Descriptive Statistics Calculator

How to use this calculator

1. Type or paste your numbers into the data box. Separate them with commas, spaces, or new lines, or mix all three; the calculator reads every valid number and ignores blanks and non-numeric text.
2. Choose whether to treat your numbers as a sample or a whole population. This only changes which variance and standard deviation pair is highlighted as your main answer; both are always shown.
3. Set how many decimal places to display, from 0 to 10. This affects the displayed values only, not the underlying math.
4. Pick the order for the value list echoed back to you: ascending, descending, or as entered.
5. Select Calculate. The tool shows the count, sum, mean, median, mode, minimum, maximum, range, quartiles Q1 and Q3, the interquartile range, and both sample and population variance and standard deviation.

How it works

This is one-variable descriptive statistics, the standard summary of a single list of numbers. The definitions follow the NIST/SEMATECH e-Handbook of Statistical Methods, with the sample-versus-population rule corroborated by Penn State STAT 200.

The basics come first. The count is how many numbers you entered. The sum adds them all up. The mean is the sum divided by the count. The minimum and maximum are the smallest and largest values, and the range is the maximum minus the minimum.

For the median, the calculator sorts the data and takes the middle value. When you have an even count, it averages the two middle values. The mode is whatever value appears most often. If several values tie, all of them are shown; if nothing repeats, it reports no mode.

For the quartiles, this calculator uses the exclusive method, also called the Moore and McCabe method. It splits the sorted data at the median, takes the median of the lower half as the first quartile (Q1), and takes the median of the upper half as the third quartile (Q3). When the count is odd, the overall median is left out of both halves. The Journal of Statistics Education describes this method and the textbooks that teach it. The interquartile range (IQR) is simply Q3 minus Q1.

Spread is reported two ways, following the NIST handbook section on measures of scale. Population variance divides the sum of squared differences from the mean by n, and population standard deviation is its square root. Sample variance divides by n minus 1 instead. That n minus 1, called Bessel’s correction, is used when your data is a sample drawn from a larger group, and sample standard deviation is its square root. Both pairs are always shown, so you never have to guess which one your assignment wants.

Examples

These two cases match the figures the calculator returns, so you can check your own entry against them.

If you enter the eight values 10, 2, 38, 23, 38, 23, 21, 23 as a sample, the tool returns a count of 8, a sum of 178, and a mean of 22.25. The median is 23 and the mode is 23, since 23 appears three times. The minimum is 2, the maximum is 38, and the range is 36. Using the exclusive method, Q1 is 15.5 and Q3 is 30.5, so the IQR is 15. The sample variance is about 151.36 with a sample standard deviation near 12.30, while the population variance is 132.4375 with a population standard deviation near 11.51.

If you enter the five values 4, 8, 15, 16, 23 as a population, the count is 5, the sum is 66, and the mean is 13.2. The median is 15 and there is no mode, since every value is unique. The minimum is 4, the maximum is 23, and the range is 19. Because the count is odd, the median is left out of both halves, giving Q1 of 6 and Q3 of 19.5, so the IQR is 13.5. The population variance is 43.76 with a population standard deviation near 6.62, and the sample variance is 54.7 with a sample standard deviation near 7.40.

Every statistic this calculator reports, defined

Each output below measures one of two things: where the data sits (its center, called a measure of location) or how spread out the data is (its spread, called a measure of scale). The terms and symbols follow the NIST/SEMATECH e-Handbook and OpenStax Introductory Statistics.

Count (n)

How many valid numbers you entered. It sets the divisor for the mean and is the n in the variance formulas.

Sum (Σx)

All your numbers added together. The symbol Σ means add up.

Mean (x̄)

The average: the sum divided by the count. It is a measure of center and is written x̄, read x-bar (NIST).

Median (Q2)

The middle value once the data is sorted, or the average of the two middle values when the count is even. It is a measure of center that is not pulled around by extreme values (NIST).

Mode

The value that appears most often. If several values tie, all are reported; if nothing repeats, there is no mode.

Minimum and maximum

The smallest and largest values in your data.

Range

The maximum minus the minimum. It is the simplest measure of spread.

First and third quartiles (Q1 and Q3)

Q1 is the 25th percentile and Q3 is the 75th percentile, the cut points that bound the middle half of the data (NIST).

Interquartile range (IQR)

Q3 minus Q1, the spread of the middle 50 percent of the data (NIST).

Variance and standard deviation

Variance is the average of the squared distances from the mean, so its units are squared. The standard deviation is the square root of the variance, which puts the spread back into the original units. The sample versions use the symbols s² and s; the population versions use σ² and σ (NIST, OpenStax).

