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Descriptive statistics calculator

Summarize a numerical data set

Enter a list of values to organize the data and calculate measures of center and spread without confusing sample statistics with population parameters.

Separate values with commas, spaces, or new lines.
Try an example

Result

The data center around 7 to 8. The value 7 occurs most often, and the values span 9 units from 4 to 13.

n=5xˉ=8x~=7mode⁡=7R=9\begin{aligned}n&=5 & \bar{x}&=8\\ \widetilde{x}&=7 & \operatorname{mode}&=7\\ R&=9\end{aligned}
Conditions
  • Each entry must be a finite real number.
  • The sample standard deviation requires at least two observations.
  • Population variance divides by N; sample variance divides by n minus 1.
  • A data set can have no mode or more than one mode, so the mode is not forced to be unique.

Steps

  1. Count and order the observations The list is already ordered, and it contains five values.4, 7, 7, 9, 13n=54,\ 7,\ 7,\ 9,\ 13 \qquad n=5
  2. Calculate the mean Add all observations and divide by their count.xˉ=4+7+7+9+135=405=8\bar{x}=\frac{4+7+7+9+13}{5}=\frac{40}{5}=8
  3. Read the median, mode, and range The middle ordered value is 7, the repeated value is 7, and the maximum minus the minimum is 9.x~=7mode⁡=7R=13−4=9\begin{aligned}\widetilde{x}&=7 & \operatorname{mode}&=7\\ R&=13-4=9\end{aligned}
  4. Measure spread about the mean The squared deviations sum to 44. Divide by 5 for the population or by 4 for a sample, then take a square root for standard deviation.∑(xi−8)2=44σ=445s=444\begin{aligned}\sum (x_i-8)^2&=44\\ \sigma&=\sqrt{\frac{44}{5}}\\ s&=\sqrt{\frac{44}{4}}\end{aligned}
Independent check

The ordered-list check confirms the median and range. The deviations sum to zero, and their squared values 16, 1, 1, 1, and 25 sum independently to 44.

Scope

What this descriptive-statistics covers

For the data 4, 7, 7, 9, 13, there are 5 observations. The mean is 8, the median is 7, the mode is 7, and the range is 9. The population standard deviation is about 2.966, while the sample standard deviation is about 3.317.

Measures of center

Calculate the arithmetic mean, median, and all values tied for the mode.

Examples: Mean of 2, 4, 6, Median of 2, 3, 7, 8, Modes of 1, 1, 2, 2, 5

Measures of spread

Calculate range, population variance and standard deviation, or sample variance and standard deviation.

Examples: Range of 4, 7, 7, 9, 13, Population spread of -3, 0, 3, Sample spread of 1, 3, 5

Ordered summaries

Sort values before identifying the median, minimum, and maximum so position-based summaries are reproducible.

Examples: 9, 2, 5, 2, 10, 12, 15, 18, 20

Exact and decimal results

Show sufficiently precise decimal results and keep the sample or population definition visible beside them.

Examples: s=\sqrt{11}, \sigma=\sqrt{6}\approx2.449

How to use it

Enter enough information for one clear task

  1. 1
    Enter the observations

    Separate numerical values with commas or spaces. Enter repeated values separately because frequency matters.

  2. 2
    Compare both data roles

    The result labels both population and sample variance and standard deviation. Use the population values only for a complete group; use the sample values when the observations come from a larger population.

  3. 3
    Read center before spread

    Compare mean, median, and mode first, then interpret range and standard deviation in the original measurement units.

  4. 4
    Check the context

    A numerical summary does not explain why a pattern exists, prove causation, or show whether the data collection process was representative.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Symmetric three-value data

Add and divide for the mean; take the middle ordered value for the median.

2, 4, 62,\ 4,\ 6

Expected result

xˉ=4,x~=4,R=4\bar{x}=4,\quad \widetilde{x}=4,\quad R=4

Data with one mode

Count each value and compare the frequencies.

1, 1, 2, 41,\ 1,\ 2,\ 4

Expected result

xˉ=2x~=1.5mode⁡=1\begin{aligned}\bar{x}&=2 & \widetilde{x}&=1.5\\ \operatorname{mode}&=1\end{aligned}

Data centered at zero

Use the population denominator N = 3 for this complete three-value population.

−3, 0, 3-3,\ 0,\ 3

Expected result

μ=0,σ2=6,σ=6\mu=0,\quad \sigma^2=6,\quad \sigma=\sqrt{6}

No variation

Every deviation from the mean is zero.

5, 5, 55,\ 5,\ 5

Expected result

μ=5,σ=0,R=0\mu=5,\quad \sigma=0,\quad R=0

Even number of observations

Average the two middle ordered values to find the median.

