Statistics Calculator
Mean, median, mode, variance, standard deviation & quartiles from a list of numbers
Last updated June 15, 2026
Method: Standard descriptive-statistics formulas. Mean = ฮฃx / n; sample variance divides the sum of squared deviations by n−1 (Bessel's correction), population variance by n; standard deviation is the square root of each. Quartiles use linear interpolation between the closest ranks of the sorted data.
Included: Count, sum, mean, median, mode (including multimodal), min, max, range, sample and population variance, sample and population standard deviation, Q1/Q2/Q3 and the interquartile range.
Not included: Inferential statistics (confidence intervals, t-tests, regression), weighted means, grouped-frequency tables, and charts. Quartile values may differ slightly from tools that use a different quartile method.
๐ Enter your data set
Separate values with commas, spaces, or new lines. Decimals and negatives are fine.
๐ Mean (average)
๐ Spread & dispersion
๐ Range & quartiles
| Statistic | Value |
|---|---|
| Count (n) | 7 |
| Sum (ฮฃx) | 127 |
| Minimum | 12 |
| Maximum | 25 |
| Range | 13 |
| Q1 (25th percentile) | 15 |
| Q2 (median) | 18 |
| Q3 (75th percentile) | 21 |
| Interquartile range (IQR) | 6 |
Statistics calculator: everything you need to know
A statistics calculator turns a list of numbers into every descriptive measure at once. Paste 12, 15, 15, 18, 20, 22, 25 and it instantly returns the mean (18.14), median (18), mode (15), range (13), sample standard deviation (4.53), and quartiles (Q1 = 15, Q3 = 21) โ no formulas, spreadsheet, or separate mean-median-mode and variance tools needed.
Working with percentages of those numbers instead of their spread? Use the Percentage Calculator for a plain percent-of-a-number, the Percentage Increase Calculator or Percentage Change Calculator to measure growth or decline between two figures, and the Discount Calculator for sale-price math. This page is the one to reach for when you want the center and spread of a whole data set.
A full worked example
Take the data set 12, 15, 15, 18, 20, 22, 25 (n = 7). Here is how every result is produced, step by step.
Mean. Add the values and divide by the count:
mean = Σx ÷ n = 127 ÷ 7 ≈ 18.142857 Median. The data is already sorted and there are 7 values (odd), so the median is the 4th value: 18. If the count were even, the median would be the average of the two middle values.
Mode. The value 15 appears twice while every other value appears once, so the mode is 15. If no value repeated, there would be no mode.
Range. Maximum minus minimum = 25 − 12 = 13.
Variance and standard deviation. First find each value's deviation from the mean, square it, and add them up. The sum of squared deviations here is about 122.857. Then:
s² = Σ(x − mean)² ÷ (n − 1) = 122.857 ÷ 6 ≈ 20.476 So the sample variance is about 20.476 and the sample standard deviation is its square root, about 4.525. Dividing instead by n (= 7) gives the population variance of about 17.551 and a population standard deviation of about 4.189. The two differ only because one divides by n−1 and the other by n.
How to calculate standard deviation step by step
The table below shows exactly how the sum of squared deviations above is built for the data 12, 15, 15, 18, 20, 22, 25 with a mean of 18.142857. Subtract the mean from each value, square the result, then add the squares.
| Value (x) | Deviation (x − mean) | Squared deviation |
|---|---|---|
| 12 | −6.1429 | 37.7347 |
| 15 | −3.1429 | 9.8776 |
| 15 | −3.1429 | 9.8776 |
| 18 | −0.1429 | 0.0204 |
| 20 | 1.8571 | 3.4490 |
| 22 | 3.8571 | 14.8776 |
| 25 | 6.8571 | 47.0204 |
| Sum | 0 | 122.8571 |
Divide the total 122.8571 by n−1 = 6 for the sample variance (20.4762, sample SD 4.5251) or by n = 7 for the population variance (17.5510, population SD 4.1894). Notice the deviations always sum to zero โ a quick check that you subtracted the mean correctly.
Quartiles. With linear interpolation, Q1 (25th percentile) is 15, Q2 (the median) is 18, and Q3 (75th percentile) is 21, giving an interquartile range of Q3 − Q1 = 6. The IQR describes the spread of the middle half of the data.
