Box Plot Calculator | Five-Number Summary, IQR & Outlier Detection
Compute the five-number summary (minimum, Q1, median, Q3, maximum) from raw data. Calculates the interquartile range (IQR), inner and outer Tukey fences, and identifies mild and extreme outliers. Shows sorted data, mean, and standard deviation alongside the box plot statistics.
What Is the Box Plot Calculator | Five-Number Summary, IQR & Outlier Detection?
A box plot (box-and-whisker plot) visualizes the distribution of a dataset using five key statistics. The box spans the interquartile range (Q1 to Q3) showing the middle 50% of the data. The line inside the box is the median. Whiskers extend to the last data point within the Tukey fences, and points beyond the fences are flagged as outliers.
Formula
Five-number summary: min, Q1 (median of lower half), Q2 (median), Q3 (median of upper half), max. IQR = Q3 − Q1. Tukey fences: inner = Q1 ± 1.5·IQR, outer = Q1 ± 3·IQR. Skewness = 3(mean − median) / σ.
How to Use
- 1
Type or paste your data values in the text area, separated by commas (e.g. 10, 20, 35, 40, 55).
- 2
At least 4 data points are required to compute meaningful quartiles.
- 3
Click Calculate Box Plot Statistics to compute all statistics.
- 4
Read the five-number summary (min, Q1, median, Q3, max) from the card row.
- 5
Check the IQR, mean, standard deviation, and range in the secondary stats row.
- 6
Look at the ASCII box plot to visualize the distribution shape at a glance.
- 7
Review the Tukey fence table and any detected mild or extreme outliers.
Enter your data values separated by commas in the text area, then click Calculate. You can paste data from a spreadsheet or type values manually.
Example Calculation
Dataset: 10, 12, 14, 15, 15, 16, 17, 18, 19, 20, 21, 55, 80. After sorting, Q1=14, median=17, Q3=20, IQR=6. Inner upper fence=20+9=29. Values 55 and 80 exceed the inner fence so both are mild or extreme outliers.
Understanding Box Plot | Five-Number Summary, IQR & Outlier Detection
Tukey Fence and Outlier Reference
| Fence | Formula | Classification | Typical Action |
|---|---|---|---|
| Inner Lower | Q1 − 1.5 × IQR | Mild outlier if below | Investigate; may be valid extreme value |
| Inner Upper | Q3 + 1.5 × IQR | Mild outlier if above | Investigate; may be valid extreme value |
| Outer Lower | Q1 − 3.0 × IQR | Extreme outlier if below | Likely data error or rare event |
| Outer Upper | Q3 + 3.0 × IQR | Extreme outlier if above | Likely data error or rare event |
| Whisker end | Last in-fence data point | Normal range | Box plot whisker extends here |
Quartile Calculation Methods Compared
| Method | Q1 Definition | Q3 Definition | Used By |
|---|---|---|---|
| Tukey / Inclusive | Median of lower half (excl. Q2 for odd n) | Median of upper half (excl. Q2 for odd n) | R, this calculator, exploratory stats |
| Exclusive | Median of lower half (always excl. median) | Median of upper half (always excl. median) | Some textbooks |
| Linear interpolation | Interpolated at 25th percentile | Interpolated at 75th percentile | Excel QUARTILE.INC, Python numpy |
| Nearest rank | Value at ⌈n/4⌉th position | Value at ⌈3n/4⌉th position | Some statistical software |
Interpreting Box Plot Skewness
- ▸Symmetric distribution: median is centered in the box; whiskers are approximately equal length; Pearson skewness ≈ 0.
- ▸Right-skewed (positive): median is closer to Q1; right whisker is longer; mean > median; Pearson skewness > 0.1.
- ▸Left-skewed (negative): median is closer to Q3; left whisker is longer; mean < median; Pearson skewness < −0.1.
- ▸Outliers on the right side of a symmetric distribution can pull the mean rightward, creating apparent positive skew.
- ▸IQR captures the middle 50% of data and is resistant to outliers, unlike range or variance.
- ▸A long box (large IQR) indicates high variability in the central data; a narrow box shows tight clustering.
- ▸When comparing multiple box plots, look for differences in median position, box width, and whisker symmetry.
Frequently Asked Questions
What is the IQR and why does it matter?
The IQR (Interquartile Range) = Q3 − Q1 covers the middle 50% of your data. It is a robust measure of spread because it is not affected by extreme values or outliers, unlike range or standard deviation. A large IQR indicates high variability in the central bulk of the data.
How are outliers defined?
This calculator uses Tukey's fence method: mild outliers are values beyond Q1 − 1.5·IQR or Q3 + 1.5·IQR but within Q1 − 3·IQR and Q3 + 3·IQR. Extreme outliers are beyond the outer fences (±3·IQR). These are not automatically wrong values — they warrant investigation but may be legitimate extreme observations.
Why does this calculator sometimes give different Q1/Q3 values than Excel?
There are multiple valid methods for computing quartiles (Tukey inclusive, exclusive, linear interpolation). Excel uses linear interpolation by default. This calculator uses Tukey's inclusive method: Q1 is the median of the lower half excluding the overall median for odd-sized datasets. The difference is small, especially for large datasets.
What does skewness tell me?
Pearson's second skewness coefficient = 3(mean − median)/σ measures asymmetry. Near 0 means symmetric; positive (> 0.1) means right-skewed with a longer right tail; negative (< −0.1) means left-skewed. Right skew is common in income, price, and wait-time data; left skew appears in test scores and age-at-death data.
Can I use this for comparing multiple groups?
This calculator processes one dataset at a time. To compare groups, run each separately and note the five-number summaries. Look for overlap (or lack thereof) in the IQR ranges: non-overlapping boxes suggest statistically meaningful group differences worth formal testing.
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