Descriptive Statistics
Enter a list of numbers to get a full descriptive summary: the mean, median and mode, the minimum, maximum and range, and both the population and sample variance and standard deviation. Each measure is shown with its formula.
Three centres, because they disagree
A list of numbers is hard to hold in mind. Descriptive statistics compress it into figures answering two questions: where is the data centred, and how spread out is it. The centre is described by the mean, median and mode; the spread by the range, the variance and the standard deviation.
This calculator computes all of them from one comma-separated list, and shows each measure with the arithmetic that produced it rather than as a bare number.
Reporting three measures of centre is not redundancy. The mean uses every value and moves when any one of them moves; the median cares only about position in the sorted order; the mode is what occurs most often. On symmetric data they nearly coincide; on skewed data they separate, which is itself informative.
The same applies to spread. The range depends entirely on the two extreme values, so a single outlier sets it. The standard deviation weighs every deviation from the mean, describing the bulk of the data rather than its edges.
How to use this calculator
- Enter your data as a list Numbers separated by commas, spaces or semicolons — all three work, and they can be mixed. Decimals and negatives are accepted.
- Check the count and sum first The first output line reports n and Σx. If n does not match how many values you meant to enter, a separator was missed or an extra one typed, and everything below is computed from the wrong data.
- Read the centre measures together Mean, median and mode are reported in sequence. Comparing them beats reading any one alone, because their disagreement measures the skew.
- Choose between the population and sample spread Both variance and standard deviation are given in two versions. Which to quote depends on whether your list is the whole group or a sample from a larger one.
The formula, and where it comes from
x̄ = Σx / n σ² = Σ(x − x̄)² / n s² = Σ(x − x̄)² / (n − 1) σ = √σ²
The mean is the total divided by the count — the balance point of the data, in the sense that the deviations above it exactly cancel those below.
Both variances start from the same quantity: the sum of squared deviations from the mean. Squaring serves two purposes. It removes the signs, so that deviations above and below do not cancel, and it penalises large departures more than small ones, which is why a single distant value moves the variance so much.
The two versions differ only in the divisor. Dividing by n gives the population variance, correct when your list is the entire group of interest. Dividing by n − 1 gives the sample variance, which is larger and is right when the list is a sample: deviations from the sample's own mean are systematically a little too small, and the reduced divisor corrects that bias.
The standard deviation is the square root of the variance, and exists mainly so the spread is expressed in the data's own units: a variance in squared centimetres is hard to interpret, a standard deviation in centimetres is not. The sample version is computed only when there are at least two values, since n − 1 would otherwise be zero.
What each input means
- x Data set — form field “Data set (comma-separated)”
- The list of observations, separated by commas, spaces or semicolons. Order does not matter: the solver sorts internally for the median. Every entry must parse as a finite number, or the list is rejected.
- n Count
- How many values were read. Reported first, as the quickest confirmation that the list parsed as intended.
- x̄ Mean
- The arithmetic average, Σx / n. Shown with both the total and the count so the division can be checked.
- σ, s Standard deviations
- Population and sample versions respectively, in the same units as the data. The sample value is always the larger of the two.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A spread-out set with no repeats
The six values 4, 8, 15, 16, 23, 42. Nothing repeats, and the last value sits well above the rest, so this shows both the no-mode case and the effect of a high outlier.
Inputs Data set (comma-separated) = 4, 8, 15, 16, 23, 42
- Count & sum n = 6, Σx = 108
- Mean x̄ = Σx / n = 108 / 6 = 18
- Median middle value of the sorted data = 15.5
- Mode no mode (all values unique)
- Range max − min = 42 − 4 = 38
- Population variance σ² = Σ(x − x̄)² / n = 151.667
- Population std dev σ = √σ² = 12.3153
- Sample variance s² = Σ(x − x̄)² / (n − 1) = 182
- Sample std dev s = √s² = 13.4907
Result mean = 18, median = 15.5, mode = no mode (all values unique), σ = 12.3153
The mode line reads “no mode (all values unique)” rather than a number. That is the correct result, not a failure: a mode exists only when some value occurs more often than the others.
