Mean, Median, Mode, and Range
Measures of central tendency summarise a dataset with a single representative value. Choosing the right measure matters — the same dataset can tell very different stories depending on which average you use.
Mean (Arithmetic Average)
Mean = Sum of all values ÷ Count
Dataset: 4, 7, 13, 2, 9
Mean = (4+7+13+2+9) ÷ 5 = 35 ÷ 5 = 7Sensitive to outliers. A single extreme value pulls the mean significantly.
Median (Middle Value)
Sort data, find middle value
Sorted: 2, 4, 7, 9, 13 → Median = 7
Even count: (4th + 5th) ÷ 2Resistant to outliers. Used for income, house prices, and skewed distributions.
Mode (Most Frequent Value)
Dataset: 3, 7, 7, 2, 7, 4 → Mode = 7
Can have 0, 1, or multiple modesUsed for categorical data (most popular product, most common shoe size).
Range
Range = Maximum − Minimum
Dataset above: 13 − 2 = 11Which to Use?
- Symmetric data without outliers → Mean
- Skewed data or outliers present → Median
- Categorical or discrete data → Mode
Calculate mean, median, mode: Free Average Calculator
Three Measures of Centre
- Mean (arithmetic average): Sum all values and divide by count. Sensitive to outliers. Best for symmetric, normally-distributed data.
- Median: Middle value when sorted. If even count, average the two middle values. Resistant to outliers. Best for skewed data or data with outliers.
- Mode: Most frequent value. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, multimodal). Used for categorical data.
When to Use Each
The choice of average matters enormously in real-world analysis. Average household income in the US is pulled upward by billionaires — the median income better represents the "typical" household. A single outlier like one student scoring 0 on a test drags the mean down while the median remains representative. For shoe sizes or colours in a retail context, mode is the only meaningful average — you want the most common size to stock, not the average of size numbers. For exam scores in a roughly normal distribution, mean, median, and mode coincide and any is appropriate.
Frequently Asked Questions
Is the mean always between the minimum and maximum?
Yes, always. The arithmetic mean of any dataset is always greater than or equal to the minimum value and less than or equal to the maximum value. This provides a quick sanity check: if your calculated mean falls outside the data range, there is an error.
Can a dataset have no mode?
Yes. If all values appear exactly once (no repetitions), the dataset has no mode. Some statisticians in this case say all values are modes; others say there is no mode. For continuous numerical data, mode is rarely useful since exact repetition is uncommon.
What is a weighted mean and when is it used?
A weighted mean accounts for the fact that not all values contribute equally. Grade point averages weight each course grade by credit hours. Investment portfolio returns weight each asset by its share of the total. Formula: weighted mean = Σ(value × weight) / Σ(weights).