2, 3, 7, 82,\ 3,\ 7,\ 8

Expected result

xˉ=5,x~=5,R=6\bar{x}=5,\quad \widetilde{x}=5,\quad R=6

Five-value summary

Use all five values for the mean and the third ordered value for the median.

10, 12, 15, 18, 2010,\ 12,\ 15,\ 18,\ 20

Expected result

n=5xˉ=15x~=15R=10\begin{aligned}n&=5 & \bar{x}&=15\\ \widetilde{x}&=15 & R&=10\end{aligned}

Complete example

Describe five quiz scores

Treat the five listed scores as a data set. We will calculate center and spread, then state exactly which denominator each spread measure uses.

4, 7, 7, 9, 134,\ 7,\ 7,\ 9,\ 13
  1. 1
    Find the mean

    The observations total 40, so their arithmetic mean is 8.

    xˉ=405=8\bar{x}=\frac{40}{5}=8
  2. 2
    Find position and frequency summaries

    The middle value and the most frequent value are both 7.

    x~=7,mode⁡=7\widetilde{x}=7,\qquad \operatorname{mode}=7
  3. 3
    Calculate squared deviations

    Subtract 8 from each observation and square each difference.

    (−4)2+(−1)2+(−1)2+12+52=44(-4)^2+(-1)^2+(-1)^2+1^2+5^2=44
  4. 4
    Apply the selected denominator

    If these five scores are the full population, divide by 5. If they are a sample, divide by 4.

    σ=445≈2.966s=444=11≈3.317\begin{aligned}\sigma&=\sqrt{\frac{44}{5}}\approx2.966\\ s&=\sqrt{\frac{44}{4}}=\sqrt{11}\approx3.317\end{aligned}
xˉ=8x~=7mode⁡=7R=9σ≈2.966s≈3.317\begin{aligned}\bar{x}&=8 & \widetilde{x}&=7\\ \operatorname{mode}&=7 & R&=9\\ \sigma&\approx2.966 & s&\approx3.317\end{aligned}

Verification: Re-adding the deviations gives -4 - 1 - 1 + 1 + 5 = 0, as deviations from the mean must. The two variances share the same squared-deviation total and differ only by the stated denominator.

Avoidable errors

Common mistakes and how to fix them

Using the unsorted list for the median

Problem: Calling the visually central input value the median.

Why it matters: The median is positional only after the values are ordered.

Better approach: Sort first, then take the middle value or average the two middle values.

Using n for every variance

Problem: \frac{\sum(x_i-\bar{x})^2}{n}

Why it matters: That denominator treats the listed values as the full population, not as a sample estimate.

Better approach: Use n minus 1 for sample variance and N for population variance.

Reporting a forced mode

Problem: Choosing one value even when every value occurs once or several values tie.

Why it matters: A mode is determined by frequency, not by position or size.

Better approach: Report no mode when no value repeats, or report every value tied for the highest frequency.

Treating a summary as an explanation

Problem: Concluding that a large mean caused the observed spread.

Why it matters: Descriptive statistics summarize a data set but do not establish cause.

Better approach: Interpret the numerical pattern separately from the study design and subject-matter explanation.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • The count, sorted order, sum, minimum, maximum, and frequency table are checked independently.
  • Deviations from the computed mean are checked to sum to zero, allowing for displayed rounding.
  • Variance and standard deviation are linked by squaring the reported standard deviation.
  • Sample and population results reuse the same squared-deviation sum with different labeled denominators.

When to stop and revise the input

  • The summary applies only to finite numerical observations entered on the page.
  • Descriptive results do not test a hypothesis, estimate a confidence interval, or prove that a sample is representative.
  • This calculator does not report quartiles because quartile conventions differ across textbooks and software; use a source that states its convention when quartiles are required.
  • Displayed decimals may hide small differences, and inputs whose totals or squared deviations exceed the browser's finite numeric range are rejected instead of returning infinity.

Common questions

Summarize a numerical data set FAQ

Should I choose sample or population statistics?

Choose population when the entered values are every member of the group you want to describe. Choose sample when the values are observations from a larger population you want to estimate.

Can a data set have more than one mode?

Yes. If two or more values share the greatest frequency, each is a mode. If no value occurs more often than another, the data set has no mode under the usual classroom convention.

Why can the mean and median be different?

The mean uses every value and moves when an extreme value changes. The median depends on ordered position, so it is often less affected by a very high or very low observation.

Does a standard deviation have squared units?

No. Variance has squared units. Standard deviation is the square root of variance, so it returns to the original units of the observations.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.