Sample vs. population: why this tool divides by n − 1

Variance and standard deviation come in two forms, and the only difference is the number you divide by. Population variance (σ²) divides the sum of squared differences from the mean by n. Sample variance (s²) divides by n minus 1 instead. That n minus 1 is called Bessel’s correction, and it gives a less biased estimate of the spread of the larger group your sample came from (OpenStax, NIST).

Here is the walkthrough on the small set 4, 8, 15, 16, 23.

1. Find the mean. The sum is 66 and the count is 5, so the mean is 13.2.
2. Add up the squared differences from the mean. They come to 218.8.
3. Divide. For the population version, divide by n, which is 5: 218.8 divided by 5 is 43.76. For the sample version, divide by n minus 1, which is 4: 218.8 divided by 4 is 54.7. The standard deviations are the square roots, about 6.62 and 7.40.

Notice the sample version is larger. Dividing by the smaller number nudges the estimate up to account for the fact that a sample usually understates the true spread.

So which do you pick? Use the population form when your data is the entire group you care about, such as the test scores of every student in one class. Use the sample form when your data is a subset meant to stand in for a larger group, such as 30 students chosen to represent a whole school. This calculator always shows both pairs and highlights the one you select, defaulting to sample, so you never have to guess which number your assignment wants. Sample variance and standard deviation are undefined for a single data point, because n minus 1 would be zero.

Mean or median? Reading the right center for skewed data

The mean and the median both describe the center of your data, but they answer to outliers very differently. The mean adds every value and divides, so one very large or very small value drags it in that direction. The median is just the middle value, so it barely moves when a few values are extreme. Statisticians call the median robust for that reason (OpenStax, NIST).

Income is the classic case. Picture a small group where almost everyone earns about 30,000 a year, but one person earns 5 million. The mean climbs into six figures, far above what anyone in the room actually earns, while the median stays near 30,000 and describes the typical person well. When a distribution has a long tail like this, the mean and the median pull apart, and the median is usually the more honest summary (OpenStax).

Use this quick guide:

• Lean on the median when the data is skewed, has outliers, or has a long tail: incomes, home prices, response times.
• Lean on the mean when the data is roughly symmetric with no extreme values, and when you need a number you can add and average further, as the variance and standard deviation do.

A fast check: compare the two. When the mean sits well above or below the median, your data is skewed, and the median is likely the safer center to report.

How to read quartiles, the IQR, and spot outliers

Quartiles split your sorted data into four parts. The table below shows what each cut point marks and how much of the data it bounds, following the NIST e-Handbook quartile and outlier section.

Statistic	Percentile	What it bounds
Minimum	0th	The smallest value
Q1 (first quartile)	25th	Lower 25 percent sits below it
Q2 (median)	50th	Half the data sits below it
Q3 (third quartile)	75th	Lower 75 percent sits below it
Maximum	100th	The largest value
IQR (Q3 − Q1)	n/a	Spread of the middle 50 percent

The IQR is also the basis for the standard outlier test used in box plots. Build a fence on each side of the middle half (NIST):

• A value below Q1 minus 1.5 times the IQR, or above Q3 plus 1.5 times the IQR, is a mild outlier.
• A value below Q1 minus 3 times the IQR, or above Q3 plus 3 times the IQR, is an extreme outlier.

For example, with Q1 of 15.5, Q3 of 30.5, and an IQR of 15, the mild outlier fences fall at 15.5 minus 22.5, which is −7, and 30.5 plus 22.5, which is 53. Any value outside that range would be flagged as a mild outlier.

What the data says

Most people land on a stats calculator with the same complaint: their answer is close but not exactly the answer key. The usual cause is not a broken tool. It is grabbing the wrong standard deviation when the screen shows two at once, the sample version (Sx, divided by n minus 1) and the population version (sigma, divided by n).

That confusion runs deep. A peer-reviewed statistics-education study found that introductory students often picture standard deviation as plain bar height or raw spread, and only grasp it as variation about the mean after deliberate practice (Statistics Education Research Journal (delMas & Liu)). So when learners ask why their sample standard deviation divides by n minus 1 instead of n, the standard textbook answer is about better estimation, not an arbitrary rule:

“The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by (n - 1) gives a better estimate of the population variance.”

OpenStax, Introductory Statistics, in Measures of the Spread of the Data.

Choosing the wrong center misleads in the same quiet way. Income is the clearest case. The U.S. Census Bureau reports a real median household income of $80,610 for 2023, not an average, because a handful of very high incomes would pull a mean upward and misrepresent the typical household (U.S. Census Bureau). The mechanism is simple: in a skewed distribution the mean is dragged toward the long tail, while the rank-based median stays put (NIST/SEMATECH e-Handbook).

So which center should you report? This table follows the NIST guidance and gives you a one-glance rule by the shape of your data (NIST/SEMATECH e-Handbook).

Distribution shape	Best location estimator	Reason
Symmetric / normal	Mean	Most efficient estimator under normality
Skewed	Median (mean also reported)	Mean is pulled toward the skew
Heavy-tailed / outliers	Median	Extreme tail values distort the mean but not the rank-based median

What this tool does that others don’t

• It returns the full one-variable summary on one screen. Some calculators stop at mean, variance, and standard deviation and skip the median, mode, range, minimum, maximum, quartiles, and IQR, so you have to finish the job somewhere else.
• It names the quartile method on the result and explains why other tools differ. The exclusive method and the inclusive method can give different Q1 and Q3 for an odd count, and most tools never say which one they use. This one states the exclusive method plainly, so your results are reproducible.
• It explains when to use sample versus population variance and standard deviation. That choice is the most common source of student error, and many tools leave it unaddressed. Here both pairs are shown and the difference is spelled out.
• It tells you how it handles ties and mixed separators. Multiple modes, no mode, and a mix of commas, spaces, and new lines are all handled the same way every time, with no silent surprises.

Frequently asked questions

Does this compute the same descriptive statistics taught in courses?

Yes. This is an independent, free calculator for one-variable descriptive statistics. It returns the same standard measures taught in introductory statistics courses: mean, median, mode, range, quartiles, the interquartile range, and both the sample and population versions of variance and standard deviation. You can use it to check the same kind of one-variable results.

What statistics does this calculator compute?

For any list of numbers it returns the count, sum, mean, median, mode, minimum, maximum, range, the first and third quartiles (Q1 and Q3), the interquartile range (IQR), and both the sample and population versions of variance and standard deviation.

How do I enter my data?

Paste or type your numbers into the box separated by commas, spaces, or new lines. You can mix separators and the calculator will still read every number correctly. Blank lines, extra spaces, and any non-numeric text are ignored.

What does the sample versus population option do?

The calculator always computes both versions of variance and standard deviation. The sample-or-population selector just chooses which pair is highlighted as your main answer. Choose population when your data is the entire group, and sample when it is a subset drawn from a larger group.

When do I divide by n versus n minus 1?

Divide by n for population variance and standard deviation, and by n minus 1 for the sample versions. If a question mentions a sample or asks you to estimate the spread of a larger group, use n minus 1. Sample statistics are undefined for a single data point because n minus 1 would be zero.

Which quartile method does this calculator use?

It uses the exclusive method: the sorted data is split at the median, then Q1 is the median of the lower half and Q3 is the median of the upper half, with the overall median left out of both halves. This matches the method taught in many courses, often called the Moore and McCabe method.

Why do other calculators give different quartiles for the same data?

There is no single universal definition of a quartile. Some tools use an inclusive method that keeps the median in each half, and spreadsheet functions use interpolation, so Q1 and Q3 can differ slightly between tools. This calculator states clearly that it uses the exclusive method so your results are reproducible.

How is the interquartile range (IQR) calculated?

The IQR is the third quartile minus the first quartile, so IQR equals Q3 minus Q1. It measures the spread of the middle 50 percent of the data and is often used to find outliers: values below Q1 minus 1.5 times the IQR, or above Q3 plus 1.5 times the IQR, are commonly flagged as potential outliers.

Sources

• NIST/SEMATECH e-Handbook, 1.3.5.1 Measures of Location. Defines the mean as the sum of the data divided by the number of points, and the median as the middle value or the average of the two middle values for an even count.
• NIST/SEMATECH e-Handbook, 1.3.5.6 Measures of Scale. Defines the variance and standard deviation, including the divide-by-n-minus-1 form for a sample.
• NIST/SEMATECH e-Handbook, 7.1.6. Defines the lower and upper quartiles as the 25th and 75th percentiles and the IQR as Q3 minus Q1.
• Penn State STAT 200, Formulas. Corroborates the population variance over N versus sample variance over n minus 1 distinction.
• Eric Langford, Quartiles in Elementary Statistics, Journal of Statistics Education 14(3), 2006. Defines the exclusive quartile method used here and lists the textbooks that teach it.