# Standard Deviation Calculator - Sample and Population σ

Paste a list of numbers and compute sample (n−1, s) or population (n, σ) standard deviation, variance, mean, min, max, and range. Runs in your browser.

### Data List

Numbers (comma, space, or newline)_2, 4, 4, 4, 5, 5, 7, 9_

Spread formula

Clear

Sample variance = Σ(x − mean)² / (n − 1), n ≥ 2. Population variance = Σ(x − mean)² / n, n ≥ 1. Default list: population SD = 2, sample SD ≈ 2.138.

### Spread ResultsCalculated Copy

Population standard deviation σ

2

Count n: 8

Mean: 5

Variance: 4

Range: 7

Min: 2

Max: 9

Σ(x − mean)² = 32. Denominator n = 8.

### What Is a Standard Deviation Calculator and Why Sample vs Population Matters

A **standard deviation calculator** measures how far a list of numbers typically sits from its mean. The statistic is not “the average difference.” It is the square root of an average of _squared_ differences, which penalizes outliers more than a plain mean absolute deviation and keeps the result in the same units as the data. If exam scores are points, σ is points. If masses are grams, s is grams.

The fork that trips homework is **sample versus population**. Population standard deviation σ uses every member of the set you actually care about and divides the sum of squared deviations by **n**. Sample standard deviation s treats the list as a draw from a larger unseen set and divides by **n − 1** (Bessel's correction) so the sample variance is unbiased for the population variance under ordinary sampling assumptions. This page makes you pick the fork before you copy a number, because silently defaulting to n − 1 when a problem said “population” is how a correct Σ(x − mean)² still loses the mark.

Computation stays in the browser. Paste commas, spaces, or a column of new lines. The default list **2, 4, 4, 4, 5, 5, 7, 9** is the well-known eight-point example whose mean is 5, whose sum of squared deviations is 32, whose **population SD is exactly 2**, and whose **sample SD is √(32/7) ≈ 2.138**. Use it as a checksum: if another site disagrees on this list, one of the two tools is using the other denominator.

### How to Use This Free Online Standard Deviation Calculator

Using this **sample and population SD tool** is a short loop:

1. **Paste numbers** separated by commas, spaces, semicolons, or new lines. A spreadsheet column usually pastes as one value per line.
2. Toggle **Sample (n−1, s)** or **Population (n, σ)**. Sample needs at least two values. Population accepts a single value (variance and SD are then 0).
3. Read **count, mean, variance, standard deviation, min, max, and range**. Σ(x − mean)² is shown so you can check the numerator by hand.
4. **Copy** the summary or **Clear** the box. The last list and toggle are stored in this browser for up to 30 days, not on a server.

**Limits:** at most 10,000 numeric tokens. Scientific notation such as `1e-3` is accepted. Grouping commas like `1,234` split into 1 and 234 because a comma is a delimiter. Letters, units glued to a number (`5cm`), and empty lists are rejected. This is a descriptive calculator: it does not compute a confidence interval, a z-score of a new observation, or a p-value.

### Population Standard Deviation σ - Formula and the Famous 2, 4, 4, 4, 5, 5, 7, 9 Example

σ = √[ Σ(xi − μ)² / n ]

Population variance = Σ(x − mean)² / n, then take the square root. n ≥ 1.

For the default eight numbers the sum is 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40, so the mean μ = 40 / 8 = **5**. Deviations from 5 are −3, −1, −1, −1, 0, 0, 2, 4. Squares are 9, 1, 1, 1, 0, 0, 4, 16. Those squares sum to **32**. Population variance is 32 / 8 = **4**. Population standard deviation is √4 = **2**. If this calculator shows anything else on that list in population mode, something is wrong - not a rounding disagreement.

Why square at all? Absolute deviations −3 and +4 would partially cancel if you added them raw; squaring makes every miss positive and weights a miss of 4 four times a miss of 2 in the variance (16 versus 4), not twice. Taking the square root after averaging returns to the original unit so you can say “typical distance from the mean is 2,” not “typical squared distance is 4.”

Population mode is the right button when the list _is_ the whole set: every tree in a 12-tree orchard, every invoice issued this Tuesday, every student in a closed class of eight. It is also the right button when a worksheet explicitly says population or σ. Pair the mean with the [average calculator](https://www.toolcomb.com/average-calculator) if you want median and mode as well; this page is built around variance and SD, not a full descriptive dump.

