Calculate population and sample standard deviation with full variance breakdown, step-by-step working, and data visualisation. Essential for stats students and data analysis.
📊 Population SD 🔬 Sample SD 📐 Variance 📈 Mean & Median 🔢 Full Breakdown Table 📉 Data Visualisation ⚡ Step-by-Step Working
Standard Deviation Calculator
·
Population & Sample SD
·
Variance Breakdown
·
Mean, Median, Mode
·
Step-by-Step Working
·
Data Visualisation
·
100% Free
·
Z-Scores & Outliers
·
Range & IQR
·
Standard Deviation Calculator
·
Population & Sample SD
·
Variance Breakdown
·
Mean, Median, Mode
·
Step-by-Step Working
·
Data Visualisation
·
100% Free
·
Z-Scores & Outliers
·
Range & IQR
·
📊 Standard Deviation & Variance Calculator
Choose Mode
Population (σ)
Use when your data IS the entire population. Divides by N.
Sample (s)
Use when your data is a sample from a larger population. Divides by N−1 (Bessel’s correction).
Enter Your Data
💡 Tip: You can paste data from a spreadsheet. Commas, spaces, tabs, and new lines all work as separators.
Quick Examples
✅
📐 Step-by-Step Working
📐 Z-Score Calculator
Enter Values
📐
📐 Step-by-Step Working
What Is Standard Deviation?
Short answer: Standard deviation (SD) measures how spread out numbers are around their average. It is the square root of the variance, reported in the same units as your data — so an SD of 12 points on a test means scores typically deviate by about 12 points from the mean. Low SD = clustered data; high SD = scattered data.
Population vs Sample Standard Deviation — Which Should You Use?
Use population SD (σ) when your data is the entire group you care about; use sample SD (s) when it’s a subset you’re generalizing from. When in doubt, use the sample version.
Population (σ)
Sample (s)
Symbol
σ (sigma)
s
Denominator
N (divide by n)
n−1 (Bessel’s correction)
When to use
You measured everyone (e.g. all 30 students in a class)
You measured a subset (e.g. 30 surveyed voters)
Typical result
Slightly smaller
Slightly larger (more conservative)
How to Calculate Standard Deviation: 5 Steps
Find the mean (μ or x̄): add all values, divide by how many there are.
Subtract the mean from each value to get the deviations.
Square each deviation (this removes negatives and penalizes large gaps).
Average the squared deviations: divide by N for population, by n−1 for a sample — that’s the variance.
Take the square root of the variance — that’s the standard deviation.
Worked Example: Test Scores 78, 88, 81, 92, 65, 58
Notice the deviations always sum to zero — a handy self-check. Try these numbers in the calculator above to confirm.
Standard Deviation in Excel, Google Sheets, Python & R
Tool
Population SD
Sample SD
Excel / Google Sheets
=STDEV.P(range)
=STDEV.S(range)
Python (NumPy)
numpy.std(x)
numpy.std(x, ddof=1)
R
sd(x)*sqrt((n-1)/n)
sd(x)
Watch out: Excel’s older STDEVP/STDEV are the same functions under legacy names, and NumPy defaults to population (ddof=0) — the single most common source of “my answer differs” confusion.
Is Your SD High or Low? How to Interpret It
There’s no universal “good” SD — it only means something relative to the mean. Compare two classes with the same average:
Class
Mean
SD
Interpretation
A
50
1.4
Scores tightly clustered — consistent performance
B
50
25
Scores wildly spread — the average hides huge variation
For a unit-free comparison, use the coefficient of variation: CV = SD ÷ mean × 100%. Class A has CV ≈ 3%, Class B ≈ 50%.
Worked Z-Score Example
A z-score tells you how many SDs a value sits from the mean: z = (x − μ) ÷ σ. Using the test scores above (μ = 77, σ ≈ 12.03):
Score of 92: z = (92 − 77) ÷ 12.03 = 1.25
→ about 1.25 SDs above average — a strong but not extreme score.
Standard Deviation vs Variance vs Range vs IQR
Measure
What it captures
Units
Outlier-robust?
