Statistics Calculator - Mean, Median, Quartiles, Variance, SD

Descriptive statistics compress a list into location (where the data sit) and spread (how widely they sit). This page is the combined report: count, sum, mean, median, mode, extrema, range, quartiles, IQR, and both the sample and population variance/SD. It is not a t-test, not a regression, and not a weighted mean.

Three sibling tools on DevOkk.com stay specialized on purpose. The average calculator is the four-number / weighted-mean page - use it when some observations count more than others. The standard deviation calculator is SD-only: pick sample or population and read s or σ with a short spread strip. The mean, median, mode calculator is the search four-pack (plus range) without quartiles or variance. Come here when one paste should print the whole descriptive table, including Tukey hinges.

Default list 12, 15, 15, 18, 20, 22, 30 has n = 7, sum = 132, mean ≈ 18.857, median 18, mode 15, min 12, max 30, range 18. Tukey hinges include the median 18 in both halves, so Q1 = 15 and Q3 = 21, IQR = 6. Exclusive-hinge software that drops the median from both halves would report Q3 = 22 instead. That difference is documented so a homework key can be matched.

Standardizing a single value after you know the mean and SD is a z-score follow-up, not a column on this table. Keep the workflow: describe the list here, then convert one observation to a z if a course asks for it.

1. Paste numbers separated by commas, spaces, semicolons, or line breaks. Scientific notation such as 1.2e3 is accepted.
2. Need n ≥ 1. Sample variance and sample SD appear only when n ≥ 2; population formulas still run at n = 1 (variance 0).
3. Copy the full report, toggle fullscreen, or drag the desktop splitter. Clear empties the box. The list is stored only in this browser for 30 days.

Quartile definitions are not unique. Spreadsheets may use a percentile with linear interpolation (Excel QUARTILE.INC vs QUARTILE.EXC). This calculator uses the common median of halves method known as Tukey hinges:

• Sort the sample x(1) ≤ … ≤ x(n).
• The overall median is the ordinary median (middle value, or average of the two central values when n is even).
• If n is even, the lower half is the first n/2 observations and the upper half is the last n/2.
• If n is odd, the overall median is included in both halves (Tukey’s inclusive hinge).
• Q1 is the median of the lower half; Q3 is the median of the upper half. IQR = Q3 − Q1.

For the default seven points, halves are 12, 15, 15, 18 and 18, 20, 22, 30. Each half has even length, so Q1 = (15+15)/2 = 15 and Q3 = (20+22)/2 = 21. A method that excludes 18 from both halves would use 12, 15, 15 and 20, 22, 30 (Q1 = 15, Q3 = 22). If your answer key disagrees by a point or two, check which hinge convention it named - not whether the list was typed wrong.

IQR is resistant: changing 30 to 300 leaves Q1, Q3, and IQR unchanged under this split, while the range and the standard deviation jump. Outlier fences Q1 − 1.5 IQR and Q3 + 1.5 IQR are not flagged automatically; compute them from the printed hinges if a boxplot problem asks.

Let SS = Σ(x − mean)². Population variance σ² = SS / n treats the list as the entire universe. Sample variance s² = SS / (n − 1) treats the list as a sample and uses Bessel’s correction so s² is unbiased for a larger population’s variance under i.i.d. normal-ish sampling. Standard deviations are the square roots. This page always prints both so you can match a prompt that says “σ” versus “s”.

A single number cannot have a sample SD: the denominator n−1 would be zero. The UI says “need n ≥ 2” rather than dividing by zero. Population SD of one point is 0, which is mathematically consistent and usually uninteresting.

If you only need s or σ and not quartiles, the standard deviation calculator is the shorter interface. If you only need the four-pack of center, use mean-median-mode. Weighted observations stay on average. z = (x − mean) / SD is the z-score calculator.

The mean is the balance point. The median splits the ordered list in half. The mode is the most frequent value (or values). On a symmetric unimodal sample they nearly agree. On the default list the mean (~18.86) sits a little above the median (18) because 30 pulls the average; the mode is 15 because that value appears twice. A right-skewed income sample would show the same pattern more dramatically - which is why housing reports quote median price.

Mode on a continuous measurement with all unique values is reported as “no unique mode.” Tying frequencies are listed as multimodal rather than picking an arbitrary winner. That matches the mean-median-mode page; this statistics page simply adds quartiles and variances around the same center measures.

Data: 12, 15, 15, 18, 20, 22, 30. Sorted, that is already ordered. n = 7. Sum = 132. Mean = 132/7 ≈ 18.857142857. Median = 18 (fourth of seven). Mode = 15. Min = 12, max = 30, range = 18. Tukey Q1 = 15, Q3 = 21, IQR = 6.

