Descriptive Statistics Calculator — Mean, SD, Quartiles (Free)
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A descriptive statistics calculator turns a list of numbers into a full summary: count, mean, median, mode, range, quartiles, IQR, variance, standard deviation, standard error, skewness and kurtosis. For the 15 test scores below, the mean is 1,205 ÷ 15 ≈ 80.33, the median is 85, the quartiles are 72 and 88, and the sample standard deviation is about 9.98.
For these 15 values, the mean is 80.3333, the median is 85 and the sample standard deviation is 9.9762.
Spread
Five-number summary and quartiles
Shape
Bars: how many values fall in each class of width 10. Box: Q1 to Q3 with the median in green, the blue diamond is the mean, whiskers reach the smallest and largest values inside the 1.5 × IQR fences.
Frequency table (grouped)
| Class | Frequency | Relative frequency | Cumulative frequency |
|---|---|---|---|
| 60 to under 70 | 2 | 13.33% | 2 |
| 70 to under 80 | 5 | 33.33% | 7 |
| 80 to under 90 | 5 | 33.33% | 12 |
| 90 to 100 | 3 | 20% | 15 |
Show steps
- Count the values and add them: n = 15 and Σx = 1,205.
- Mean: x̄ = Σx ÷ n = 1,205 ÷ 15 = 80.3333.
- Sort the data: 61, 66, 70, 72, 74, 78, 79, 85, 85, 85, 85, 88, 90, 92, 95. The smallest value is 61 and the largest is 95, so the range is 95 − 61 = 34.
- Median: n is odd, so the median is the middle value, number 8 in the sorted list: 85.
- Mode: the value appearing most often (4 times): 85.
- Quartiles by median of halves (the median is left out of both halves when n is odd). Lower half: 61, 66, 70, 72, 74, 78, 79, so Q1 = 72. Upper half: 85, 85, 85, 88, 90, 92, 95, so Q3 = 88.
- IQR = Q3 − Q1 = 88 − 72 = 16. Outlier fences: Q1 − 1.5 × IQR = 48 and Q3 + 1.5 × IQR = 112. No values fall outside.
- Subtract the mean from each value, square the results and add them: Σ(x − x̄)² = 1,393.3333.
- Sample variance: s² = 1,393.3333 ÷ (n − 1) = 1,393.3333 ÷ 14 = 99.5238, and s = √99.5238 = 9.9762. Population variance: σ² = 1,393.3333 ÷ 15 = 92.8889, and σ = 9.6379.
- Standard error: s ÷ √n = 9.9762 ÷ √15 = 2.5758. Coefficient of variation: s ÷ |x̄| × 100 = 12.4185%.
- Skewness (sample): n ÷ ((n − 1)(n − 2)) × Σ((x − x̄) ÷ s)³ = −0.4664. A value near 0 means the data are roughly symmetric; positive means a longer right tail.
Use the sample formulas (n − 1) when your data are a sample from a bigger group, which is the usual case in class and in research.
How to use the descriptive statistics calculator
- Paste or type your numbers, separated by commas, spaces or new lines. If your data come as a frequency table, use the Values + frequencies tab instead.
- Choose Sample or Population. This decides whether the headline standard deviation, the coefficient of variation, skewness and kurtosis use n − 1 or n.
- Pick the quartile method your class, textbook or software uses. Results update as you type.
- Read the summary, then check the histogram, box plot and frequency table. Open Show steps to see every calculation with your numbers.
Need only the average, middle and most common value? The simpler mean, median and mode calculator is quicker. For a detailed table of deviations, use the standard deviation calculator, and for a printable chart, the box and whisker plot maker.
Measures of center: mean, median and mode
- Mean x̄ = Σx ÷ n, the sum divided by the count. It uses every value, so one extreme value can pull it a long way.
- Median: the middle value of the sorted list, or the average of the two middle values when n is even. It ignores how extreme the ends are.
- Mode: the value that appears most often. A data set can have several modes or none.
In the test-score example the mean (80.33) is below the median (85), because a few low scores (61, 66) pull the mean down. That matches the negative skewness of about −0.47.
Measures of spread: range, IQR, variance and standard deviation
- Range = maximum − minimum.
- Interquartile range IQR = Q3 − Q1, the spread of the middle half of the data.
- Sample variance s² = Σ(x − x̄)² ÷ (n − 1) and sample standard deviation s = √s².
- Population variance σ² = Σ(x − x̄)² ÷ n and σ = √σ².
- Standard error of the mean SE = s ÷ √n, how much the sample mean would vary from sample to sample.
- Coefficient of variation CV = standard deviation ÷ |mean| × 100%, spread relative to the size of the numbers.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
n = 8 and Σx = 40, so the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which add to 32. Population variance: 32 ÷ 8 = 4, so σ = 2. Sample variance: 32 ÷ 7 ≈ 4.5714, so s ≈ 2.1381. The median is (4 + 5) ÷ 2 = 4.5, the mode is 4, and the range is 9 − 2 = 7.
