How to Calculate Mean, Median, Mode & Standard Deviation
Descriptive statistics summarise a data set with a handful of numbers that reveal its centre and spread. Mean, median, and mode each describe the typical value differently, while standard deviation quantifies how spread out data is. Choosing the right measure matters — a single outlier can skew the mean dramatically while leaving the median unchanged.
Last updated: March 31, 2026
The Formula
Mean = Σx / n Median = middle value when sorted (average of two middles if n is even) Mode = most frequently occurring value Range = Max − Min Variance = Σ(x − Mean)² / n Std Dev = √Variance
Variable Definitions
| Symbol | Name | Description |
|---|---|---|
| Σx | Sum | All data points added together |
| n | Count | Total number of data points |
| x̄ | Mean | The arithmetic average |
Step-by-Step Example
Seven students scored: 72, 85, 90, 85, 68, 92, 78. Find mean, median, mode, and standard deviation.
Given
Solution
- 1Sort the data:
68, 72, 78, 85, 85, 90, 92 - 2Calculate mean:
(68+72+78+85+85+90+92) / 7 = 570 / 7 ≈ 81.43 - 3Find median (position 4 of 7):
Median = 85 - 4Find mode (most frequent):
Mode = 85 (appears twice) - 5Calculate variance:
Σ(x−81.43)² / 7 ≈ 74.53 - 6Standard deviation:
√74.53 ≈ 8.63
Mean = 81.43, Median = 85, Mode = 85, Std Dev ≈ 8.63.
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Common Mistakes to Avoid
Forgetting to sort before finding the median — it is position-based, not value-based.
Using population formula (÷n) on sample data — use (÷n−1) when data is a subset of a larger group.
Assuming there is always one mode — data can be bimodal, multimodal, or have no mode at all.
Reporting variance instead of standard deviation — variance is in squared units, std dev is in the original units.