Descriptive statistics for common data sets
Here is the full set of mean, median, mode, range, standard deviation and quartiles computed for six typical data sets, so you can see how the measures move together. Sample SD divides by n−1; population SD divides by n.
| Data set | n | Mean | Median | Mode | Range | Sample SD | Pop. SD | Q1 | Q3 | IQR |
|---|---|---|---|---|---|---|---|---|---|---|
| 72, 85, 90, 78, 88, 95, 82 | 7 | 84.29 | 85 | None | 23 | 7.72 | 7.15 | 80 | 89 | 9 |
| 12, 15, 15, 18, 20, 22, 25 | 7 | 18.14 | 18 | 15 | 13 | 4.53 | 4.19 | 15 | 21 | 6 |
| 120, 135, 150, 145, 160, 155 | 6 | 144.17 | 147.5 | None | 40 | 14.63 | 13.36 | 137.5 | 153.75 | 16.25 |
| 4, 8, 15, 16, 23, 42 | 6 | 18 | 15.5 | None | 38 | 13.49 | 12.32 | 9.75 | 21.25 | 11.5 |
| 68, 70, 72, 71, 69, 73, 70, 68 | 8 | 70.12 | 70 | 68, 70 | 5 | 1.81 | 1.69 | 68.75 | 71.25 | 2.5 |
| 200, 210, 205, 800, 215 | 5 | 326 | 210 | None | 600 | 265.03 | 237.05 | 205 | 215 | 10 |
The last row shows why the median matters: one outlier (800) drags the mean to 326 while the median stays at 210, and the range explodes to 600 even though the IQR is just 10.
The formulas, in plain notation
- Mean:
mean = Σx / n - Population variance:
σ² = Σ(x − mean)² / n - Sample variance:
s² = Σ(x − mean)² / (n − 1) - Standard deviation: the square root of the matching variance (σ or s).
- Range:
max − min - IQR:
Q3 − Q1
How to use this statistics calculator
- Enter your numbers: type them or paste a column straight from a spreadsheet. Separate values with commas, spaces, tabs, semicolons, or new lines.
- Check the count: the label shows how many valid values were detected, so you can confirm nothing was dropped or duplicated.
- Press โCalculate statisticsโ: the results appear in three cards โ central tendency, spread, and range/quartiles.
- Pick sample or population: read the sample column (n−1) for data that is a sample, and the population column (n) for data that is the whole group.
- Try the example buttons (test scores, daily sales, a small set) to see how the measures change with different data, then clear and enter your own.
Who this calculator is for
- Students in a statistics or math class checking homework answers for mean, median, mode, variance, and standard deviation.
- Teachers building or verifying problem sets and worked examples.
- Researchers and analysts who want a quick descriptive summary of a column before deeper analysis.
- Business users summarizing sales, response times, ratings, or survey scores without opening spreadsheet software.
- Anyone who has a list of numbers and wants the center and spread at a glance.
Key statistics terms explained
- Mean: the arithmetic average; the sum divided by the count. Sensitive to outliers.
- Median: the middle value of sorted data. Resistant to outliers, which makes it useful for skewed data like incomes.
- Mode: the most frequent value (or values). A data set can have one mode, several, or none.
- Variance: the average squared distance from the mean; spread in squared units.
- Standard deviation: the square root of variance; spread in the original units.
- Quartiles (Q1, Q2, Q3): the values that split sorted data into four equal parts; Q2 is the median.
- Interquartile range (IQR): Q3 − Q1, the spread of the middle 50% of values.
A second example: when the mean and median disagree
Consider the data 4, 8, 15, 16, 23, 42 (n = 6). The sum is 108, so the mean is 108 / 6 = 18. Because there are six values (even), the median is the average of the 3rd and 4th values: (15 + 16) / 2 = 15.5. The mean (18) sits above the median (15.5) because the large value 42 pulls the average upward. No value repeats, so there is no mode. This gap between the mean and median is a quick signal that the data is skewed โ when they are far apart, the median often describes the โtypicalโ value better than the mean.
Scenarios: same center, different spread
Two data sets can share a mean yet behave very differently. Both 49, 50, 51 and 10, 50, 90 have a mean of 50, but the first has a tiny sample standard deviation (about 1.0) while the second has a large one (about 40.0). That is the whole point of measuring spread: the center alone hides how consistent or volatile the values are. When you compare two groups โ say, two machines, two classes, or two months of sales โ look at the standard deviation alongside the mean, not just the average.
What changes the result the most
- Outliers: a single extreme value can swing the mean, range, and standard deviation sharply, while barely moving the median.
- Sample vs. population: for small data sets the n vs. n−1 choice noticeably changes the variance and standard deviation; for large sets the difference shrinks.
- Count (n): more values generally produce a more stable, trustworthy estimate of center and spread.
- Repeated values: duplicates create or shift the mode and can pull the mean and median toward the repeated value.
