Standard Deviation, Mean, Median and Mode Calculator

Paste a list of numbers to get the mean, median, mode(s), range, variance and standard deviation — both the population formula (÷ n) and the sample formula (÷ n − 1) — with a table of deviations. For grouped data, enter class intervals and frequencies to use the direct, assumed mean or step-deviation method from NCERT Class 10 and 11, plus the median and mode formulas.

Numbers

Separate values with commas, spaces or new lines (you can paste a column from a spreadsheet).

Population SD (σ)

Sample SD (s)

2.1381

Mean

Median

4.5

Mode

Range

Variance σ²

Variance s²

4.5714

Count n

Sum

Coefficient of variation (σ ÷ mean × 100) = 40%

Step-by-step solution

Sum = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40
Count = 8
Average (mean) = Sum ÷ Count = 40 ÷ 8 = 5
Median: average of the 4th and 5th sorted values = (4 + 5) ÷ 2 = 4.5
Mode: 4 (appears 3 times)
Range = Max − Min = 9 − 2 = 7
Deviations from the mean (x − x̄) and their squares:
2 − 5 = -3 → squared 9
4 − 5 = -1 → squared 1
4 − 5 = -1 → squared 1
4 − 5 = -1 → squared 1
5 − 5 = 0 → squared 0
5 − 5 = 0 → squared 0
7 − 5 = 2 → squared 4
9 − 5 = 4 → squared 16
Σ(x − x̄)² = 32
Population variance σ² = Σ(x − x̄)² ÷ n = 32 ÷ 8 = 4
Population SD σ = √4 = 2
Sample variance s² = Σ(x − x̄)² ÷ (n − 1) = 32 ÷ 7 = 4.5714
Sample SD s = √4.5714 = 2.1381

How to use the standard deviation calculator

Choose “Raw data” for a list of values or “Grouped data” for class intervals with frequencies.
Raw data: type or paste the numbers separated by commas, spaces or new lines.
Grouped data: type the classes (0-10, 10-20, …) and their frequencies in the same order, and pick a method.
Read the results and open the steps for the deviation table and every formula with numbers filled in.

Formulas

Mean	`x̄ = Σx ÷ n`
Population variance and SD	`σ² = Σ(x − x̄)² ÷ n, σ = √σ²`
Sample variance and SD	`s² = Σ(x − x̄)² ÷ (n − 1), s = √s²` Use when your data is a sample from a bigger group.
Grouped mean (step deviation)	`x̄ = a + h × Σfu ÷ Σf, u = (x − a) ÷ h`
Grouped SD (step deviation)	`σ = h × √(Σfu² ÷ N − (Σfu ÷ N)²)`
Grouped median	`l + ((N/2 − cf) ÷ f) × h`
Grouped mode	`l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h`

Solved examples

Example 1: Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

Mean = 40 ÷ 8 = 5
Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 → sum 32
Population: σ² = 32 ÷ 8 = 4, σ = 2
Sample: s² = 32 ÷ 7 ≈ 4.571, s ≈ 2.138

Answer: σ = 2 (population), s ≈ 2.138 (sample)

Example 2: Find the mean of the grouped data 0-10 (5), 10-20 (8), 20-30 (15), 30-40 (16), 40-50 (6).

Class marks: 5, 15, 25, 35, 45; a = 25, h = 10
u = −2, −1, 0, 1, 2; Σfu = −10 − 8 + 0 + 16 + 12 = 10
x̄ = 25 + 10 × 10 ÷ 50

Answer: Mean = 27

Example 3: Find the median and mode of 3, 7, 7, 2, 9, 7, 4.

Sorted: 2, 3, 4, 7, 7, 7, 9 — the 4th value is the middle
7 appears most often (3 times)

Answer: Median 7, mode 7

What standard deviation tells you

The mean tells you the centre of the data; the standard deviation tells you how spread out the values are around it. A small SD means most values are close to the mean, a large SD means they are scattered. Two classes can have the same average mark of 70 — one where everybody scored 65–75 (small SD) and one with marks from 30 to 100 (large SD).

Variance is the square of the standard deviation. SD is in the same units as the data (marks, cm, ₹), which is why it is the number usually reported.

Population or sample — which one?

If your numbers are the whole group you care about (all 40 students of your class), use the population formula, dividing by n. If they are a sample used to estimate a bigger group (40 students out of the whole school), divide by n − 1 to get the sample SD — it corrects the tendency of a sample to underestimate the spread. NCERT Class 11 uses the population formula; statistics courses and spreadsheet functions such as STDEV.S use the sample one.

Frequently asked questions

How do I calculate standard deviation?

Find the mean, subtract it from each value, square the results, add them up, divide by n (population) or n − 1 (sample), and take the square root.

What is the difference between sample and population standard deviation?

Population SD divides by n and describes the whole group; sample SD divides by n − 1 and estimates the spread of a bigger group from a sample. The sample SD is always a little larger.

How do I find the mean, median and mode?

Mean = sum ÷ count. Median = middle value of the sorted list (average of the two middle values if the count is even). Mode = the value that appears most often; there can be more than one, or none.

Which method should I use for grouped data?

All three give the same mean. The direct method uses Σfx ÷ Σf; the assumed mean method uses smaller numbers d = x − a; the step-deviation method divides d by the class width h, which keeps the numbers smallest. NCERT accepts any of them.

What is the empirical relation between mean, median and mode?

3 × median ≈ mode + 2 × mean, for moderately skewed data. The calculator checks this for grouped data.