Standard Deviation Calculator
Free descriptive statistics tool — compute standard deviation, variance, IQR, coefficient of variation, SEM, MAD, and outlier detection for any dataset.
Data Input
Enter numbers separated by commas or line breaks. Non-numeric values are ignored.
Supports .csv, .txt, .xlsx, .xls — headers detected automatically.
Enter individual values; click + to add more rows.
When to Use Standard Deviation
When to Use This Tool
The standard deviation and variance calculator is ideal whenever you need to measure and report the spread of data in descriptive statistics. Use it when you need to:
- Summarise variability in continuous data (test scores, measurements, survey ratings)
- Check whether data has high or low dispersion before running inferential tests
- Detect outliers using z-scores or Tukey fences
- Compare variability across datasets using the coefficient of variation
- Report descriptive statistics in APA 7th edition format
- Calculate confidence intervals for the mean
Sample vs Population SD — Decision Guide
✅ Use Sample SD (n−1)
When your data is a sample drawn from a larger population — most research scenarios.
When your data is a sample drawn from a larger population — most research scenarios.
🔵 Use Population SD (n)
When your data is the entire population — e.g., all students in a single class, all branches of a company.
When your data is the entire population — e.g., all students in a single class, all branches of a company.
How to Use This Calculator
Step-by-Step Guide
- Enter your data — paste numbers in the Type/Paste tab, upload a .csv/.xlsx file, or add values manually.
- Load a sample dataset (optional) — choose from the dropdown to explore a worked example.
- Choose data type — Sample (n−1) for most research; Population (n) when you have all observations.
- Select confidence level — 95% is standard; use 99% for clinical or high-stakes research.
- Click "Calculate Standard Deviation" — results appear instantly.
- Read the KPI cards — SD, variance, CV, SEM, IQR at a glance.
- Explore the charts — Chart 1 shows the distribution with ±1 SD / ±2 SD bands; Chart 2 shows a box plot with Tukey fences.
- Check the outlier table — every value is flagged if it falls beyond Tukey fences or |z| > 2.
- Copy the APA writing example — pre-filled with your computed values.
- Export — download as .txt, .xlsx, .docx, or print as PDF.
Example: Dataset = {45, 52, 55, 60, 63, 67, 68, 70, 72, 74, 75, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 98}. SD ≈ 15.2, Variance ≈ 230.8, IQR ≈ 22, CV ≈ 20.7%, SEM ≈ 3.24.
Frequently Asked Questions
FAQ — Standard Deviation & Variance
What is standard deviation?
Standard deviation (SD) measures how spread out values are around the mean. A low SD means values cluster close to the mean; a high SD means they are widely scattered. It is the square root of variance and expressed in the same units as the original data.
How do you calculate standard deviation step by step?
1) Find the mean (x̄). 2) Subtract x̄ from each value and square the result: (xᵢ − x̄)². 3) Sum all squared differences. 4) Divide by n−1 (sample) or n (population). 5) Take the square root. This gives the standard deviation.
What is the difference between sample and population standard deviation?
Sample SD divides by n−1 (Bessel's correction) to give an unbiased estimate of the population SD. Population SD divides by n and is used only when you have data for the entire population. In most research, use sample SD.
What is a good standard deviation?
There is no universally "good" SD — it depends on the scale and context. Use the coefficient of variation (CV = SD/Mean × 100%) to compare across datasets. CV < 15% is usually low variability; CV > 30% is high. For exam scores out of 100, an SD of 10–15 is typical.
What is variance and how does it relate to standard deviation?
Variance is the average squared deviation from the mean. Standard deviation is the square root of variance, returning the result to the original units. Variance is useful in mathematical derivations (e.g., ANOVA); SD is more interpretable for reporting.
What is the standard error of the mean (SEM)?
SEM = SD / √n. It estimates how much the sample mean would vary if you drew many samples. SEM is always smaller than SD and is used for confidence intervals around the mean. It decreases as sample size increases; SD does not.
When should I use IQR instead of standard deviation?
Use IQR (Q3 − Q1) when data is skewed or contains outliers. IQR only looks at the middle 50% of the data and is robust to extreme values. SD is preferred for roughly symmetric, normally distributed data. Always report both in full descriptive statistics.
What is the coefficient of variation (CV)?
CV = (SD / Mean) × 100%. It expresses standard deviation as a percentage of the mean. CV is useful when comparing variability across datasets with different units or scales — for example, comparing the variability of weight (kg) vs height (cm).
How do I report standard deviation in APA format?
Report as: M = [mean], SD = [value]. Example: "The sample had a mean score of M = 72.4 (SD = 8.3, n = 45)." Always pair SD with the mean. Include n. Do not use ± unless comparing groups. Round to 2 decimal places in text.
What are Tukey fences and how do they detect outliers?
Tukey fences are computed from the IQR: Lower fence = Q1 − 1.5×IQR; Upper fence = Q3 + 1.5×IQR. Values outside these are mild outliers. Values beyond Q1 − 3×IQR or Q3 + 3×IQR are extreme outliers. This method is robust to non-normality.
What is mean absolute deviation (MAD)?
MAD = Σ|xᵢ − x̄| / n. It is the average absolute difference from the mean. Unlike variance, it does not square the deviations, so it is more interpretable and less sensitive to outliers than SD. It is reported in the same units as the data.
Can I upload a CSV or Excel file?
Yes. Click the Upload File tab, select a .csv, .xlsx, or .xls file. The tool auto-detects numeric columns, lets you pick the column, previews the first 8 rows with your selection highlighted, then loads the data into the calculator.
Is this standard deviation calculator free?
Yes. STATS UNLOCK's standard deviation and variance calculator is completely free, with no registration, no limits, and no watermarks. Results can be downloaded as .txt, .xlsx, .docx, or printed as PDF.
What is Bessel's correction?
Bessel's correction uses n−1 in the denominator of sample variance instead of n. Dividing by n underestimates the true population variance; n−1 corrects this bias. Named after Friedrich Bessel who first described it.
What does a z-score tell me about my data?
A z-score = (value − mean) / SD. It shows how many standard deviations a value lies from the mean. z = +2 means 2 SDs above the mean. |z| > 2 is often used as a threshold for potential outliers; |z| > 3 for extreme outliers.
References
The standard deviation calculator, variance formulas, and outlier detection methods implemented here follow the descriptive statistics methodology cited in the references below.
- Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). SAGE Publications.
- Hays, W. L. (1994). Statistics (5th ed.). Harcourt Brace College Publishers.
- Howell, D. C. (2012). Statistical Methods for Psychology (8th ed.). Cengage Learning.
- Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley.
- Bessel, F. W. (1838). Untersuchung über die Wahrscheinlichkeit der Beobachtungsfehler. Astronomische Nachrichten, 15(358), 369–404.
- Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.
- Nunnally, J. C., & Bernstein, I. H. (1994). Psychometric Theory (3rd ed.). McGraw-Hill.
- Tabachnick, B. G., & Fidell, L. S. (2019). Using Multivariate Statistics (7th ed.). Pearson.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum. https://doi.org/10.4324/9780203771587
- American Psychological Association. (2020). Publication Manual of the American Psychological Association (7th ed.). APA. https://doi.org/10.1037/0000165-000
- Sokal, R. R., & Rohlf, F. J. (2012). Biometry (4th ed.). W. H. Freeman.
- R Core Team. (2024). R: A Language and Environment for Statistical Computing. R Foundation. https://www.r-project.org