Descriptive Statistics
Descriptive statistics summarizes and organizes data sets through measures of central tendency (mean, median, mode) and measures of dispersion (range, variance, standard deviation).
Mean: x̄ = Σxᵢ ÷ n Variance: σ² = Σ(xᵢ − x̄)² ÷ n Standard deviation: σ = √σ² Range: R = max − min
Median — step by step
- Sort the data in ascending order
- If n is odd: median = the middle element (position (n+1)/2)
- If n is even: median = the mean of the two middle elements
When should you use each measure?
- Mean: symmetric data without extreme outliers (test scores)
- Median: data with outliers or skewed data (salaries, house prices)
- Mode: categorical data, or when the most common value matters (clothing size)
- Standard deviation: measures how far the data spread from the mean — the smaller, the more concentrated
Frequently asked questions
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations from the mean. The standard deviation is the square root of the variance — it is in the same unit as the original data, which makes it easier to interpret.
Can a data set have more than one mode?
Yes! If two values share the highest frequency, the set is bimodal (two modes). With three or more it is multimodal. If no value repeats, the set has no mode.
What is an outlier and how does it affect the mean?
An outlier is a value far from the rest (e.g. 100 in {1, 2, 3, 100}). The mean is very sensitive to outliers and can be misleading. The median is more robust in those cases.
What is the difference between population and sample variance?
Population variance divides by n. Sample variance divides by (n−1), correcting the bias when estimating a population variance from a sample. This calculator uses the population variance.