Sample vs. population standard deviation: how much do they differ?
Sample standard deviation (divide by n−1) is always slightly larger than population standard deviation (divide by n) for the same data, by an exact factor of √(n / (n−1)). This quick-reference shows how the gap shrinks as your data set grows, so you know when the choice actually matters.
| Count (n) | Multiply pop. SD by | Sample SD is larger by |
|---|---|---|
| 2 | 1.4142 | 41.42% |
| 3 | 1.2247 | 22.47% |
| 5 | 1.1180 | 11.80% |
| 10 | 1.0541 | 5.41% |
| 20 | 1.0260 | 2.60% |
| 30 | 1.0171 | 1.71% |
| 50 | 1.0102 | 1.02% |
| 100 | 1.0050 | 0.50% |
With only 2 values the sample SD is over 40% larger than the population SD, but by n = 100 the two agree to within half a percent. For small homework data sets the n vs. n−1 choice clearly matters; for large data sets it rarely changes a reported figure.
Tips for getting reliable statistics
- Confirm the count first. If the detected count does not match how many numbers you expected, look for a missing comma or a non-numeric label that was ignored.
- Decide sample or population before you read the answer so you quote the right variance and standard deviation.
- Report the median for skewed data (incomes, prices, response times) and the mean for roughly symmetric data.
- Pair a center with a spread. โMean 50, SD 2โ tells a very different story from โmean 50, SD 40.โ
- Keep full precision for intermediate steps and round only the final answer to avoid rounding errors compounding.
Limitations and assumptions
- This is a descriptive calculator. It summarizes the data you enter; it does not perform inferential tests, confidence intervals, hypothesis tests, or regression.
- Quartile method matters. This tool uses linear interpolation between ranks; spreadsheets and textbooks sometimes use a different rule, so Q1/Q3 can vary by a small amount.
- It treats each number as equally weighted โ there is no weighted mean or grouped-frequency support.
- Non-numeric entries are ignored rather than flagged, so always cross-check the count.
- Sample variance and sample standard deviation require at least two values; with a single number they are undefined and shown as โโโ.
How it compares to related calculators
This page answers โwhat are all the descriptive statistics for my list of numbers?โ If your question is narrower, a focused tool may be quicker:
- For just the center, use the Average Calculator (mean, median, mode).
- For spread on its own, use the Standard Deviation Calculator.
- To see how far one value sits from the mean in standard deviations, use the Z-Score Calculator.
- For the chance of events, use the Probability Calculator.
- For percent questions about your numbers, use the Percentage Calculator.
About this formula
The measures here are standard, universally defined descriptive statistics, so they need no external source โ the math is deterministic and identical across any correct implementation. The only convention worth stating is the one this tool follows: sample variance and standard deviation use Bessel's correction (divide by n−1), population versions divide by n, and quartiles use linear interpolation between the closest ranks of the sorted data. Other software may use a different quartile rule, which is the usual reason Q1 and Q3 differ slightly between tools.
Sources
This calculator uses standard descriptive-statistics formulas, which are deterministic and require no external data source. See the โAbout this formulaโ note above for the exact conventions used (Bessel's correction for sample statistics and linear-interpolation quartiles).
โ ๏ธ Common mistakes & edge cases
Using population stats when you have a sample
If your numbers are a sample of a larger group, divide by n−1, not n. Using the population formula on a sample underestimates the true spread. Read the โsampleโ column for s and sยฒ.
Letting an outlier hide behind the mean
One extreme value can move the mean a lot while the median barely changes. If the mean and median disagree, the data is skewed โ report the median or note the outlier.
Confusing variance with standard deviation
Variance is in squared units; standard deviation is its square root, back in the original units. Quote the standard deviation when you want a number comparable to the data itself.
Forgetting that quartile rules differ
Different tools use different quartile definitions, so Q1 and Q3 can differ by a little between this calculator and a spreadsheet. Match the method your assignment or software expects.
❓ Frequently asked questions
What does this statistics calculator compute?
This statistics calculator takes a list of numbers and returns a full set of descriptive statistics: the count, sum, mean (average), median, mode, minimum, maximum, range, sample and population variance, sample and population standard deviation, and the three quartiles (Q1, Q2, Q3) plus the interquartile range. You paste or type your data once and get every common summary measure at the same time.
How is the mean calculated?
The mean (arithmetic average) is the sum of all values divided by how many values there are: mean = ฮฃx / n. For example, the data 12, 15, 15, 18, 20, 22, 25 sums to 127 across 7 values, so the mean is 127 / 7 โ 18.14. The mean uses every data point, which is why a single very large or very small value can pull it noticeably up or down.
