Standard deviation in one sentence

Standard deviation is the typical distance from the mean — calculated by measuring each value’s deviation from the average, squaring those deviations, averaging them (using n or n − 1), and taking the square root to return to the original units.

• Same units as your data: If your data is in dollars, std dev is in dollars (unlike variance).
• Quick spread summary: One number that captures how “consistent” vs “volatile” a dataset is.
• Works everywhere: Grades, salaries, stock returns, manufacturing tolerances, experiments, analytics.
• Pairs well with the mean: Mean tells the center; standard deviation tells the spread.

• If two people have the same average sleep time, the one with bigger std dev has the more chaotic schedule.
• If two products have the same average rating, the one with bigger std dev is more “love-it-or-hate-it”.
• If two teams score the same points on average, the one with bigger std dev is more streaky.

Cheat sheet: formulas & symbols

The calculator supports both common definitions. The only difference is the denominator: population uses n; sample uses n − 1.

• Mean: μ = (Σx) / n
• Variance: σ² = (Σ(x − μ)²) / n
• Std dev: σ = √σ²

• Mean: x̄ = (Σx) / n
• Variance: s² = (Σ(x − x̄)²) / (n − 1)
• Std dev: s = √s²

How the calculator works (step-by-step)

This section is intentionally detailed so you can understand (and explain) the math. If you’re new to stats, don’t worry — the idea is simple: measure how far values are from the average, then summarize those distances in a stable way.

Your input is treated as a list. We accept commas, spaces, tabs, and new lines. This means you can paste data straight from a spreadsheet column. The calculator ignores blank entries and validates that every token is a real number. If anything looks suspicious (like letters), you’ll see a clear error message so you can fix it fast.

The mean is the “center of gravity” of the dataset: add up all values and divide by how many values there are. We show the count n and the mean because they’re the foundation for everything that follows.

For each value x, compute its deviation: x − mean. A value above the mean has a positive deviation; below the mean is negative. If we simply averaged these deviations, they’d cancel out to zero — which is why the next step matters.

Squaring makes every deviation positive and emphasizes larger gaps. This is where the term “variance” comes from: it’s literally the average squared distance from the mean (with a denominator choice).

Here we choose Population vs Sample:

• Population: divide by n because you have the whole group.
• Sample: divide by n − 1 because you’re estimating the population variance from a sample. Using n − 1 is called Bessel’s correction, and it reduces bias.

Variance is in “squared units” (e.g., squared dollars), which is hard to interpret. Taking the square root returns you to the original units, producing standard deviation. That’s why standard deviation is often the go-to spread metric.

When you calculate, we print a readable breakdown: your cleaned list, the mean, each squared deviation, the sum of squared deviations, the denominator used, the variance, and finally the standard deviation. You can copy/paste it into assignments or documentation.

Worked examples (copy-friendly)

Data: 10, 12, 9, 11, 13

• Mean = (10+12+9+11+13)/5 = 55/5 = 11
• Deviations: −1, +1, −2, 0, +2
• Squared deviations: 1, 1, 4, 0, 4 → sum = 10
• Population variance = 10/5 = 2 → σ = √2 ≈ 1.414
• Sample variance = 10/4 = 2.5 → s = √2.5 ≈ 1.581

Notice the sample standard deviation is slightly bigger. That’s expected because dividing by n − 1 compensates for sampling uncertainty.

Dataset A: 9, 10, 11 (mean 10) vs Dataset B: 0, 10, 20 (mean 10)

• Both have the same average.
• Dataset A is tight: std dev is small.
• Dataset B is wild: std dev is much larger.

This is why standard deviation is so useful: the mean alone can hide how chaotic the data is.

Imagine a factory targets a bolt length of 50mm. Two machines both average 50mm, but: Machine 1 has std dev 0.2mm (very consistent), while Machine 2 has std dev 1.5mm (much more variation). Even if averages match, the second machine will produce more out-of-tolerance parts.

How to interpret your result (without overthinking)

A standard deviation number only makes sense relative to your mean and your context. Here are practical rules of thumb that help most people:

• If std dev is tiny compared to the mean, values cluster tightly.
• If std dev is large compared to the mean, values vary widely.

Standard deviation has the same units as your data. That’s why it’s great in everyday settings: “Typical variation is about ±2 points,” or “Typical variation is about ±$50.”

Outliers can inflate standard deviation. If you suspect outliers, consider checking the median and IQR too. But std dev is still a fantastic “first-pass” spread measure.

In a bell-curve-ish distribution, about 68% of values are within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. This is the classic “68–95–99.7 rule”. Use it only if the distribution is reasonably symmetric and not extremely skewed.

Frequently Asked Questions

• What’s the difference between variance and standard deviation?

  Variance is the average squared distance from the mean. Standard deviation is the square root of variance. People usually prefer standard deviation because it’s in the same units as the original data.

• When should I use sample standard deviation (n − 1)?

  Use sample standard deviation when your list is a subset used to estimate variability in a larger population. The n − 1 denominator (Bessel’s correction) reduces bias in the variance estimate.

• Do negative numbers work?

  Yes. The calculator supports negative values, decimals, and scientific notation (like 1.2e-3). Spread works the same way — we square deviations, so distances stay positive.

• Why is my standard deviation “0”?

  Standard deviation is 0 only when every number is exactly the same (all values equal the mean). If you expected a non-zero result, double-check your list for repeated values or parsing issues.

• Is standard deviation the same as “average deviation”?

  Not exactly. Some people mean mean absolute deviation (MAD) when they say “average deviation”. Standard deviation uses squared deviations and has stronger mathematical properties, especially in normal distributions.

• Can I compute standard deviation from a frequency table?

  This page is built for raw lists. If you have a frequency table, expand it into a list (repeat values by frequency), or use a dedicated frequency-table stats calculator.

• What’s a “good” standard deviation?

  There is no universal “good” value. A good or bad spread depends on what you’re measuring: tight tolerances in manufacturing might demand a tiny std dev, while creative performance metrics might naturally vary a lot. Compare against goals, historical baselines, or benchmarks.