Reading the numbers: a walkthrough of the default dataset
The preloaded data — 12, 15, 15, 18, 21, 24, 24, 24, 30 — is small enough to check by hand, which makes it a good tour of what each tile means. Nine values sum to 183, so the mean is 20.33. The median, the middle value once sorted, is 21 — close to the mean here because the data is fairly balanced. The mode is 24, the only value that repeats more than any other. Q1 is 15 and Q3 is 24, so the interquartile range is 9: the middle half of the data spans nine units, a spread measure that ignores whatever the extremes are doing.
The two standard deviations differ on purpose. Population SD (5.44) divides the squared deviations by n and describes exactly the numbers you typed. Sample SD (5.77) divides by n − 1 and answers a different question: if these nine values are a sample from something larger, what is our best estimate of that larger population’s spread? Dividing by n − 1 — Bessel’s correction — compensates for the fact that deviations measured around the sample’s own mean are systematically a little too small.
Mean, median, or mode — which one to report
The mean uses every value, which is both its strength and its weakness: one outlier moves it. The median only cares about order, so it holds steady when the extremes misbehave. For anything with a long right tail — salaries, house prices, page-load times, hospital bills — the median usually tells the truer story, and a mean sitting far above the median is itself a finding: it says a small number of large values dominate. The mode matters mostly for discrete choices (most common shoe size, most frequent rating) and is deliberately blank when nothing repeats, rather than pretending every unique value is a mode.
A practical habit: report a center and a spread together. “Median 21, IQR 9” or “mean 20.3, SD 5.8” each paint a picture; a center alone hides whether the data huddles tightly or sprawls.
Z-scores and percentile ranks: locating one value in the crowd
A z-score re-expresses a value as “how many standard deviations from the mean.” In the default data, the value 20 scores z = −0.06 — essentially dead on the mean. Values beyond ±2 are unusual for roughly bell-shaped data, and beyond ±3 are rare enough to warrant a second look at either the observation or the data entry. Z-scores also make different scales comparable: a student’s z of +1.5 in math and +0.3 in reading says something no pair of raw scores can.
Percentile rank answers the complementary question — what share of the dataset sits below a value. It uses the midpoint convention for ties: 24 appears three times with five values below it, giving (5 + 1.5) / 9 ≈ the 72.2nd percentile. Note this is the empirical rank within your data, not a claim about any wider population.
The confidence interval, and its honest limits
The interval uses the classic mean ± z × (sample SD / √n) construction with z at 1.645, 1.96, or 2.576 for 90%, 95%, and 99%. For the default data the 95% interval is 16.57 to 24.10 — a margin of ±3.77 around the mean. Widening confidence costs width: the same data gives a ±3.16 margin at 90% and ±4.95 at 99%. Certainty is bought with vagueness.
Two caveats keep this honest. First, the z-based formula assumes the sample is reasonably large or the underlying data roughly normal; below a few dozen observations, a t-distribution interval would be somewhat wider, so treat small-sample intervals here as slightly optimistic. Second, the interval says nothing about individual values — it brackets the mean, not where the next observation will land. Shrinking the interval by collecting more data follows a square-root law: quadrupling n only halves the margin.
Frequently asked questions
Should I use population SD or sample SD?
Use population SD when your list is the entire group you care about — every student in the class, every order this month. Use sample SD when the list is a subset standing in for something bigger — 9 measurements representing a machine’s typical output. Sample SD divides by n − 1 instead of n, which makes it slightly larger and corrects the bias that comes from measuring spread around the sample’s own mean.
Why are my mean and median so different?
A gap between them is a skew detector. The mean is pulled toward extreme values; the median is not. If the mean sits well above the median, a few large values are dragging it up — typical for incomes, house prices, and response times. When that happens, the median is usually the more honest single summary.
How does percentile rank handle values that appear multiple times?
It uses the midpoint convention: values strictly below the target count fully, and ties count half. In the default dataset, 24 appears three times with five values below it, so its rank is (5 + 1.5) / 9 = 72.2nd percentile. This avoids the ambiguity of a tied value being simultaneously "above" and "at" its own position.
What does a 95% confidence interval actually mean?
If you repeated the same sampling process many times and built an interval each time, about 95% of those intervals would contain the true population mean. It is a statement about the reliability of the procedure, not a 95% probability that the true mean sits inside this particular interval — a subtle but real distinction that trips up many reports.
Why do my quartiles differ from Excel or my textbook?
There are several accepted ways to compute quartiles. This tool uses linear interpolation on the sorted positions (the same method as Excel’s QUARTILE.INC). Textbooks that use the median-of-halves method can give slightly different Q1 and Q3 for the same data — neither is wrong, they are different conventions.
What input formats does the parser accept?
Paste numbers separated by commas, spaces, tabs, or line breaks — straight from a spreadsheet column works. Negatives, decimals, and scientific notation like 2.5e3 are all recognized; any surrounding text is ignored.