Statistics Calculator
Mean, median, mode, quartiles, variance and standard deviation for raw data, frequency tables and grouped data — with the full NCERT-style working, a histogram, an ogive and a box plot.
Standard deviation measures how far values sit from the mean: σ = √(Σ(x − x̄)² ÷ n). For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, Σ(x − x̄)² = 32 and n = 8, so σ = √4 = 2. For grouped data use σ = √(Σf(x − x̄)² ÷ Σf).
Your data
Separate values with commas, spaces or new lines.
| Value x | Frequency f |
|---|
| Class (lower – upper) | f |
|---|
Step-by-step working
Real numbers from your data at every step.
Charts
Frequency table
Data in ascending order
Averages and spread, explained
Measures of central tendency tell you where the data is centred; measures of dispersion tell you how spread out it is. You need both to describe a data set.
Pick the data type
Raw list of values, a table of values with frequencies, or class intervals with frequencies.
Enter the data
Paste numbers or fill the table. Inclusive classes like 10–19 are converted automatically.
Read the results
All measures update instantly. Switch between population and sample variance as needed.
Study the working
Open each tab for the formula, the substituted values and the final answer — plus charts to print.
Mean of grouped data
x̄ = Σfx ÷ Σf (direct)
x̄ = a + Σfd ÷ Σf (assumed mean, d = x − a)
x̄ = a + h × Σfu ÷ Σf (step deviation, u = (x − a) ÷ h)
Median & mode of grouped data
Median = l + ((n/2 − cf) ÷ f) × h
Mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h
Empirical relation: Mode ≈ 3 Median − 2 Mean.
Variance & standard deviation
σ² = Σf(x − x̄)² ÷ n
σ = √σ² is in the same unit as the data. Divide by n − 1 instead (s²) when your data is a sample from a bigger population.
Quartiles & outliers
Q1, Q2 (median) and Q3 split sorted data into four equal parts. IQR = Q3 − Q1 holds the middle 50%.
A value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is treated as an outlier.
Statistics Calculator — your questions
How do you calculate mean, median and mode?
The mean is the total divided by how many values there are. The median is the middle value after sorting, or the average of the two middle values when the count is even. The mode is the value that appears most often. For 3, 5, 5, 7, 10 the mean is 6, the median is 5 and the mode is 5.
What is the formula for the median of grouped data?
Median = l + ((n/2 − cf) ÷ f) × h, where l is the lower limit of the median class, n is the total frequency, cf is the cumulative frequency before that class, f is its frequency and h is the class width. Find the median class first by locating n/2 in the cumulative frequency column.
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations from the mean, so its unit is the square of the data's unit. Standard deviation is the square root of the variance, which brings it back to the original unit and makes it easier to read. A larger standard deviation means the data is more spread out.
When do I divide by n and when by n − 1?
Divide by n when your numbers are the whole population you care about — the marks of all 40 students in a class, for instance. Divide by n − 1 when they are a sample drawn from a larger group, because dividing by n would underestimate the real spread. This calculator shows both.
What is the empirical relation between mean, median and mode?
For a moderately skewed distribution, Mode is approximately 3 × Median − 2 × Mean. CBSE often gives two of the three measures and asks for the third using this relation. It is an approximation, not an exact rule, so it will not match a calculated mode perfectly.
What is the step deviation method?
It makes grouped mean calculations easier by shrinking the numbers. Pick an assumed mean a near the middle, set u = (x − a) ÷ h where h is the class width, then x̄ = a + h × (Σfu ÷ Σf). The answer is identical to the direct method but the arithmetic is far smaller.
How do I find quartiles and spot outliers?
Q1, Q2 and Q3 split the sorted data into four equal parts, and the interquartile range is IQR = Q3 − Q1, which holds the middle 50% of values. Any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is treated as an outlier and is shown separately on the box plot.