A scientific calculator plus five focused math tools — roots, GCD/LCM, primes, quadratics and statistics — in one place.
Descriptive statistics turn a long list of numbers into a few that summarise it. Mean, median and mode describe the centre of the data. Range, variance and standard deviation describe its spread. Knowing which to use, and when one misleads, is a genuinely useful skill.
We’ll use this dataset throughout: 4, 8, 15, 16, 23, 42, 8
Measures of centre
Mean (average)
Add everything, divide by the count.
(4 + 8 + 15 + 16 + 23 + 42 + 8) ÷ 7 = 116 ÷ 7 = 16.57
Median (middle value)
Sort the data and take the middle value: 4, 8, 8, 15, 16, 23, 42. The median is 15. For an even count, average the two middle values.
Mode (most frequent)
The value that appears most often: 8 (twice). A dataset can have several modes, or none if every value is unique.
Mean or median?
The mean uses every value, so a single extreme value (an outlier) drags it around. The median ignores how extreme the edges are.
Imagine 9 people earning $40,000 and one earning $1,000,000:
- Mean = $136,000. Nobody actually earns that.
- Median = $40,000. Typical.
Rule of thumb: use the median for skewed data like incomes, house prices and response times; use the mean for roughly symmetric data like heights or test scores. When the two differ a lot, the data is skewed.
Measures of spread
Range
Max − min = 42 − 4 = 38. Simple, but determined by just two values.
Variance
The average of the squared distances from the mean. Squaring makes every distance positive and weights large deviations more heavily.
Standard deviation
The square root of the variance, back in the original units. Roughly, it’s the typical distance of a value from the mean.
For our dataset: population SD ≈ 11.93, sample SD ≈ 12.88.
Population vs sample: n or n − 1?
| Divide by | Use when | |
|---|---|---|
| Population variance / SD | n | Your data is the entire group (every student in the class) |
| Sample variance / SD | n − 1 | Your data is a sample from a larger group (100 customers out of thousands) |
Dividing by n − 1 (Bessel’s correction) compensates for the fact that a sample tends to underestimate the true spread. With large datasets the difference becomes negligible. MathKit shows both, so you can pick the one that fits.
Interpreting standard deviation
For data that is roughly bell-shaped (normally distributed):
- about 68% of values lie within 1 SD of the mean
- about 95% within 2 SD
- about 99.7% within 3 SD
If exam scores have a mean of 70 and SD of 10, roughly two-thirds of students scored between 60 and 80, and a score of 95 is unusually high.
A worked comparison
Two classes both average 70%:
- Class A: 68, 69, 70, 71, 72 → SD ≈ 1.4
- Class B: 40, 55, 70, 85, 100 → SD ≈ 21.2
Same mean, very different stories. Class B has students struggling and excelling. That’s why you should always report a measure of spread alongside the average.
Summary
| Measure | Tells you | Watch out for |
|---|---|---|
| Mean | Arithmetic centre | Outliers |
| Median | Typical value | Ignores the magnitude of extremes |
| Mode | Most common value | May not exist or may be several |
| Range | Total spread | Set by two values only |
| Standard deviation | Typical distance from the mean | Use sample vs population correctly |
Paste your own data into MathKit’s Statistics tab to get all of these at once. For percentage-based comparisons of results, use the percentage calculator, and for exam scoring with penalties, the negative marks calculator.
A scientific calculator plus five focused math tools — roots, GCD/LCM, primes, quadratics and statistics — in one place.