### Sample Standard Deviation s and Bessel's Correction (n − 1)

s = √[ Σ(xi − x̄)² / (n − 1) ]

Sample variance = Σ(x − mean)² / (n − 1). Requires n ≥ 2.

Same eight numbers, same Σ(x − mean)² = 32, but now the denominator is 7. Sample variance = 32 / 7 ≈ **4.571429**. Sample standard deviation s = √(32/7) ≈ **2.138090**. That is not a different data set. It is a different claim: you are estimating the spread of a larger population from eight observations, and the sample mean already used those eight numbers, which systematically shrinks the apparent scatter if you still divide by n.

**Bessel's correction** is the name for dividing by n − 1 rather than n when computing sample variance. Degrees of freedom language: one degree of freedom was spent estimating the mean, so n − 1 remain for spread. If n = 1, n − 1 = 0 and sample variance is undefined; this calculator refuses that case instead of printing Infinity or silently switching to population. A single number has no sample spread around a mean that was forced to equal that number.

Unbiasedness applies to the _variance_ s², not automatically to s itself. The square root is a nonlinear function, so E[s] is still a slightly biased estimator of σ in small samples. Intro stats courses still report s as “the” sample standard deviation, and so does this tool. If you need a bias-corrected σ̂ with extra constants, you are past what a list-paste widget should silently apply.

### Variance vs Standard Deviation - Same Information, Different Units

**Variance** is the mean of squared deviations (with n or n − 1 in the denominator). **Standard deviation** is its square root. They rank data sets the same way: larger variance always means larger SD because square root is increasing on [0, ∞). What changes is interpretability. Variance of 4 (points²) is awkward in a parent newsletter. SD of 2 points is a sentence.

Algebra likes variance. Variances of independent random variables add; standard deviations do not. If you later build a standard error of a mean, you will take s / √n, which starts from sample SD, not from a hand-wavy “average difference.” If you later want a p-value for a mean, you still need a test statistic and a reference distribution - the [p-value calculator](https://www.toolcomb.com/p-value-calculator) - not only s. This page stops at the descriptive layer: count, mean, variance, SD, min, max, range.

Range is max − min. For the default list, min 2, max 9, range **7**. Range uses two points and ignores everything in the middle, so it is a poor substitute for SD, but it is a superb typo detector (a 90 that should have been 9). Min and max are printed so you can see which tail moved.

### When to Use Sample Standard Deviation vs Population σ

Use **sample (s, n − 1)** when the numbers are a sample and the goal is to estimate the spread of a larger population: eight scores drawn from a national exam, 40 reaction times from a lab that will run hundreds more, a random subset of invoices. Inferential formulas you meet next (t intervals, many software defaults) assume s, not σ.

Use **population (σ, n)** when the list is the entire population of interest, or when the problem statement says population. Every student in a class of eight is a population for “this class.” Those same eight scores are a sample for “all students who took the exam nationwide.” The arithmetic does not know your research question; the toggle is the research question.

Do not mix denominators mid-writeup. Quoting σ = 2 in one sentence and s ≈ 2.138 in the next for the same eight numbers without naming the formula looks like two data sets. If a rubric says “show the formula,” copy Σ(x − mean)² from the result panel and write the denominator you toggled. If you need a percent-style error against a known reference value rather than spread around a mean, that is a different statistic - the [percent error calculator](https://www.toolcomb.com/percent-error-calculator), not SD.

### How to Calculate Standard Deviation by Hand - Step Checklist

1. Count n. Confirm no extra blank that became a zero.
2. Sum the values. Mean = sum / n.
3. For each x, compute (x − mean), then square it. Add those squares. That total is Σ(x − mean)².
4. Population: divide by n. Sample: divide by n − 1. That quotient is the variance.
5. Square root of the variance is the standard deviation.

A common slip is averaging the deviations before squaring (they can cancel toward zero) or taking the square root of Σ(x − mean)² without dividing. Another is using n − 1 on a population problem because a graphing calculator defaulted to sample. Walk the default list until 2 and ≈2.138 feel inevitable, then paste your real data.

