Measures of central tendency
- Mean (x̄): the sum of all values divided by how many there are. It is sensitive to extreme values.
- Median: the middle value of the sorted data. With an even number of values, the mean of the two middle ones. Outliers do not affect it.
- Mode: the most frequent value. There can be several, or none.
Measures of spread
- Range: maximum minus minimum.
- Variance: the average of the squared distances from the mean.
- Standard deviation: the square root of the variance, measured in the same units as the data.
- Coefficient of variation: standard deviation / mean. Useful for comparing the spread of data sets on different scales.
Example · scores 4, 6, 7, 7, 8, 10
Mean = 42 / 6 = 7. Median = (7 + 7) / 2 = 7. Mode = 7. Squared deviations: 9, 1, 0, 0, 1, 9 → sum 20. Population variance = 20/6 ≈ 3.33 and standard deviation ≈ 1.83. As a sample: 20/5 = 4 and s = 2.
Population or sample standard deviation?
If you have every value you care about (the grades of your whole class), it is a population and you divide by n. If the data is only part of something larger that you want to estimate (100 people surveyed in a whole city), it is a sample and you divide by n − 1 (Bessel's correction), so the estimated variance is not biased downward. Scientific calculators label them σₙ and σₙ₋₁ (or sx).
Quartiles and outliers
Quartiles split the sorted data into four equal parts. This tool uses linear interpolation (the same method as QUARTILE.INC in Excel and Google Sheets); some textbooks use slightly different methods that can give different values for small data sets. A value is flagged as an outlier if it lies more than 1.5 times the interquartile range below Q₁ or above Q₃.
Frequently asked questions
When should I use the mean and when the median?
For symmetric data without outliers, the mean. With extreme values (salaries, house prices), the median better represents a typical value.
Can I paste data from Excel or Google Sheets?
Yes: copy a column or row and paste it. Line breaks and tabs are recognized as separators.
How do I calculate a weighted average?
Multiply each value by its weight, add them up and divide by the sum of the weights. For grades, use the grade calculator.