What is the difference between mean, median, and mode?
The mean is the average of all values. The median is the middle value when the data is sorted (or the average of the two middle values for an even count). The mode is the value that appears most often. They are all measures of the 'center' of a data set, but they answer slightly different questions: the median resists outliers, the mean reflects every value, and the mode shows the most common result.
What is the difference between sample and population standard deviation?
Population standard deviation (ฯ) divides the sum of squared deviations by n and is used when your data is the entire group you care about. Sample standard deviation (s) divides by n โ 1 (Bessel's correction) and is used when your data is a sample drawn from a larger population; dividing by n โ 1 corrects the tendency of a sample to underestimate the true spread. This calculator shows both so you can pick the right one.
When should I use sample vs. population statistics?
Use population formulas (divide by n) when your numbers represent every member of the group โ for example, the test scores of every student in one class you are describing. Use sample formulas (divide by n โ 1) when your numbers are a subset used to estimate a larger group โ for example, 30 customers surveyed to estimate all customers. When in doubt in research or statistics coursework, the sample versions are usually expected.
How are the quartiles calculated?
This calculator sorts the data and uses linear interpolation between the closest ranks (the same approach as the common 'inclusive' or linear method). Q1 is the value at the 25th percentile, Q2 is the median (50th percentile), and Q3 is the 75th percentile. The interquartile range (IQR) is Q3 โ Q1 and describes the spread of the middle 50% of the data. Note that other software sometimes uses slightly different quartile rules, so values can differ by a small amount between tools.
What is variance and how does it relate to standard deviation?
Variance is the average of the squared deviations from the mean โ it measures how spread out the data is, but in squared units. Standard deviation is simply the square root of the variance, which returns the measure to the original units of the data. A larger variance or standard deviation means the values are more spread out around the mean; a value near zero means they cluster tightly.
Can the data set have more than one mode?
Yes. A data set is bimodal when two values tie for the most frequent, and multimodal when three or more tie. This calculator lists every value that shares the highest frequency. If every value appears exactly once, there is no mode and the calculator reports 'None', because no value is more common than the others.
What input formats does the calculator accept?
You can separate your numbers with commas, spaces, tabs, semicolons, or new lines โ paste a column from a spreadsheet or type a comma-separated list, either works. Decimals (like 3.14) and negative numbers (like โ5) are supported. Any text that is not a valid number is ignored, so a stray label will not break the calculation.
Is this a free statistics calculator?
Yes. This is a completely free statistics calculator with no sign-up, no fee, and no limit on the number of values or how many data sets you run. Everything is computed instantly in your browser, so your numbers are never uploaded to a server.
What is the range and why is it a weak measure of spread?
The range is the difference between the largest and smallest values (max โ min). It is the simplest measure of spread, but it depends only on the two extreme values, so a single outlier can make it misleading. Standard deviation and the interquartile range describe spread more reliably because they use more of the data.
Does this calculator round the results?
Results are shown to up to four decimal places and trailing zeros are trimmed for readability, but the underlying math is done in full double precision before display. If you need more decimal places for a homework answer, keep the intermediate sum and sum-of-squares from the formulas rather than re-rounding the displayed figures.
How do I calculate standard deviation step by step?
Four steps: (1) find the mean; (2) subtract the mean from each value to get its deviation; (3) square each deviation and add them up to get the sum of squared deviations; (4) divide by nโ1 (sample) or n (population) to get the variance, then take the square root. For the data 12, 15, 15, 18, 20, 22, 25 the mean is 18.142857, the squared deviations sum to 122.8571, so the sample variance is 122.8571 / 6 โ 20.4762 and the sample standard deviation is โ20.4762 โ 4.5251.
Can I paste data from Excel or Google Sheets?
Yes. Copy a column of numbers from Excel, Google Sheets, or a CSV and paste it straight into the input โ values on separate lines are read correctly, as are commas, spaces, tabs, and semicolons. Row labels, currency symbols, and other non-numeric text are ignored, so always check that the detected count matches the number of cells you copied.
๐ก Good to know
The mean is not always the โtypicalโ value
For skewed data like home prices or salaries, a few large values inflate the mean. The median usually describes the typical case better, which is why income statistics are almost always reported as medians.
Standard deviation has an intuitive meaning
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two. That โempirical ruleโ turns a single number into a quick sense of how spread out your data is.
Your data stays in your browser
Every calculation runs locally with JavaScript โ nothing you type is uploaded. You can run as many data sets as you like, for free, with no sign-up.
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