Standard deviation
Typical distance from the mean
Same as data
No
Variance
Average squared distance from the mean
Squared units
No
Range
Max − min
Same as data
No (worst)
IQR
Middle 50% spread (Q3 − Q1)
Same as data
Yes
MAD
Median absolute deviation
Same as data
Yes
CV
Relative spread (SD ÷ mean)
Percentage
No
For skewed data or heavy outliers, prefer the median with IQR over mean with SD.
5 Common Standard Deviation Mistakes
Dividing by n instead of n−1 for a sample — the classic error; it understates the true spread.
Forgetting the square root and reporting the variance as if it were the SD.
Applying the 68–95–99.7 rule to non-normal data — it only holds for roughly bell-shaped distributions.
Comparing SDs across different units without the coefficient of variation.
Ignoring outliers first — one extreme value inflates SD dramatically because deviations are squared.
📐 Standard Deviation Formulas Reference
Population SD (σ): σ = √[ Σ(xᵢ − μ)² / N ]
Sample SD (s): s = √[ Σ(xᵢ − x̄)² / (N−1) ]
Population Var (σ²): σ² = Σ(xᵢ − μ)² / N
Sample Var (s²): s² = Σ(xᵢ − x̄)² / (N−1)
Mean: x̄ = Σxᵢ / N
Z-Score: z = (x − μ) / σ
Coefficient of CV = (σ / μ) × 100%
Variation (CV):
Range: max(x) − min(x)
The key difference between population and sample standard deviation is the denominator: population divides by N (all data points), while sample divides by N−1. This correction (Bessel’s correction) adjusts for the fact that a sample tends to underestimate the true spread of the full population. When in doubt about which to use: if your data is a subset drawn from a larger group, use sample SD.
💡 How to Use This Calculator
Standard Deviation: Enter your numbers separated by commas or spaces, choose population or sample mode, and click Calculate. You’ll see the SD, variance, mean, median, range, and a full breakdown table showing how each value contributes to the spread.
Z-Score Lookup: Enter a single data value, the mean, and the standard deviation to find how many standard deviations that value lies from the mean. Useful for identifying outliers, standardising scores, and comparing across different distributions.
Which mode to use? Use Population (σ) when your dataset contains every member of the group you’re studying. Use Sample (s) when your dataset is a subset taken from a larger group — this is by far the more common scenario in research and data analysis.
💎 6 Key Rules for Standard Deviation
1
Standard deviation measures spread, not size. A larger SD means data points are more spread out from the mean; a smaller SD means they are clustered closely. Two datasets can have the same mean but very different SDs — always report both.
2
Use sample SD (s) for most real-world work. Unless you have data on every single member of a population (rare!), you’re working with a sample. Sample SD uses N−1 in the denominator (Bessel’s correction) to give an unbiased estimate of the true population spread.
3
Variance is just standard deviation squared. Variance (σ² or s²) is the average of squared deviations from the mean. It’s useful in formulas and theoretical work, but SD is easier to interpret because it’s in the same units as your original data (e.g. cm, dollars, seconds).
4
The 68–95–99.7 rule (Empirical Rule) applies to normal distributions. For a normally distributed dataset, about 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. A Z-score beyond ±3 is typically considered an outlier.
5
Outliers inflate standard deviation significantly. Because deviations are squared, extreme values have a disproportionately large effect on SD. Always inspect your data for outliers before reporting — even a single extreme value can make SD misleadingly large.
6
The Coefficient of Variation (CV) makes SDs comparable. Divide the SD by the mean and multiply by 100 to get CV as a percentage. This lets you compare the relative spread of datasets measured in different units or with very different means — e.g. comparing variability in test scores vs. heights.
Frequently Asked Questions
Standard deviation (SD) is a measure of how spread out the values in a dataset are around the mean. A low SD means the values are clustered close to the average; a high SD means they are spread widely. For example, if two classes both average 70% on a test but one has SD = 5 and the other SD = 20, the second class has much more varied performance. SD is reported in the same units as the original data, making it intuitive to interpret.
Population SD (σ) is used when your dataset contains every member of the group you’re studying (e.g. all employees at a company, all students in a class). It divides the sum of squared deviations by N. Sample SD (s) is used when your data is a subset of a larger population (e.g. a survey of 500 people from a city of millions). It divides by N−1, which is called Bessel’s correction — this adjustment makes the sample SD an unbiased estimator of the true population SD. In practice, use sample SD unless you explicitly have the complete population.
