviation calculator page keeps teaching.

The mean tells you where the center is; the standard deviation tells you how far the values typically wander from it. It is a typical distance from the mean, in the data's own units.

Two data sets, both with mean 5:

• Set A: 4, 4, 5, 5, 7 — every value sits within 2 of the mean.
• Set B: 1, 4, 5, 6, 9 — same mean, but the ends reach 4 away.

Same center, different consistency — and only the spread row would show it. Type both into the standard deviation calculator and watch the numbers separate. A small $s$ means values huddle (consistent manufacturing, reliable free throws); a large $s$ means values scatter.

One more reading, for bell-shaped data: the 68-95-99.7 rule. Roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. A mean of 70 with $s = 3$ predicts most scores between 64 and 76. The rule is exact only for normal distributions — the normal distribution page on this site draws the full curve — but it is a good rough guide whenever data piles up in one hump.

Every standard deviation calculator runs one formula:

$s=\sqrt{\dfrac{\sum(x_i-\bar{x})^2}{n-1}}$

The recipe behind it, in four moves:

1. Find the mean $\bar{x}$ of the data.
2. Square each deviation $(x_i - \bar{x})^2$ and add them up.
3. Divide by $n-1$ — that quotient is the sample variance $s^2$.
4. Take the square root — you are back in the original units; that is $s$.

Why square at all? Raw deviations always sum to zero — the positives and the negatives cancel. Squaring makes every deviation positive, so spread cannot hide. The final square root undoes the squaring of the units.

The default data set, every step visible. Data: 2, 4, 4, 4, 5, 5, 7, 9.

1. Mean: the sum is $40$ over $n = 8$ values, so $\bar{x} = 40 \div 8 = 5$.
2. Deviations from 5: $-3, -1, -1, -1, 0, 0, 2, 4$.
3. Square them: $9, 1, 1, 1, 0, 0, 4, 16$. Sum: $9+1+1+1+0+0+4+16 = 32$.
4. Sample variance: $s^2 = 32 \div (8-1) = 32/7 \approx 4.5714$.
5. Sample standard deviation: $s = \sqrt{4.5714} \approx 2.1381$.

A three-column table — value, deviation, squared deviation — keeps the bookkeeping tidy, and that table is exactly what the AI box shows if you ask it to walk the same data.

In words: these eight values average 5, and they typically sit about 2.14 away from 5. The standard deviation calculator above confirms both decimals in its first two rows.

Divide by $n-1$ instead of $n$? The adjustment has a name — Bessel's correction — and a one-sentence reason: a sample understates the true spread, so the estimate gets nudged up.

Why does a sample understate? Deviations are measured from the sample's own mean $\bar{x}$, and no number sits closer to your sample than its own average — the sum of squared deviations around $\bar{x}$ is the smallest it can be. The true center $\mu$ usually lies a bit further from your sample's values, so deviations measured around $\mu$ would be larger. Dividing by $n-1$ instead of $n$ compensates.

When would you divide by $n$? When your data is the whole population — every fifth grader in one school, not a sample of fifth graders. Then there is no larger group being estimated, and the population formula $\sigma=\sqrt{\sum(x_i-\mu)^2/n}$ is the right one.

The standard deviation calculator on this page reports the sample version, $n-1$ — the safer default whenever data is drawn from something bigger. The gap shrinks as samples grow: with $n = 100$, the correction barely matters.

1. Stopping at the variance. After the division, the number is in squared units — $s^2 \approx 4.5714$ for the example data. If the answer must sit beside the data in its own units (points, dollars, degrees), take the square root: $s \approx 2.1381$. The standard deviation calculator lists both rows precisely to keep them apart.
2. Dividing by $n$ on a sample. For sample data the denominator is $n-1$. Dividing by $n$ shrinks the result — for the example, $32 \div 8 = 4$ instead of $32 \div 7 \approx 4.5714$ — and understates the spread.
3. Skipping the squaring. The raw deviations $-3, -1, -1, -1, 0, 0, 2, 4$ sum to exactly 0 — always. Without squaring, every data set looks perfectly concentrated. The squares are what make spread measurable.
4. Deviating from the wrong center. Deviations come from the mean of the same data set — not from 0, and not from a target value.

Run your hand computation past the standard deviation calculator — each of these errors appears as a mismatch in the first row.