Mean and Standard Deviation Formulas
Statistics formulas for test readings: mean, median, mode, range, standard deviation, variance and standard error, with a worked voltage example.
The formulas
| To find | Formula |
|---|---|
| Mean (average) | x̄ = (x₁ + x₂ + … + xₙ) / n |
| Median | the middle value of the sorted readings; the average of the two middle values when n is even |
| Mode | the value that occurs most often |
| Range | largest reading − smallest reading |
| Population variance | σ² = Σ(x − x̄)² / n |
| Sample variance | s² = Σ(x − x̄)² / (n − 1) |
| Standard deviation | the square root of the variance: σ or s |
| Standard error of the mean | s / √n |
Use the sample formulas (n − 1) when the readings are a sample of a larger set, which is the normal case for repeated test readings. Use the population formulas only when the readings are the whole set. A small standard deviation shows that the readings agree with each other; it does not show that they are correct, because a consistent offset from the true value gives a small spread and a large error.
Worked example: six voltage readings
Six readings of a nominal 230 V supply: 230.1, 229.8, 230.4, 229.8, 231.0 and 230.2 V.
- Sum = 1,381 V, so the mean = 1,381 / 6 = 230.2 V
- Sorted: 229.8, 229.8, 230.1, 230.2, 230.4, 231. With six readings the median is the average of the third and fourth: (230.1 + 230.2) / 2 = 230.1 V
- The mode is 229.8 V, which occurs twice
- Σ(x − x̄)² = 1.008
- Population variance = 1.008 / 6 = 0.1681, so σ = 0.4099 V
- Sample variance = 1.008 / 5 = 0.2017, so s = 0.4491 V
- Standard error of the mean = 0.4491 / √6 = 0.1833 V
Frequently asked questions
Should I use the sample or the population standard deviation?
For repeated readings of the same quantity, use the sample standard deviation (divide by n − 1). The population value divides by n and suits a complete set, such as every unit in a small batch.
What is the difference between the mean and the median?
The mean uses every reading, so one bad reading moves it. The median is the middle reading and is little affected by a single outlier. A large gap between them suggests an outlier or a drifting reading.
What if no value is repeated?
Then there is no mode. If two or more values tie for most frequent, all of them are modes.
Does a small standard deviation mean the readings are accurate?
No. It means they are repeatable. Accuracy is the offset from the true value, which the percentage error calculator gives.
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