Sample or population?
Use the population formulas only when your list contains every member of the group you want to describe, such as the scores of all 24 students in one class when you care only about that class. If the numbers are a sample used to say something about a larger group, divide by n − 1. Dividing by n − 1 makes the variance slightly larger, which corrects for the fact that a sample tends to underestimate the spread of the whole population.
Quartile methods give different answers
There is no single rule for quartiles, which is why two tools can disagree. This calculator lets you choose. Here is the same data set, 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49 (n = 11), under each method:
| Method | Used by | Q1 | Q3 | IQR |
|---|---|---|---|---|
| Median of halves (median left out) | TI-83/84, most textbooks | 15 | 43 | 28 |
| Tukey hinges (median in both halves) | Some textbooks, box plots | 25.5 | 42.5 | 17 |
| Interpolation, position (n − 1)p + 1 | Excel QUARTILE.INC, Google Sheets, R | 25.5 | 42.5 | 17 |
| Interpolation, position (n + 1)p | Excel QUARTILE.EXC, Minitab, SPSS | 15 | 43 | 28 |
With a large data set the methods agree closely. With a small one, use the method your teacher or software uses so your answers match.
Shape: skewness, kurtosis and outliers
Skewness measures lopsidedness. Near 0 means roughly symmetric, positive means a longer right tail (a few large values), and negative means a longer left tail. A common rule of thumb treats values between −0.5 and 0.5 as roughly symmetric. Excess kurtosis compares how heavy the tails are with a normal distribution, which has excess kurtosis 0. The sample versions here match Excel’s SKEW and KURT functions.
Outliers are flagged with the 1.5 × IQR rule: any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. In 1, 2, 3, 4, 5, 6, 7, 8, 9, 50, the quartiles are 3 and 8, so the upper fence is 8 + 7.5 = 15.5 and 50 is an outlier. It drags the mean up to 9.5 while the median stays at 5.5.
Common mistakes with descriptive statistics
| Mistake | How to catch it |
|---|---|
| Using n instead of n − 1 for a sample | Check the Sample/Population setting. s is always a little larger than σ. |
| Finding the median of unsorted data | Sort first. The steps show the sorted list. |
| Rounding the mean before computing deviations | Keep full precision until the final answer. |
| Quartiles that do not match the answer key | Switch the quartile method; the key likely uses another rule. |
| A typo such as 850 instead of 85 | Look at the minimum, maximum, outliers and histogram. A strange bar or red circle points to it. |
| Reporting the mean for skewed data | When skewness is large or there are outliers, report the median and IQR as well. |
Descriptive statistics in Excel and by hand
In Excel, AVERAGE, MEDIAN, MODE.SNGL, STDEV.S, STDEV.P, VAR.S, QUARTILE.INC, SKEW and KURT give the same values as the matching settings here. The Analysis ToolPak’s Descriptive Statistics tool prints most of them at once. Work one small data set by hand to learn the formulas, then use the calculator to check homework and to handle long lists quickly.
Descriptive statistics practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Sort first: 1, 2, 4, 6, 7, 9. The two middle values are 4 and 6, so the median is 5.
The mean is 5 and the squared deviations add to 32. σ² = 32 ÷ 8 = 4, so σ = 2.
Lower half 1, 3, 5, 7 gives Q1 = 4; upper half 9, 11, 13, 15 gives Q3 = 12. IQR = 12 − 4 = 8.
IQR = 10, so the fences are 20 − 15 = 5 and 30 + 15 = 45. Only 46 is outside.
A long right tail pulls the mean up but barely moves the median, and the skewness is positive.
Quiz complete
Frequently asked questions
What is a descriptive statistic?
A number that summarizes a data set without drawing conclusions beyond it: a measure of center (mean, median, mode), of spread (range, IQR, standard deviation) or of shape (skewness, kurtosis). Inferential statistics, such as confidence intervals and tests, go a step further and generalize to a population.
What does a summary statistics calculator show?
At least the count, mean, median, mode, minimum, maximum, range, quartiles and standard deviation. This one adds both sample and population variance, standard error, coefficient of variation, mean absolute deviation, skewness, kurtosis, outliers, a histogram, a box plot and a frequency table.
Is there a descriptive statistics calculator with steps?
Yes. Open Show steps under the results. It shows the sum and mean, the sorted list, how the median and quartiles were found, the sum of squares, both variances and standard deviations, the standard error and the skewness.
Why are my quartiles different from my calculator or Excel?
Different tools use different quartile rules. A TI-84 leaves the median out of both halves, while Excel's QUARTILE.INC interpolates. Choose the matching method in the Quartile method menu.
Should I use the mean or the median?
Use the mean for roughly symmetric data without outliers. Use the median when the data are skewed or contain outliers, as with incomes or house prices. Reporting both is often best.
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