Statistical Measures of Central Tendency
In statistics, central tendency describes the central or typical value for a probability distribution. The three most common measures are:
- Mean (Average): The sum of all observations divided by the total number of items: $\bar{x} = \frac{\sum x}{n}$.
- Median: The middle score in a dataset arranged in ascending order. If $n$ is even, it is the average of the two middle scores.
- Mode: The value that occurs with the highest frequency in the dataset.
- Range: The difference between the highest and lowest scores ($Max - Min$), describing statistical dispersion.
• Median (Odd n) = Value at position (n + 1) ÷ 2
• Median (Even n) = [Value at (n÷2) + Value at (n÷2 + 1)] ÷ 2
• Range = Maximum Value − Minimum Value
Worked Example with Outliers
Dataset: 15, 18, 22, 22, 35, 40, 55, 92
- Count ($n$) = 8 items
- Sorted order: 15, 18, 22, 22, 35, 40, 55, 92
- Sum = 15 + 18 + 22 + 22 + 35 + 40 + 55 + 92 = 299
- Mean = $299 \div 8 =$ 37.38
- Median (middle 2 values: 22 and 35) = $(22 + 35) \div 2 =$ 28.50
- Mode = 22 (occurs twice)
- Range = $92 - 15 =$ 77
Notice how the outlier 92 pulls the Mean (37.38) significantly higher than the Median (28.50). In skewed datasets, the median provides a more realistic representation of typical values.
Frequently Asked Questions
The Mean is the arithmetic sum of numbers divided by the count. The Median is the middle value when numbers are sorted in order. The Mode is the number that appears most frequently in the dataset.
The median is preferred when data contains significant extreme outliers or is heavily skewed (such as income distributions or home prices), because outliers distort the arithmetic mean.
Yes. A dataset can have one mode (unimodal), two modes (bimodal), multiple modes (multimodal), or no mode at all if all values occur with equal frequency.