If you need to draw a comparison group or a simulated sample of the same size, the [random number generator](https://www.toolcomb.com/random-number-generator) can produce integers or decimals in a range. That is sampling practice, not a substitute for measuring the spread of the list you already have.

### Honest Limits - What This SD Calculator Does Not Do

It does not compute a weighted standard deviation, a grouped (frequency table) SD, interquartile range, or median absolute deviation. It does not drop outliers unless you delete them from the paste. It does not implement Bessel correction variants other than n versus n − 1. It does not replace a statistics package for huge data files: the cap is 10,000 points, and extremely large values with tiny differences can lose precision.

Parsing is delimiter-based. A US thousands separator is a split, not grouping. Locale decimals that use a comma as the decimal mark will split as well. Convert to a plain period decimal before pasting. The tool will not infer units or convert centimeters to meters; inconsistent units make a numerically valid SD that is scientifically meaningless.

### Spreadsheets, STDEV.S vs STDEV.P, and Checking This Calculator

Excel and Google Sheets split the same fork this page names in words. `STDEV.S` (or older `STDEV`) is sample, denominator n − 1. `STDEV.P` is population, denominator n. `VAR.S` and `VAR.P` are the matching variances. If a sheet shows 2.138 on 2, 4, 4, 4, 5, 5, 7, 9 you are in sample mode; if it shows 2.000 you are in population mode. Pasting the same eight cells into this textarea should match the function you actually called, not the function you assumed.

Different spreadsheet and statistics programs can choose sample or population settings by default. The important step is to name the denominator. This calculator labels a copied result as Sample (n−1, s) or Population (n, σ), instead of leaving a bare 2.14 that could mean either.

For ordinary homework and small data sets, the displayed statistics are appropriate. Data with extremely large values and tiny differences can lose precision; center or rescale those values before comparing results. Ten thousand points is the paste cap to keep the page responsive, not a statistical limit on the number of observations.

### Frequently Asked Questions (FAQ) - Sample vs Population Standard Deviation

#### What is the difference between sample and population standard deviation?

Population standard deviation (σ) divides the sum of squared deviations by n, the number of values in the whole population. Sample standard deviation (s) divides by n−1 (Bessel's correction) so that s² is an unbiased estimator of the population variance when the list is a sample. Toggle Sample (n−1, s) or Population (n, σ) on this calculator before you copy a result.

#### Why does sample standard deviation use n−1 (Bessel's correction)?

A sample mean is fitted to the same data you then measure spread around, which underestimates variance if you still divide by n. Dividing the sum of squared deviations by n−1 (Bessel's correction) corrects that bias for variance. You need at least two values; n−1 would be zero.

#### How do I calculate standard deviation by hand?

Find the mean. Subtract the mean from each value, square those differences, and add them. For a population, divide that sum by n and take the square root. For a sample, divide by n−1 and take the square root. The default list 2, 4, 4, 4, 5, 5, 7, 9 has mean 5, sum of squares 32, population SD 2, and sample SD √(32/7) ≈ 2.138.

#### What is the difference between variance and standard deviation?

Variance is the mean of squared deviations (with n or n−1 in the denominator). Standard deviation is the square root of variance, so it is in the same units as the data. This calculator reports both, plus count, mean, min, max, and range.

#### When should I use sample standard deviation instead of population?

Use sample (s, n−1) when the numbers are a sample and you want to estimate the spread of a larger population. Use population (σ, n) when the list is the entire population, or when a homework problem says “population.” Classroom data of every student in a class of 8 is a population for that class; 8 scores drawn from a national exam are a sample.

#### Does this standard deviation calculator upload my numbers?

No. Count, mean, variance, and standard deviation are computed in your browser with JavaScript. Nothing is sent to a server. After the page loads, the tool still works if the network drops.

### Why Choose Our Standard Deviation Calculator?

- **Explicit sample vs population** - n−1 (s) or n (σ), never a hidden default you cannot see.
- **Famous checksum list** - 2, 4, 4, 4, 5, 5, 7, 9 → population SD 2, sample SD ≈ 2.138.
- **Variance, mean, min, max, range, and Σ(x − mean)²** in one panel so the numerator is checkable.
- **Private.** All arithmetic stays in the browser.
- **Honest limits** - 10,000 values, no weighted or grouped SD, no inference.
- **Copy and Clear** with a 30-day local draft.

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