The mean lands above the median, pulled up by 42. With an even count the median is the average of the two middle values after sorting, here 15 and 16. The gap between the two centre measures is the signature of a right-skewed set.
A set with a clear mode
The eight values 2, 3, 3, 5, 7, 7, 7, 9. Here 7 appears three times and 3 appears twice, so the most frequent value is unambiguous.
Inputs Data set (comma-separated) = 2, 3, 3, 5, 7, 7, 7, 9
- Count & sum n = 8, Σx = 43
- Mean x̄ = Σx / n = 43 / 8 = 5.375
- Median middle value of the sorted data = 6
- Mode 7
- Range max − min = 9 − 2 = 7
- Population variance σ² = Σ(x − x̄)² / n = 5.48438
- Population std dev σ = √σ² = 2.34187
- Sample variance s² = Σ(x − x̄)² / (n − 1) = 6.26786
- Sample std dev s = √s² = 2.50357
Result mean = 5.375, median = 6, mode = 7, σ = 2.34187
The mode is 7 alone, since it is strictly more frequent than anything else. Had 3 also appeared three times, both would have been reported: the solver lists every value tied at the maximum frequency.
Comparing the two standard deviations shows the effect of the divisor. With n = 8 the sample version divides by 7 instead of 8, making it roughly seven percent larger — a difference that shrinks steadily as the data set grows.
Reading the result
What the gap between mean and median tells you
When they are close, the data is roughly symmetric. When the mean is noticeably higher, a few large values are pulling it up; when it is lower, a few small ones are pulling it down. For skewed data the median is usually the more honest single summary, which is why incomes are reported as medians.
Reading the standard deviation
It is a typical distance from the mean, in the data's own units. Roughly two thirds of values fall within one standard deviation for approximately bell-shaped data, but that rule of thumb comes from the normal distribution and fails for strongly skewed sets.
When you would use this
Summarising a set of measurements
Test scores, response times, yields from repeated trials: a mean with a standard deviation attached distinguishes a consistent process from an erratic one with the same average.
Checking work before further analysis
Most statistical procedures take a mean and a standard deviation as inputs. Computing them here first, with the count visible, catches transcription errors before they reach a confidence interval or a hypothesis test.
Assumptions and limitations
What this calculator assumes
- Every entry in the list parses as a finite number; a stray letter or symbol causes the whole list to be rejected rather than silently skipped.
- The data is numeric and at least interval-scaled — the mean of a set of category codes is arithmetic without meaning.
Where it stops being the right tool
- Quartiles, the interquartile range and percentiles are not computed, so there is no five-number summary.
- No grouped or weighted data: frequencies must be expanded into repeated values before entry.
- One variable at a time. Correlation, covariance and regression between two lists are outside this solver.
Common mistakes
Quoting the population standard deviation for sample data
Why it happens. It is the first of the two reported and the smaller of them, so it reads as the default answer. On a genuine sample it understates the spread, and every interval built from it comes out too narrow.
How to avoid it. Ask what the list represents. If it is a subset standing in for a larger group, use the s and s² lines, not σ and σ².
Reporting the mean for visibly skewed data
Why it happens. The mean is the familiar average, and it is what “average” is usually taken to mean. When a few extreme values dominate, it describes no typical member of the set.
How to avoid it. Compare the mean with the median. If they differ substantially, quote the median as the centre and say the data is skewed rather than presenting a single figure.
Losing values to a malformed separator
Why it happens. A missing comma joins two numbers into one, and a decimal point used as a thousands mark changes how many values are read.
How to avoid it. Check the reported n against how many values you intended. That one number verifies the entire input in a single glance.
Key terms
Frequently asked questions
What is the difference between the population and sample standard deviation?
The divisor. The population version divides the sum of squared deviations by n; the sample version divides by n − 1, making it slightly larger. The correction compensates for measuring deviations from the sample's own mean.
What happens if no value repeats?
The result is reported as having no mode. A mode requires some value to occur more often than the rest, so a list of distinct values has none. If several values tie at the highest frequency, all of them are listed.
How should I format my data?
As a list separated by commas, spaces or semicolons — any of the three, and they can be mixed. Decimals and negatives are fine, and the order does not matter.