Statistics Calculator
Free online statistics calculator. Find mean, median, mode, range, variance, standard deviation, and sum from any data set with step-by-step working.
Statistics Calculator
Comprehensive Guide & Practice
Master the mathematical principles, formulas, and step-by-step methods with worked examples and FAQs.
1. What Are Mean, Median, and Mode?
Mean, median, and mode are three measures of central tendency — they each describe the centre or typical value of a data set in a different way. Understanding all three gives you a complete picture of any numerical data.
2. The Mean (Arithmetic Average)
The mean is the sum of all values divided by the count of values. Formula: Mean = Σx / n. Example: for the data set {12, 15, 18, 15, 20}, the mean = (12+15+18+15+20)/5 = 80/5 = 16.
The mean is sensitive to outliers. A single extreme value can pull it far from the centre of the data.
3. The Median (Middle Value)
The median is the middle value when data is sorted. If there are two middle values (even count), average them. Example: sorted data {12, 15, 15, 18, 20} — the middle value is the 3rd = 15. The median is resistant to outliers, making it useful for skewed data like income distributions.
4. The Mode (Most Common Value)
The mode is the value that appears most frequently. In {12, 15, 15, 18, 20}, the mode is 15 (appears twice). A data set can have multiple modes (bimodal, trimodal) or no mode if all values are unique.
5. Range and Standard Deviation
The range = Maximum − Minimum. It measures the spread of data. The standard deviation (σ) measures how spread out values are from the mean. A low σ means values cluster near the mean; a high σ means they are spread out. Formula: σ = √(Σ(x − mean)² / n).
6. When to Use Each Measure
- Use the mean when data is symmetric and has no extreme outliers.
- Use the median when data is skewed or has outliers (e.g., house prices).
- Use the mode for categorical data (e.g., most popular shoe size).
See also: Probability Calculator | Full Mean, Median, Mode Guide | Probability Basics
7. Frequently Asked Questions
How do you find the mean of a set of numbers?
Add all numbers together, then divide by the count. For {4, 8, 6, 5, 7}: sum = 30, count = 5, mean = 30 ÷ 5 = 6.
What if the data has no mode?
If every value appears exactly once, the data set has no mode. This is perfectly normal for continuous data like exact measurements.
What is a bimodal distribution?
A bimodal distribution has two modes — two values both appearing with the highest frequency. Example: {2, 3, 3, 5, 5, 7} has modes 3 and 5.