Standard Deviation Calculator — Mean & Variance

Dataset

Separate numbers with commas, spaces, semicolons, or new lines

Mean (μ)

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Population σ

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Sample s

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What is a Std Dev?

The Standard Deviation Calculator computes population and sample standard deviation, mean, median, variance, count, sum, minimum, maximum, and range from a list of numbers you enter. Paste any comma-, semicolon-, or space-delimited list of numbers and all statistics appear instantly — no need to format data or set up a spreadsheet.

Standard deviation is the most widely used measure of data dispersion — how spread out values are around their mean. In everyday terms, it answers "how consistent is this data?" A cricket team with a batting average of 40 and standard deviation of 3 is far more consistent than one with the same average and standard deviation of 25. In finance, standard deviation of returns is the standard measure of volatility and investment risk.

In the Indian academic context, standard deviation is a mandatory topic in CBSE Class 11 Mathematics (Statistics chapter) and is central to the undergraduate statistics syllabus. Beyond academics, it appears in manufacturing quality control, clinical research, business analytics, and any data analysis task — understanding it conceptually, and being able to compute it for real datasets without error, is a fundamental quantitative skill.

This calculator computes both population standard deviation (σ, divides by n) and sample standard deviation (s, divides by n−1) simultaneously, so you can use the correct version for your context. For deeper analysis of any individual value within a dataset, pair this with the Z-Score Calculator to compute how many standard deviations that value sits from the mean.

How to use this Std Dev calculator

Enter your numbers — type or paste your dataset into the text area. Values can be separated by commas (12, 15, 11, 18), semicolons (12; 15; 11), or spaces (12 15 11 18). Mixed delimiters are also accepted. Non-numeric characters are silently ignored.
Check the Count — verify the displayed count matches the number of values you intended to enter. If it doesn't, check for accidental extra characters or missing values in your input.
Read Population or Sample SD — for a full population dataset, use Population Standard Deviation (σ). For a sample from a larger population, use Sample Standard Deviation (s). When in doubt, using sample SD (n−1) is the conservative choice for most real-world analysis.
Compare Mean and Median — if they are close, the distribution is roughly symmetric. If they differ substantially (mean much higher than median, for example), the data is skewed and the SD should be interpreted cautiously.
Use the Min, Max, Range — spot potential outliers or data entry errors. If the range seems disproportionately large relative to the mean, investigate whether an extreme value is valid or erroneous.
Take the Z-Score further — copy the mean and SD values and enter them into the Z-Score Calculator along with any specific data point to compute its percentile rank within this distribution.

Formula & Methodology

Mean:μ = Σx / n

Population Variance:σ² = Σ(xᵢ − μ)² / n

Population Standard Deviation:σ = √(σ²)

Sample Variance:s² = Σ(xᵢ − x̄)² / (n − 1)

Sample Standard Deviation:s = √(s²)

Median:Sort values in ascending order. If n is odd, the median is the middle value. If n is even, the median is the average of the two middle values.

Variable definitions:
- n — count of values
- xᵢ — individual value
- μ / x̄ — mean of all values
- Σ — sum over all i from 1 to n

Worked example — monthly sales figures for a retail store:

Monthly sales (₹ lakh) over 10 months: 12, 15, 11, 18, 17, 13, 16, 14, 19, 10

Step 1 — Mean: Σx = 145; μ = 145/10 = 14.5

Step 2 — Median: Sorted: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19; middle two = 14 and 15; Median = 14.5

Step 3 — Squared deviations from mean (14.5):(12−14.5)² = 6.25; (15−14.5)² = 0.25; (11−14.5)² = 12.25; (18−14.5)² = 12.25; (17−14.5)² = 6.25; (13−14.5)² = 2.25; (16−14.5)² = 2.25; (14−14.5)² = 0.25; (19−14.5)² = 20.25; (10−14.5)² = 20.25

Step 4 — Sum of squared deviations: 82.50

Step 5 — Population variance σ² = 82.50 / 10 = 8.25

Step 6 — Population SD: σ = √8.25 ≈ 2.87 lakh

Step 7 — Sample SD: s = √(82.50/9) = √9.17 ≈ 3.03 lakh

The mean and median both equal ₹14.5 lakh, confirming the distribution is symmetric. The SD of ~₹2.87 lakh means about 68% of months had sales between ₹11.63 and ₹17.37 lakh.

Assumption: The standard deviation formulas assume all input values are numeric and finite. Infinite values, NaN, or text entries are ignored. The calculator does not apply any grouping or frequency weighting — all entries are treated as individual data points with equal weight.

Frequently Asked Questions

What is standard deviation and what does it measure?

