"The average is 18.7" sounds like a fact, but whether it is a useful fact depends on the data behind it. This article takes one small, realistic data set through our statistics calculator, shows what one or two extreme values do to each summary number, and ends with a rule of thumb for what to report. Every number below was computed by the tool and checked against NumPy and SciPy.
The data
Thirteen delivery times in minutes:
12, 15, 11, 18, 22, 15, 14, 30, 13, 16, 17, 15, 45
Most deliveries take 11 to 22 minutes. One took 30, and one took 45 (a wrong address, say). Paste the list into the calculator and it reports:
| Summary | Value |
|---|---|
| Mean | 18.69 |
| Median | 15 |
| Mode | 15 (three times) |
| Sample standard deviation | 9.31 |
| Q1, Q3 (Tukey's hinges) | 14, 18 |
| Outliers (beyond 1.5 × IQR) | 30, 45 |
| 95% confidence interval for the mean | 13.06 to 24.32 |
Already the story is visible: the mean is almost four minutes above the median, the standard deviation is larger than the whole range of the typical deliveries would suggest, and the confidence interval is eleven minutes wide.
What one value does
Remove the 45 and recompute; then remove the 30 as well:
| Data | n | Mean | Median | SD | 95% CI for the mean |
|---|---|---|---|---|---|
| All 13 | 13 | 18.69 | 15 | 9.31 | 13.06 to 24.32 |
| Without 45 | 12 | 16.50 | 15 | 5.14 | 13.23 to 19.77 |
| Without 30 and 45 | 11 | 15.27 | 15 | 3.04 | 13.23 to 17.31 |
The median does not move. The mean drops by more than three minutes. The standard deviation falls by two thirds, because it squares each distance from the mean: the 45 is 26 minutes from the mean, and 26² = 676 dominates the sum. That is the core difference between the two families of statistics:
- Mean and standard deviation use every value's size. They are efficient when the data are roughly symmetric, and they are what most formulas (confidence intervals, t-tests, regression) are built on.
- Median and quartiles use only the order of the values. They are robust: a single wild value cannot drag them far.
This does not mean the outliers should be deleted. The 45-minute delivery happened; a customer waited for it. If you are planning staffing, the mean is the right number, because total time is mean × count. If you are telling a customer how long a delivery usually takes, the median is the honest number.
Which spread to report
Report spread in the same family as the centre:
- Mean ± standard deviation for roughly symmetric data: "16.5 ± 5.1 minutes".
- Median with the quartiles (or the interquartile range) for skewed data: "median 15 minutes (IQR 14 to 18)".
The calculator's box plot shows the choice at a glance. Here the box (Q1 to Q3) is narrow, the median sits inside it, and two circles far to the right are the outliers. A long whisker or circles on one side mean skew, and skew means the median is the safer headline.
Two more measures help: skewness is 2.25 for the full data (strongly right-skewed; 0 would be symmetric) and the coefficient of variation (SD ÷ mean) is 50%, which says the spread is large relative to the typical value.
Sample or population, and the standard error
The tool asks whether the data are a sample or the whole population. Thirteen deliveries out of all deliveries are a sample, so the standard deviation divides by n − 1 (9.31) rather than n (8.95). The difference shrinks as n grows, but it matters for small data sets and for the interval below.
Do not confuse the standard deviation with the standard error of the mean. The SD (9.31) describes how much individual deliveries vary. The standard error, SD ÷ √n = 9.31 ÷ √13 = 2.58, describes how precisely the mean is estimated. The 95% confidence interval is mean ± t × SE, where t comes from Student's t distribution with 12 degrees of freedom (2.179), not the 1.96 of the normal distribution: 18.69 ± 2.179 × 2.58 gives 13.06 to 24.32. With only a handful of values, using 1.96 would make the interval too narrow.
Quartiles depend on the method
Small data sets have no single "right" quartile. For 2, 4, 4, 4, 5, 5, 7, 9, the third quartile is:
- 6 with Tukey's hinges (the median of the upper half: 5, 5, 7, 9);
- 5.5 with the inclusive method used by Excel's QUARTILE.INC and NumPy;
- 6.5 with the exclusive method of Excel's QUARTILE.EXC and Minitab.
None is wrong. Say which you used, especially when comparing with numbers from another program. The calculator names its method next to the result.
Common mistakes
- Reporting the mean of skewed data (incomes, house prices, waiting times) as if it were typical.
- Deleting outliers because they are inconvenient. Investigate them; remove one only if it is an error, and say so.
- Mixing families: a median with a standard deviation, or a mean with an IQR.
- Using the population SD for a sample, which understates the spread.
- Quoting the standard error as if it were the spread of the data. It is much smaller and answers a different question.
- Using 1.96 for a small sample's confidence interval. With 13 values the right multiplier is 2.179.
- Comparing quartiles from programs that use different methods.
A rule of thumb
Look at the box plot first. If it is roughly symmetric with no outliers, report the mean and standard deviation (and a confidence interval if you are estimating something). If it is skewed or has outliers, report the median and quartiles, and mention the mean if totals matter. When in doubt, report both; it costs one line and hides nothing. For quick arithmetic on any of these numbers, the scientific calculator is a tab away.