Descriptive statistics lab
Paste your numbers and explore summary measures and a histogram. Ten lessons help you calculate and interpret the results. Your data stays in this browser.
Calculate from your data
Quartiles use linear interpolation at (n−1)p. Population variance divides by n; the sample estimator divides by n−1. Histogram bins are left-closed, right-open, except for the final inclusive bin.
How do I use this page?
- Read a lesson and its worked example.
- Enter an answer, check it and compare the solution steps.
- Generate another exercise or take the 10-question test.
What this page does not do: predict future lottery outcomes.
Course: 10 lessons
- Data and sample size
- Sum and units
- Arithmetic mean
- Median
- Minimum, maximum and range
- Interpolated quartiles
- Interquartile range
- Population and sample variance
- Standard deviation
- Outliers and histograms
For non-terminating decimals, round to three decimal places or enter a fraction such as 1/3.
Optional local storage only. No account or server submission. Turn off to remove saved progress.
1. Data and sample size
An observation is one measured value. Count all observations, including repeated values. First check units and missing data.
Worked example
Data 2; 2; 8 have n = 3 despite only two distinct values.
Watch out
A missing measurement is not zero. Do not replace unknown values with zero.
2. Sum and units
The sum adds all observations and retains their unit. Only add comparable quantities.
Worked example
2 kg + 3 kg + 5 kg = 10 kg.
Watch out
Convert grams to kilograms before addition. A temperature sum is not an amount of heat.
3. Arithmetic mean
The mean is sum divided by count. It uses every value and is therefore sensitive to extremes.
Worked example
For 2; 4; 9 the mean is 15/3 = 5.
Watch out
A mean of group means needs weights unless group sizes are equal.
4. Median
Sort the data. For an odd count take the middle value; for an even count average the two middle values.
Worked example
1; 3; 8; 10: median = (3 + 8)/2 = 5.5.
Watch out
Sort before finding the middle. Median and mean are different.
5. Minimum, maximum and range
Range is maximum minus minimum. It describes the full width but uses only two values.
Worked example
Data 3; 4; 12: range = 12 − 3 = 9.
Watch out
Two datasets with the same range can have very different distributions.
6. Interpolated quartiles
Use h = (n−1)p in the sorted zero-indexed list. The fractional part interpolates between neighbours. Q1 uses p = 0.25 and Q3 uses 0.75.
Worked example
For 0; 10; 20; 30: Q1 = 7.5, Q3 = 22.5.
Watch out
Software uses different quartile conventions. Match the method before comparing results.
7. Interquartile range
IQR = Q3 − Q1 describes the spread of the middle portion. It is less sensitive to isolated extremes than the full range.
Worked example
Q1 = 4 and Q3 = 12 give IQR = 8.
Watch out
IQR is not a percentage. It retains the original measurement unit.
8. Population and sample variance
Subtract the mean, square each difference and sum. Divide by n for the described population, or usually by n−1 to estimate variance from a sample.
Worked example
For 2; 4; 6 squared deviations total 8. Population variance = 8/3; sample variance = 4.
Watch out
For one observation the n−1 estimator is undefined; the calculator shows no result.
9. Standard deviation
Standard deviation is the square root of variance. It returns to the original unit and describes spread around the mean.
Worked example
Variance 9 cm² gives standard deviation 3 cm.
Watch out
Normal-distribution percentage rules do not apply to every dataset.
10. Outliers and histograms
Tukey fences are Q1 − 1.5 IQR and Q3 + 1.5 IQR. Investigate values beyond them. Histograms count observations in bins; bin width changes their appearance.
Worked example
Q1 = 2, Q3 = 6: IQR = 4 and upper fence = 12. Value 13 is flagged.
Watch out
An outlier is not automatically an error. Check its source before removing it.
Final test: 10 questions
One new question from each lesson. No time limit. Answers and explanations appear after the test. You may stop and restart.
Further reading
Original exercises by Pewniaki. The mental-maths identities are explained algebraically; no claim of historical origin or guaranteed speed advantage is made.