Standard deviation measures how spread out the values in a dataset are around their mean (average). A small standard deviation means most values cluster close to the mean; a large standard deviation means values are widely dispersed. For example, two cricket players might both average 50 runs per innings, but one might consistently score 40–60 while the other swings between 0 and 100 — the second player has a higher standard deviation, indicating greater inconsistency.

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

Population standard deviation (σ) is used when you have data for every member of the group you are studying. Sample standard deviation (s) is used when your data is a subset (sample) drawn from a larger population, and it uses n−1 in the denominator (Bessel's correction) to produce an unbiased estimate. Our calculator computes both. As the sample size grows, sample and population standard deviations converge. For n > 30, the difference is usually negligible.

How do you calculate standard deviation step by step?

Step 1: Compute the mean of all values. Step 2: Subtract the mean from each value and square the result (squared deviations). Step 3: Sum all squared deviations. Step 4: Divide by n (for population SD) or n−1 (for sample SD) to get the variance. Step 5: Take the square root of the variance to get the standard deviation. Our calculator shows all intermediate statistics — mean, variance, count, min, max, and range — alongside both standard deviation values.

What is variance and how does it relate to standard deviation?

Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance, which brings the result back to the original unit of measurement. Variance is mathematically easier to work with in theoretical statistics (it is additive for independent variables), while standard deviation is more interpretable — a standard deviation of ₹5,000 is directly comparable to the mean salary, whereas a variance of 25,000,000 is not.

What does a high or low standard deviation mean?

A high standard deviation means data points are widely spread from the mean — high variability. A low standard deviation means data points are concentrated near the mean — low variability. In investment analysis, a mutual fund with a high standard deviation in annual returns is more volatile and risky than one with a low standard deviation. In manufacturing, a low standard deviation means consistent product quality. Relative interpretation matters: a standard deviation of 5 is low if the mean is 500, but high if the mean is 6.

What is the empirical rule (68-95-99.7) for standard deviation?

For a normally distributed dataset: approximately 68% of values fall within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations. This rule helps quickly assess what proportion of data falls in a given range. For example, if the mean monthly income in a city is ₹35,000 with a standard deviation of ₹8,000, approximately 95% of residents earn between ₹19,000 and ₹51,000. Use the [Z-Score Calculator](/z-score-calculator/) to compute exact percentiles for any specific value.

Is standard deviation part of the CBSE syllabus?

Yes — standard deviation is covered in CBSE Class 11 Mathematics under the chapter 'Statistics' (Chapter 15 in NCERT). Students learn the mean deviation and standard deviation of grouped and ungrouped data, and the coefficient of variation. The formula is also used implicitly in Class 12 probability distributions. Standard deviation appears in competitive exams (CAT, GMAT, GATE) and is a core concept in undergraduate statistics and data science programmes.

How many numbers can I enter in the Standard Deviation Calculator?

There is no fixed limit on the number of values — you can enter as few as 2 numbers or hundreds of values. Enter them in the text area separated by commas, semicolons, or spaces (any of these delimiters work). For very large datasets, the calculator processes them in-browser with no server round-trip. The count, sum, min, max, range, mean, median, population variance, sample variance, population standard deviation, and sample standard deviation are all computed together.

What is the coefficient of variation and when should I use it?

The coefficient of variation (CV) = (standard deviation / mean) × 100%, which expresses variability as a percentage of the mean. It is useful for comparing variability between datasets with different units or very different means. For example, comparing the consistency of stock A (mean ₹100, SD ₹10, CV = 10%) versus stock B (mean ₹2,000, SD ₹150, CV = 7.5%) shows B is actually less variable proportionally despite its higher absolute SD. Our calculator does not compute CV directly, but you can divide the SD result by the mean.

What is the median and how does it differ from the mean?

The mean is the sum of all values divided by the count — it is sensitive to extreme outliers. The median is the middle value when data is sorted — it is resistant to outliers. For incomes in India, the mean is pulled up by very high earners, so the median income (the midpoint salary) better represents the typical person. Our calculator reports both the mean and median. If your mean and median differ significantly, the distribution is skewed and the standard deviation may not fully capture the spread.

What is the range and how does it compare to standard deviation?

Range = maximum − minimum. It is the simplest measure of spread but is highly sensitive to single outliers. A dataset like {1, 5, 5, 5, 5, 5, 100} has a range of 99 but a standard deviation of about 34 — the range is dominated by the single outlier 100. Standard deviation accounts for how all values deviate from the mean, making it a more robust and informative measure of spread. Use range as a quick sanity check and standard deviation as the primary spread statistic.