Deviations from the mean: about −6.857, −3.857, −3.857, −0.857, +1.143, +3.143, +11.143. SS ≈ 204.857. Population variance ≈ 29.265, population SD ≈ 5.410. Sample variance ≈ 34.143, sample SD ≈ 5.843. Use those figures to check a hand-worked table; slight rounding in a textbook (mean kept at 18.86) will move SS by a hair.

A classical five-number summary is min, Q1, median, Q3, max. This table prints those five plus mean, mode, range, IQR, and both SD conventions. A boxplot whisker is often drawn to the last point inside 1.5 IQR of the hinges, with points beyond plotted as outliers. Because hinges differ by algorithm, two boxplots of the same list can look slightly different. If a lab uses SPSS, R’s type 7 quantile, or Excel QUARTILE.INC, do not expect bitwise agreement with Tukey hinges on every n.

Hyndman and Fan catalogued many quantile definitions (R types 1 through 9). Inclusive Tukey hinges are closest in spirit to a “median of the half, keep the middle” classroom method, not to a continuous percentile of (n−1)p + 1. For n = 4 the even split is unambiguous: lower two vs upper two. For n = 5 the inclusive method reuses the median in both halves; exclusive hinges would leave three and three after dropping the middle. Neither is “wrong”; they answer slightly different geometric questions about the ordered sample.

Range versus IQR versus SD: range uses two points and explodes with one typo. IQR uses the middle half and ignores the tails by construction. SD uses every squared deviation from the mean, so it is the right companion to the mean and to z-scores, and the wrong companion when a single corrupted sensor reading should not dominate a QC chart. Reporting all three on this page is deliberate: you can see when they tell different stories.

Frequency tables (10 students scored 70, 4 scored 80) are weighted data. Expand them into a long list or use the average calculator for a weighted mean. This paste box will treat “70, 10” as two numbers 70 and 10, not as a score of 70 with frequency 10. Histogram bin midpoints as if they were raw microdata will also bias the SD; that approximation is a methods choice, not something this tool infers from commas.

Inferential machinery - confidence intervals, p-values, two-sample tests - needs a model on top of the descriptive table. A confidence interval for a mean uses the sample SD and a t or z critical value; compute s here, then move to the z-score calculator or a dedicated interval page. Do not read “population SD” on a sample of 12 students as if you had measured every student in the country; the n-denominator is a formula choice, not a claim about the universe.

Missing values and non-numeric tokens are rejected rather than silently dropped. A blank between two commas is not a zero. If a CSV export includes a header word, delete it before pasting. Ten thousand points is a practical cap for in-browser sort and SS accumulation; larger files belong in a spreadsheet or a statistics package.

Recompute SS with a slightly rounded mean 18.86 instead of 132/7 and you will not match this page. Keep the exact mean 18.857142… or keep fractions: mean = 132/7, SS = Σ(x − 132/7)². Population variance is SS/7, sample variance SS/6. If a teacher’s key used the n denominator and called it “standard deviation” without saying sample or population, compare both printed SD values before arguing the key is wrong.

Mode on this default is uniquely 15. Change the 30 to 15 and the list 12, 15, 15, 18, 20, 22, 15 becomes trimodal-or-not: 15 now appears three times and is the unique mode; Q1/Q3 also move because the ordered sample changed. Duplicate values are not errors. They are the point of a mode.

Scientific notation in the paste (1.5e2) is a number 150, not a token error. A thousands separator written as 1,500 with a comma will split into 1 and 500 unless you remove the comma grouping - the parser splits on commas. Use 1500 or spaces as thousands separators if you must, or paste one number per line without grouping commas.

Copy-paste from a PDF sometimes injects Unicode minus signs or thin spaces. If a token fails, retype the minus as a keyboard hyphen. Mixed lists of integers and decimals are fine: 12, 15.5, 18 is a legal sample. Empty lines are ignored. A trailing comma does not insert a missing zero.

Paste once, read the whole descriptive story: where the sample sits, how wide the middle half is, and how s versus σ differ on the same SS. That is the job of a statistics calculator as a homework companion. It will not invent a p-value or a weighted mean, and it will tell you which hinge convention produced Q1 and Q3 so you can defend the numbers against a spreadsheet that used a different quantile.

• Full descriptive table from one paste - center, quartiles, and both SD conventions.
• Documented Tukey hinges so Q1/Q3 are not a mystery Excel percentile.
• Clear split from the average, SD-only, and mean-median-mode tools.
• Private. The list never leaves the browser.