There are five steps: (1) Find the mean — add all values and divide by N. (2) Find each deviation — subtract the mean from each value (xᵢ − x̄). (3) Square each deviation — (xᵢ − x̄)². (4) Find the average of squared deviations — sum them and divide by N (population) or N−1 (sample); this is the variance. (5) Take the square root of the variance — this is the standard deviation. This calculator performs all five steps and shows every calculation in the breakdown table.
A Z-score (or standard score) tells you how many standard deviations a particular value is above or below the mean. Formula: Z = (x − μ) / σ. A Z-score of 0 means the value equals the mean; +1 means one SD above; −2 means two SDs below. Z-scores are used to: compare values from different datasets, identify outliers (|Z| > 3 is a common threshold), convert raw scores to percentiles using the standard normal distribution, and standardise data for statistical tests.
Variance is the average of the squared deviations from the mean. Standard deviation is simply the square root of variance: SD = √Variance. Variance is always in squared units (e.g. cm², dollars²), which makes it hard to interpret directly but very useful in mathematical formulas, statistical tests like ANOVA, and probability theory. Standard deviation “undoes” the squaring and returns a value in the original units, making it much easier to communicate and understand in practical contexts.
The Empirical Rule applies to data that follows a normal (bell-curve) distribution. It states that: approximately 68% of data falls within 1 standard deviation of the mean (between μ−σ and μ+σ); approximately 95% falls within 2 standard deviations; and approximately 99.7% falls within 3 standard deviations. This rule is a quick way to understand the spread of normally distributed data — for instance, if test scores have a mean of 70 and SD of 10, about 95% of students scored between 50 and 90.
Outliers can dramatically inflate standard deviation because deviations are squared before averaging — so a value far from the mean contributes disproportionately to the total. For example, the dataset [2, 3, 4, 5, 6] has SD ≈ 1.41, but adding a single outlier to get [2, 3, 4, 5, 6, 100] raises the SD to about 34.9. This sensitivity to extremes means SD should always be considered alongside a check for outliers (e.g. using Z-scores or box plots). The median and IQR are more robust measures of spread when outliers are present.
The Coefficient of Variation (CV) expresses standard deviation as a percentage of the mean: CV = (SD / Mean) × 100%. It is a relative measure of spread that allows you to compare variability across datasets with different units or scales. For example, if Dataset A has mean = 10 and SD = 2 (CV = 20%), and Dataset B has mean = 1000 and SD = 200 (CV = 20%), both have the same relative spread despite very different absolute values. CV is widely used in finance (volatility), biology, and quality control.
No — never. SD is the square root of a sum of squared deviations, so it is always zero or positive. An SD of exactly 0 means every value in the dataset is identical.
There is no universal “good” value — it depends on the mean and context. A useful rule: compare SD relative to the mean using the coefficient of variation (SD ÷ mean × 100%). A CV under ~10% usually signals consistent, tightly clustered data.
That’s Bessel’s correction. A sample’s own mean sits closer to its data points than the true population mean does, so dividing by n would systematically underestimate the spread. Dividing by n−1 corrects that bias.
Use =STDEV.P(range) for population SD and =STDEV.S(range) for sample SD. The older STDEVP/STDEV names are the same functions.
SD describes how spread out your data is; standard error (SD ÷ √n) describes how precisely your sample mean estimates the population mean. Bigger sample → smaller standard error, same SD.
No — shifting every value up or down leaves the spread unchanged. But multiplying every value by k multiplies the SD by |k|.
For heavily skewed data or data with strong outliers, SD misleads — use the median with the interquartile range (IQR) instead. Also avoid the 68–95–99.7 rule unless the distribution is roughly normal.
Methodology & Sources
All statistics on this page are computed in double-precision JavaScript and cross-checked against Excel’s STDEV.P/STDEV.S; results round to two decimals for display. Definitions and methods follow the NIST/SEMATECH e-Handbook of Statistical Methods and OpenStax Introductory Statistics.
Last updated: October 2026. Educational content — standard deviation assumes roughly symmetric data; always inspect for outliers first.
