How to use the statistics calculator
- Enter your numbers, separated by commas, spaces or new lines.
- Pick a quartile method. Schools and TI calculators usually use the median of each half; spreadsheets use linear interpolation. They can give slightly different quartiles for the same data.
- Read the mean at the top, the key figures below it, and the full summary table.
Formulas used
The median is the middle value of the sorted list, or the average of the two middle values when the count is even. The mode is the value that appears most often; there can be several, or none if nothing repeats. Q1 and Q3 split the sorted data into quarters. The standard error estimates how far the sample mean is likely to be from the true mean. Skewness is the adjusted Fisher-Pearson coefficient, the same one Excel's SKEW function returns: positive means a long tail to the right.
Worked example
The calculator starts with the waiting times, in minutes, of 12 customers: 12, 15, 11, 18, 14, 15, 20, 13, 15, 17, 16, 42.
- Sorted: 11, 12, 13, 14, 15, 15, 15, 16, 17, 18, 20, 42
- Sum 208, so the mean is 208 ÷ 12 = 17.3333
- Median: the 6th and 7th values are both 15, so the median is 15. The mode is also 15 (three times).
- Range: 42 − 11 = 31
- Quartiles (median of each half): the lower half 11–15 has median (13 + 14) ÷ 2 = 13.5, and the upper half 15–42 has median (17 + 18) ÷ 2 = 17.5. IQR = 4.
- Fences: 13.5 − 6 = 7.5 and 17.5 + 6 = 23.5, so 42 is an outlier
- Sample standard deviation: 8.16125; standard error 8.16125 ÷ √12 = 2.35595
- Skewness: 2.9025, a strong right tail caused by the 42
With linear interpolation, Q1 is 13.75 and Q3 is 17.25 instead.
Mean or median?
The one long wait pulls the mean up to 17.3 minutes even though 11 of the 12 customers waited 20 minutes or less. The median, 15, ignores how extreme the 42 is, so it better describes a typical customer. Use the median when the data is skewed or has outliers (incomes, house prices, waiting times) and the mean when the data is roughly symmetric. A skewness between about −0.5 and 0.5 suggests the two will be close.
| Skewness | Shape |
|---|---|
| Below −1 | Strong left tail |
| −1 to −0.5 | Moderate left tail |
| −0.5 to 0.5 | Roughly symmetric |
| 0.5 to 1 | Moderate right tail |
| Above 1 | Strong right tail |
An outlier flagged by the 1.5 × IQR rule is worth checking, not automatically deleting. It may be a typing mistake, or it may be the most interesting value in the list.
Frequently asked questions
Why do different calculators give different quartiles?
There is no single agreed definition. The median-of-halves method leaves the middle value out when the count is odd; Excel's QUARTILE.INC interpolates between ranks. Both are correct, and the difference shrinks as the list gets longer. Report which method you used.
What does it mean if there is no mode?
Every value appears the same number of times, so none is more common than the others. A list where each value appears once has no mode. A list can also have two modes (bimodal) or more.
What is the standard error of the mean?
It is the standard deviation divided by the square root of the count. It measures how much the mean would vary between repeated samples, and it is the building block of a confidence interval.
Should I use the sample or population figures?
Use the sample figures if your list is part of a larger group you want to describe, which is the usual case. Use the population figures only when the list contains every member of the group. The standard deviation calculator explains the difference step by step.
Related calculators
- Standard Deviation Calculator — Sample and population standard deviation, variance and mean of a list of numbers, with every step shown.
- Mean, Median, Mode & Range Calculator — The mean, median, mode and range of a list of numbers, with the sorted list and each step explained.
- Z-Score Calculator — The z-score and percentile of a value, the probability between two values, and the value at any percentile.
- Confidence Interval Calculator — Confidence interval and margin of error for a mean (t or z) or a proportion, at any confidence level.
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