Statistics Calculator
Enter a set of numbers to calculate summary statistics, dispersion metrics, and quartiles.
Dataset Summary Statistics
Count (\(N\))
5
Sum (\(\Sigma x\))
150
Mean (Average, \(\bar{x}\))
30
Median (\(Q_2\))
30
Mode
No mode (all unique)
Minimum
10
Maximum
50
Range
40
Sample Std Deviation (\(s\))
15.811388
Population Std Deviation (\(\sigma\))
14.142136
Sample Variance (\(s^2\))
250
Population Variance (\(\sigma^2\))
200
Quartile 1 (\(Q_1\))
20
Quartile 3 (\(Q_3\))
40
Interquartile Range (\(IQR\))
20
Mean Absolute Deviation (MAD)
12
Formulas & Definitions
View All Math Formulas Handbook →Mean (Average) Formula
The sum of all values divided by count $n$.
$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \dots + x_n}{n} $$
How to calculate:
$$ \text{Sum} = \sum_{i=1}^{n} x_i, \quad \bar{x} = \frac{\text{Sum}}{n} $$
Sample Standard Deviation Formula
Uses $n - 1$ degrees of freedom for an unbiased estimator.
$$ s = \sqrt{\frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2} $$
How to calculate:
$$ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}, \quad s = \sqrt{s^2} $$
Population Standard Deviation Formula
Used when the dataset represents the entire population $N$.
$$ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} $$
How to calculate:
$$ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N}, \quad \sigma = \sqrt{\sigma^2} $$
Interquartile Range & MAD Formulas
Spread metrics between 75th and 25th percentiles.
$$ IQR = Q_3 - Q_1 $$
$$ MAD = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}| $$
How to calculate:
$$ Q_1 = \text{25th Percentile}, \quad Q_3 = \text{75th Percentile} $$
Frequently Asked Questions
When should I use sample vs. population standard deviation?
Use sample standard deviation (\(s\)) when your data is a sample of a larger population (e.g. testing 50 patients out of millions). Use population standard deviation (\(\sigma\)) if you have measured every member of the group you are studying.