Statistics Calculator
Get a full statistical summary of a data set: mean, median, mode, range, quartiles, IQR, five-number summary, standard deviation and variance, skewness and outliers.
Results
| Item | Value |
|---|
Quick Answer
For the data 12, 15, 11, 18, 15, 22, 14, 15, 19, 40, the mean is 18.1, the median is 15 and the mode is 15. The sample standard deviation is 8.3593, the quartiles are Q1 = 13.5 and Q3 = 19.75, and 40 lies beyond Q3 + 1.5 × IQR, so it is flagged as an outlier.
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Table of Contents
- Statistics Calculator
- Quick Answer
- Related Calculators
- How This Statistics Calculator Works
- How to Use the Statistics Calculator
- Statistics Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Statistics Calculator vs Average Calculator
- Common Descriptive Statistics
- Sample or Population?
- Mean vs Median When Data Is Skewed
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Statistics Calculator Works
Enter a list of numbers and press Calculate. The calculator sorts the data and works out every common descriptive statistic in one go: measures of center (mean, median, mode), measures of spread (range, quartiles, interquartile range, variance, standard deviation, standard error and coefficient of variation), the five-number summary used for box plots, the skewness of the distribution, and any outliers.
Quick explanation: descriptive statistics summarize a data set.
- Center: mean, median and mode describe a typical value.
- Spread: range, IQR and standard deviation describe how varied the data is.
- Shape: skewness and outliers describe the tails.
How to Use the Statistics Calculator
- Type or paste your numbers into the box.
- Separate them with commas, spaces or new lines.
- Press Calculate.
- Read the center, spread and five-number summary.
- Check the outliers and skewness to understand the shape of the data.
Statistics Formulas
- Mean: Ȳ = ΣY ÷ N
- Median: the middle value, or the average of the two middle values
- Sample variance: s² = Σ(Y − Ȳ)² ÷ (N − 1); population variance: σ² = Σ(Y − Ȳ)² ÷ N
- Quartiles: for percentile p, set p(N + 1) = k + d, then Y(p) = Y[k] + d(Y[k+1] − Y[k])
- IQR: Q3 − Q1; outlier fences: Q1 − 1.5 × IQR and Q3 + 1.5 × IQR
- Skewness (adjusted): G1 = √(N(N − 1)) ÷ (N − 2) × [Σ(Y − Ȳ)³ ÷ N] ÷ σ³
Key Terms
- Five-number summary: minimum, first quartile, median, third quartile and maximum.
- Interquartile range: Q3 minus Q1, the spread of the middle half of the data.
- Standard error: the standard deviation divided by √N, the uncertainty of the mean.
- Coefficient of variation: standard deviation as a percentage of the mean.
- Outlier: a value far from the rest, here beyond 1.5 × IQR from the quartiles.
Worked Example
Ten test times in minutes: 12, 15, 11, 18, 15, 22, 14, 15, 19, 40.
- Sorted: 11, 12, 14, 15, 15, 15, 18, 19, 22, 40
- Mean = 181 ÷ 10 = 18.1; median = (15 + 15) ÷ 2 = 15; mode = 15
- Q1: 0.25 × 11 = 2.75, so Q1 = 12 + 0.75 × (14 − 12) = 13.5; Q3: 0.75 × 11 = 8.25, so Q3 = 19 + 0.25 × (22 − 19) = 19.75
- IQR = 6.25; upper fence = 19.75 + 1.5 × 6.25 = 29.125, so 40 is an outlier
Real-Life Examples
- Test times: the example data has a mean of 18.1, a median of 15 and one outlier (40).
- Class scores: 72, 85, 90, 66, 78, 88, 95, 81 have a mean of 81.875 and a sample SD of 9.64.
- Box plot: the five-number summary gives the minimum, Q1, median, Q3 and maximum to draw it.
- Quality control: the coefficient of variation compares spread across products with different averages.
Statistics Calculator vs Average Calculator
- Average Calculator: quick mean, median, mode and range.
- Standard Deviation Calculator: spread with the steps for SD.
- Statistics Calculator: everything at once, plus quartiles, the five-number summary, skewness and outliers.
Common Descriptive Statistics
| Statistic | What it measures | Affected by outliers? |
|---|---|---|
| Mean | Arithmetic average | Yes |
| Median | Middle value | No |
| Mode | Most common value | No |
| Range | Max − min | Yes |
| IQR | Spread of the middle 50% | No |
| Standard deviation | Typical distance from the mean | Yes |
| Skewness | Lopsidedness of the distribution | Yes |
Sample or Population?
Use the sample standard deviation and variance (dividing by N − 1) when your numbers are a sample from a bigger group, such as 30 customers out of thousands. Use the population versions (dividing by N) only when your data includes every member of the group, such as the scores of every student in one class. The sample versions are slightly larger, which corrects for the fact that a sample tends to underestimate the spread.
Mean vs Median When Data Is Skewed
When data has a long tail or an outlier, the mean is pulled toward it, while the median stays in the middle. In the example, the one 40-minute time raises the mean to 18.1 even though most times are near 15. That is why incomes and house prices are usually reported as medians. A skewness near 0 means the data is fairly symmetric; positive skewness means a long right tail, and negative skewness a long left tail.
Common Mistakes to Avoid
- Using thousands separators. 12,500 is read as two numbers, 12 and 500.
- Mixing sample and population formulas. Use N − 1 for samples.
- Reporting only the mean for skewed data. Give the median too.
- Deleting outliers without a reason. Check whether they are errors or real values.
- Comparing quartiles from different tools. Methods differ slightly between software.
Frequently Asked Questions
Enter your numbers; the calculator gives mean, median, mode, spread, quartiles, skewness and outliers.
Sample divides by N − 1 and population divides by N.
With the NIST method: for percentile p, find position p(N + 1) and interpolate between the two nearest values.
Values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR are mild outliers; beyond 3 × IQR they are extreme.
How lopsided the data is: positive means a long right tail, negative a long left tail.
If no value repeats, there is no mode.
Up to 10,000.
Excel’s QUARTILE.INC uses a different interpolation; QUARTILE.EXC matches the NIST method used here.
s ÷ √N, used to build confidence intervals for the mean.
References & Sources
| Source | Used For | Link |
|---|---|---|
| NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.1 Measures of Location | Mean, median and mode. | https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm |
| NIST/SEMATECH e-Handbook, 1.3.5.6 Measures of Scale | Variance, standard deviation, range and IQR. | https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm |
| NIST/SEMATECH e-Handbook, 7.2.6.2 Percentiles | The p(N + 1) percentile method. | https://www.itl.nist.gov/div898/handbook/prc/section2/prc262.htm |
| NIST/SEMATECH e-Handbook, 7.1.6 What are outliers in the data? | Inner and outer fences at 1.5 and 3 × IQR. | https://www.itl.nist.gov/div898/handbook/prc/section1/prc16.htm |
| NIST/SEMATECH e-Handbook, 1.3.5.11 Measures of Skewness and Kurtosis | Fisher-Pearson and adjusted skewness. | https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm |