In descriptive statistics, a box plot (also called a boxplot or box and whisker plot) is a chart used in exploratory data analysis. It shows how a set of scores is spread out.
Box plots show the distribution and skewness of numerical data. They do this by displaying the quartiles (or percentiles) and the median, the middle score.
Box plots show the five-number summary of a set of data: including the minimum score, first (lower) quartile, median, third (upper) quartile, and maximum score.
How to read a boxplot
- Interquartile Range (IQR): The length of the box, from the 25th to the 75th percentile. It spans the middle 50% of scores.
- Whiskers: The upper and lower whiskers represent scores outside the middle 50% (i.e., the lower 25% of scores and the upper 25% of scores).
- Minimum Score: The lowest score, excluding outliers (shown at the end of the left whisker).
- Lower Quartile: Twenty-five percent of scores fall below the lower quartile value (also known as the first quartile).
- Median: The median marks the mid-point of the data and is shown by the line that divides the box into two parts (sometimes known as the second quartile). Half the scores are greater than or equal to this value, and half are less.
- Upper Quartile: Seventy-five percent of the scores fall below the upper quartile value (also known as the third quartile). Thus, 25% of data are above this value.
- Maximum Score: The highest score, excluding outliers (shown at the end of the right whisker).
- Outliers: Points beyond the whiskers, indicating potential extreme values.
How to Interpret a Box Plot
A Quick Reading Guide
Each of the four segments of a box plot holds about a quarter of the data: the two whiskers and the two halves of the box.
A short segment means scores are crowded into a narrow band. A long one means the same number of scores is stretched thinly. The lengths are the information.
The table gives a working checklist. Treat it as a guide to what to look at, not a verdict. In small samples, a gap of a few units between segments can arise by chance.
| What you see | What it suggests |
|---|---|
| Median near the middle of the box; whiskers of about equal length | Roughly symmetric distribution |
| Median nearer the bottom of the box; lower whisker shorter; upper whisker long or points above it | Positive (right) skew: a tail toward high scores |
| Median nearer the top of the box; upper whisker shorter; lower whisker long or points below it | Negative (left) skew: a tail toward low scores |
| One very long whisker, or points beyond only one whisker | A long tail on that side, or isolated unusual scores: check for errors |
| Long box | Scores in the middle half are widely spread (large IQR) |
| Short box with long whiskers | Scores concentrated in the middle with stretched tails (large range relative to the IQR) |
| Box and whiskers lean in opposite directions | No clear skew: mixed evidence is common in small samples |
Reading Skew: Three Worked Examples
Here are three sets of fifteen scores. Each box plot is built with the hand method described under step-by-step construction below.
- Symmetric: 40, 44, 47, 49, 51, 52, 54, 55, 56, 58, 59, 61, 63, 66, 70
- Right-skewed: 12, 13, 14, 14, 15, 16, 17, 18, 20, 22, 26, 31, 38, 47, 66
- Left-skewed: 34, 53, 62, 69, 74, 78, 80, 82, 83, 84, 85, 86, 86, 87, 88
| Data set | Min | Q1 | Median | Q3 | Max | Flagged | Mean | Quartile skewness |
|---|---|---|---|---|---|---|---|---|
| Symmetric | 40 | 49 | 55 | 61 | 70 | None | 55.0 | 0.00 |
| Right-skewed | 12 | 14 | 18 | 31 | 66 | 66 | 24.6 | +0.53 |
| Left-skewed | 34 | 69 | 82 | 86 | 88 | 34 | 75.4 | −0.53 |
In the symmetric set, the box is split evenly (6 and 6) and the whiskers extend 9 on each side.
In the right-skewed set, the median is only 4 above Q1, but Q3 is 13 above the median.
The upper whisker is eight times as long as the lower one, which spans only two units. The score of 66 is an outlier. It lies beyond the upper fence of 56.5 and is drawn as a point.
The mean (24.6) sits well above the median (18). The left-skewed set is its mirror image.
The pattern to remember: the tail lies on the side with the longer whisker and the longer half-box. The mean is pulled toward the tail, while the median is not.
A simple number captures the lean of the middle half. Quartile skewness is (Q3 + Q1 − 2 × median) ÷ IQR.
It is zero when the median sits exactly in the middle of the box, positive when it sits low and negative when it sits high. It is a rough descriptor, not a test.
Two cautions apply when you read skew:
- One value can mislead. A long whisker or a flagged point reflects only the most extreme scores, so a single unusual value can make a distribution look more skewed than the bulk of the data are.
- Box and whiskers can disagree. In Practice Question 1 below, the median is nearer the bottom of the box, yet the lower whisker is longer. That is mixed evidence: roughly symmetric, with no clear skew.
Common Mistakes When Reading a Box Plot
These slips are easy to make, and most are avoidable by checking the legend:
- Reading the median as the mean. The line in the box is the median. The mean is not shown unless a separate marker is added.
- Assuming whiskers always mean minimum and maximum. On a schematic plot, whiskers stop at the most extreme unflagged score and other points are drawn separately.
- Drawing the whisker to the fence. The fence is a cutoff, not a data value. The whisker ends at the last actual score inside it.
- Treating every flagged point as an error. A flag means "look at this," not "delete this." It may be an error, a faulty measurement or a genuine, interesting case.
- Reading box size as sample size. In a standard box plot, box width means nothing and box length is spread, not amount of data.
- Treating overlap as a significance test. Overlapping boxes are a rough guide to similarity. They are neither a p-value nor a confidence interval.
Why are box plots useful?
Quick summary of key statistics
By displaying the median, interquartile range, and outliers at a glance, boxplots provide a concise snapshot of a dataset’s center and spread.
Box plots divide the data into sections containing approximately 25% of the data in that set.
Box plots are useful as they provide a visual summary of the data enabling researchers to quickly identify median values, the dispersion of the data set, and signs of skewness.
The image above shows data with a perfect normal distribution, where each quartile is the same length. Most real box plots will not be this symmetrical.
Median Score of a Data Set
The median is the middle score in a data set. It is shown by the line that divides the box into two parts.
Half the scores are greater than or equal to this value, and half are less.
The median is not the mean. Unless a plot adds a separate marker, such as a plus sign or a dot, the mean is not shown at all.
The median resists extreme scores, which is why box plots use it. Take fifteen memory-test scores from 12 to 30, with a mean of 21 and a median of 20.
Suppose the top score of 30 were mis-keyed as 300. The mean would jump to 39.
The median would stay at 20 and the quartiles would not move, so the box would look the same. The stray score would simply appear as a separate point.
Skewness of a data set
The box plot shape will show if a statistical data set is normally distributed or skewed.
When the median is in the middle of the box, and the whiskers are about the same on both sides of the box, then the distribution is symmetric.
When the median is closer to the bottom of the box and the lower whisker is shorter, the distribution is positively skewed (skewed right). The longer tail points toward the high scores.
When the median is closer to the top of the box and the upper whisker is shorter, the distribution is negatively skewed (skewed left). The longer tail points toward the low scores.
Dispersion of a data set
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed.
A box plot shows two nested measures of spread. The box gives the IQR, the spread of the middle half of the scores. The ends of the whiskers give the overall range, once any flagged outliers are set aside.
Having both lets you see whether a wide range reflects a genuinely wide spread or only a few distant scores.
The interquartile range (IQR) is the length of the box. It spans the middle 50% of scores and is calculated by subtracting the lower quartile from the upper quartile (Q3 − Q1).
Show outliers within a data set
An outlier is an observation that is numerically distant from the rest of the data.
When reviewing a box plot, an outlier is defined as a data point that is located outside the whiskers of the box plot.
Outliers can be genuine rare events (e.g., a patient who recovers extraordinarily quickly) or the result of data-entry mistakes, faulty instruments, or other errors.
Source: https://towardsdatascience.com/understanding-boxplots-5e2df7bcbd51
Here is the usual rule (Tukey, 1977). A point is flagged if it lies more than 1.5 × IQR below Q1 or above Q3. The fences are therefore Q1 − 1.5 × IQR and Q3 + 1.5 × IQR.
The whisker then ends at the most extreme score inside the fence, not at the fence itself.
A flag is a prompt, not a verdict. In a large normal sample the rule flags only about 0.7% of scores, but small samples flag more (Hoaglin et al., 1986). Skewed or heavy-tailed data flag far more.
Box plots are useful as they show simple, side-by-side comparisons
Boxplots make group comparison easy. The groups can be different populations or experimental conditions. Align the boxes on a common scale, and differences in medians, overall spread and outlier prevalence stand out.
A bar graph of group means shows one number per group. Each box shows five, plus any outliers.
Side-by-side boxes treat each group as an independent batch. If the same people were measured twice, such as before and after therapy, the boxes ignore which score belongs to whom. A consistent but modest within-person change can then be nearly invisible when the boxes overlap.
A visible difference in medians is also not statistical significance. That needs a test or an interval estimate.
How to compare box plots
Box plots are a useful way to visualize differences among samples or groups. They pack in a lot of statistical information, including medians, ranges and outliers. Use a common scale, then work through four steps.
Step 1: Compare the medians of box plots
Compare the respective medians of each box plot. If the median line of a box plot lies outside the box of a comparison box plot, there is likely to be a difference between the two groups.
This is a rough visual heuristic, not a hypothesis test. With small groups it can fail in either direction.
A notched box plot adds a rough check. A notch cut into the box around the median gives a sign of whether two medians differ by more than chance variation would suggest.
Where two notches do not overlap, the medians probably differ (McGill et al., 1978). With very small groups the notch grows wider than the box, a warning that the sample is too small to be informative.
Source: https://blog.bioturing.com/2018/05/22/how-to-compare-box-plots/
Step 2: Compare the interquartile ranges and whiskers of box plots
Compare the interquartile ranges (that is, the box lengths) to examine how the data is dispersed between each sample. The longer the box, the more dispersed the data. The smaller, the less dispersed the data.
Next, look at the overall spread as shown by the extreme values at the end of two whiskers.
This shows the range of scores (another type of dispersion). Larger ranges indicate wider distribution, that is, more scattered data.
Step 3: Look for potential outliers (see the above image)
Outliers are the points plotted beyond the whiskers. Check whether the groups differ in how many they have, and on which side of the box they fall.
Because the mean is sensitive to extreme values, a single outlier can substantially shift the average, potentially giving a misleading picture of the dataset.
Recognizing this effect is essential for accurate data interpretation.
Step 4: Look for signs of skewness
If the data do not appear to be symmetric, does each sample show the same kind of asymmetry?
Two groups can share a median and still differ in shape. If both lean the same way, a difference in medians is easier to interpret than if one is symmetric and the other skewed.
In each group, note which side holds the longer whisker. The tail lies there.
Step-by-Step Construction
Follow these seven steps to build a box plot by hand. The worked example uses seven scores: 3, 5, 7, 10, 12, 14 and 18.
- Gather and sort your data:
- Start by listing all the numbers in your dataset. Example: 3, 5, 7, 10, 12, 14, 18.
- Now, put those numbers in order from smallest to largest. This makes it much easier to find the important values.
- Find the minimum and maximum:
- The minimum is the smallest number in your list.
- The maximum is the largest number.
- In our example (3, 5, 7, 10, 12, 14, 18), the minimum is 3 and the maximum is 18.
- Calculate the median (the middle value):
- The median is the middle number when your data is sorted.
- If you have an odd number of values, it’s the one right in the center.
- If you have an even number, you average the two middle numbers.
- In our example, there are 7 numbers, so the middle one is the 4th one. The median is 10.
- Find the first quartile (Q1) and third quartile (Q3):
- Q1 is the median of the lower half of your data (the numbers below the overall median).
- Q3 is the median of the upper half of your data (the numbers above the overall median).
- Important: when finding Q1 and Q3, if there is an even amount of numbers in the upper or lower half, you must average the two center numbers.
- In our example, the lower half is 3, 5, 7 (below the median 10). Q1 is the middle of these numbers, so Q1 = 5.
- The upper half is 12, 14, 18 (above the median 10). Q3 is the middle of these numbers, so Q3 = 14.
- Calculate the interquartile range (IQR):
- The IQR tells you how spread out the middle 50% of your data is.
- IQR = Q3 – Q1
- In our example, Q3 = 14 and Q1 = 5, so IQR = 14 – 5 = 9.
- Identify outliers (using the 1.5 × IQR rule):
- Outliers are numbers that are very different from the rest.
- Lower outlier cutoff = Q1 – (1.5 × IQR)
- Upper outlier cutoff = Q3 + (1.5 × IQR)
- Any numbers below the lower cutoff or above the upper cutoff are outliers.
- In our example, the lower cutoff = 5 – (1.5 × 9) = 5 – 13.5 = –8.5. The upper cutoff = 14 + (1.5 × 9) = 14 + 13.5 = 27.5.
- Since our minimum is 3 and our maximum is 18, we have no outliers.
- Draw the boxplot:
- Draw a number line that covers the range of your data (from minimum to maximum).
- Mark Q1, the median, and Q3 above the number line to create the “box.”
- Draw the “whiskers.”
- The left whisker goes from Q1 to the smallest value within the lower outlier cutoff.
- The right whisker goes from Q3 to the largest value within the upper outlier cutoff.
- If you have outliers, plot them as individual points beyond the whiskers.
- Remember to use visual aids along with these steps.
A Second Example: An Even Sample With an Outlier
The worked example above has an odd number of scores and no outlier. Take sixteen memory-test scores: 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25, 27, 28, 29, 30 and 90. An even sample changes how you find the median and the quartiles:
- Median: With an even number of values, average the 8th and 9th scores: (20 + 22) ÷ 2 = 21.
- Quartiles: The lower half is the first eight values, whose middle pair is 16 and 17, so Q1 = 16.5. The upper half is the last eight, whose middle pair is 27 and 28, so Q3 = 27.5.
- Fences: The IQR is 27.5 − 16.5 = 11. The upper fence is 27.5 + 16.5 = 44 and the lower fence is 16.5 − 16.5 = 0.
- Whiskers and outlier: The score of 90 lies beyond the upper fence, so it becomes a lone point. The upper whisker stops at 30, the largest score inside the fence, and the lower whisker reaches 12.
The mean of these sixteen scores is 25.3, pulled up by the 90. The median (21) has moved only one point from its value before the 90 was added.
Why Software Draws Slightly Different Boxes
The hand method above takes the median of each half and leaves the overall median out of both halves when the sample size is odd. It is only one of several accepted ways to estimate a quartile, and the choice matters most for small samples.
| Quartile rule | n = 7 (3, 5, 7, 10, 12, 14, 18): Q1, Q3, IQR | n = 15 memory scores: Q1, Q3, IQR | n = 10 "before" anxiety scores: Q1, Q3, IQR |
|---|---|---|---|
| Median of each half, overall median left out (hand method used here) | 5, 14, 9 | 16, 27, 11 | 15, 28, 13 |
| Tukey’s hinges (overall median included in both halves when n is odd) | 6, 13, 7 | 16.5, 26, 9.5 | 15, 28, 13 |
| Linear interpolation (default in R and numpy; Excel QUARTILE.INC) | 6, 13, 7 | 16.5, 26, 9.5 | 15, 27, 12 |
| Interpolation at (n + 1) × p (Excel QUARTILE.EXC; Python’s statistics module) | 5, 14, 9 | 16, 27, 11 | 14.25, 28.75, 14.5 |
The difference can change a verdict, not just the IQR. Take eleven scores: 1, 2, 3, 5, 11, 18, 20, 24, 31, 35, 64.
By the median-of-halves rule, Q1 = 3, Q3 = 31 and the upper fence is 73, so 64 is not flagged. By Tukey’s hinges or linear interpolation, Q1 = 4, Q3 = 27.5 and the upper fence is 62.75, so 64 is flagged as an outlier.
Same data, same 1.5 × IQR rule, different verdict. The cause is a software default.
Hyndman and Fan (1996) catalogued these rules and noted that one package may use one definition for quantiles and another for drawing a boxplot. Frigge et al. (1989) found similar inconsistency in how packages built the box and the fences.
The remedy is simple. State which quartile rule produced a figure, and treat a flag that appears under one rule and vanishes under another as borderline.
Whisker Conventions
Check the figure legend, because box plots are not all drawn the same way:
- Skeletal plots: Whiskers run to the minimum and maximum, so outliers can only be hinted at, for example by one very long whisker.
- Schematic (fenced) plots: Whiskers stop at the most extreme unflagged score and the rest are plotted as points. Most software draws this form.
- Percentile whiskers: Some fields end the whiskers at fixed percentiles, such as the 5th and 95th, or at the full range.
Practice Questions
Tips for Success
- Always sort your data before finding quartiles.
- Use (Q3 − Q1) for the IQR, then multiply by 1.5 to find the outlier “cut-off.”
- If an outlier exists, plot it as a separate point outside the whiskers.
- Label your axes carefully and include a title for clarity.
Question 1: Memory Test Scores
A researcher measures short-term memory test scores for 15 participants. Each participant completes a recall test scored out of 30 points. The data are:
23, 16, 19, 27, 14, 29, 22, 18, 12, 30, 25, 20, 15, 28, 17
- Sort the data.
- Find the minimum, Q1, median, Q3, and maximum.
- Calculate the interquartile range (IQR).
- Determine if there are any outliers using the 1.5 × IQR rule.
- Draw a boxplot of the data.
- Interpret your findings.
- Are the scores clustered, spread out, or skewed?
Answer to Question 1
There are no outliers, and the evidence for skew is mixed. Here is the working:
- Sorted data: 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25, 27, 28, 29, 30.
- Five-number summary: The minimum is 12, Q1 = 16, the median (the 8th value) is 20, Q3 = 27 and the maximum is 30.
- IQR and fences: IQR = 27 − 16 = 11. The cutoffs are 16 − 16.5 = −0.5 and 27 + 16.5 = 43.5, so nothing is flagged and the whiskers run from 12 to 30.
- Interpretation: The box extends 4 below the median and 7 above it, but the lower whisker (4) is slightly longer than the upper (3). That is mixed evidence, so there is no clear skew. The mean is 21.
Question 2: Stress Scale Scores
A psychologist uses a stress scale ranging from 0 (no stress) to 40 (extreme stress). Fifteen patients reported the following scores:
6, 9, 15, 15, 18, 22, 5, 27, 32, 33, 10, 14, 16, 8, 40
- Order the scores and calculate the median.
- Find Q1 and Q3, then compute the IQR.
- Determine lower and upper outlier boundaries (Q1 − 1.5 × IQR, Q3 + 1.5 × IQR).
- Draw a boxplot, labeling any outliers as individual points.
- Discuss any potential reasons for outliers on a stress scale (e.g., unique personal situations).
Answer to Question 2
No scores are outliers, and the distribution is moderately positively skewed. Here is the working:
- Sorted data: 5, 6, 8, 9, 10, 14, 15, 15, 16, 18, 22, 27, 32, 33, 40.
- Five-number summary: The median is 15, Q1 = 9 and Q3 = 27, so the IQR is 18. The minimum is 5 and the maximum is 40.
- Outlier boundaries: 9 − 27 = −18 and 27 + 27 = 54. Neither the lowest score (5) nor the highest (40) is beyond a boundary, so the whiskers reach 5 and 40.
- Skew: The median is 6 above Q1, but Q3 is 12 above the median. The upper whisker (13) is longer than the lower (4), and the mean (18) exceeds the median (15).
- Reasons for outliers: There are none to explain. It is still worth noticing that 40 is the highest score the scale allows, so the longest whisker ends at the ceiling of the scale.
Question 3: Anxiety Scores Before and After Therapy
A clinical psychologist measures anxiety in a group of clients before and after a short therapy program. Anxiety is scored from 0 to 50. The data are:
Before: 10, 12, 15, 15, 24, 20, 28, 31, 35, 15
After: 7, 10, 10, 17, 20, 15, 21, 30, 28, 10
- Construct a boxplot for the “Before” scores.
- Construct a separate boxplot for the “After” scores (on the same scale).
- Compare the distributions:
- Which group has a higher median?
- Which group has a wider spread (IQR)?
- Are there any outliers in either group?
- Interpret the difference in anxiety levels. Based on the boxplots, does therapy appear to have an effect?
Answer to Question 3
The two boxes overlap substantially, so they give little visual evidence of a difference. Here is the working:
- Before: Sorted, the scores run 10, 12, 15, 15, 15, 20, 24, 28, 31, 35. The median is 17.5, Q1 = 15, Q3 = 28 and the IQR is 13. There are no outliers.
- After: Sorted, the scores run 7, 10, 10, 10, 15, 17, 20, 21, 28, 30. The median is 16, Q1 = 10, Q3 = 21 and the IQR is 11. There are no outliers.
- Comparison: The "before" group has the higher median (17.5 against 16) and the wider spread (IQR 13 against 11). Each group’s median lies inside the other group’s box.
So does therapy appear to have an effect? On these boxes alone, only slightly, and ten clients with no comparison group cannot support a causal claim.
But the boxes hide something. Suppose each position in the two lists is the same client. Then nine of the ten clients scored lower after therapy.
The median change (after minus before) is −4.5 and the mean change is −3.7.
The boxes overlap because people differ more from one another than they changed. For before-and-after data, a paired analysis and a plot that connects each person’s two scores show the change far more clearly.
Limitations
- Lack of Detail About Distribution Shape: Although boxplots can show skewness to some extent (through asymmetry in the box or whiskers), they do not reveal whether a distribution is unimodal, bimodal, or has other distinct peaks.
- Limited Information on Data Density: A boxplot will not show you how many data points lie at different values within each quartile. Two datasets with the same boxplot could still have very different internal distributions.
- Context for Outliers: While outliers are marked, boxplots alone do not explain why those points are far from the bulk of the data. Context or other visualizations might be needed to understand anomalies.
- No Sample Size: Five numbers from n = 8 look the same as five from n = 8,000. Only widths, labels or an overlay of the points reveal the difference.
- Little to Show for Very Small Samples: With five or six scores, the box is made of the few points themselves. The choice of quartile rule changes it noticeably.
The Same Box, Three Different Distributions
The first two limitations are easy to demonstrate. The three data sets below each have 21 scores and the same five-number summary: minimum 0, Q1 8, median 15, Q3 22 and maximum 30.
| Data set | The 21 scores | Mean | SD | Scores in each band: 0–4, 5–9, 10–14, 15–19, 20–24, 25–30 |
|---|---|---|---|---|
| A: spread evenly | 0, 3, 5, 7, 8, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 22, 24, 26, 28, 30 | 15.1 | 8.4 | 2, 4, 4, 4, 4, 3 |
| B: two clusters | 0, 6, 7, 8, 8, 8, 8, 8, 9, 9, 15, 21, 21, 22, 22, 22, 22, 22, 23, 28, 30 | 15.2 | 8.5 | 1, 9, 0, 1, 8, 2 |
| C: heaped on a few values | 0, then 8 (five times), 15 (nine times), 22 (five times), 30 | 15.0 | 6.9 | 1, 5, 0, 9, 5, 1 |
Drawn as box plots, the three would be identical.
Data set B has two humps with a hole at its median, and set C is spiked on three values. Nothing in the box shows either feature. A histogram makes the difference obvious at once.
The standard deviations (8.4, 8.5 and 6.9) differ even though the boxes do not. Matejka and Fitzmaurice (2017) made the same point for summary statistics in general. They generated data sets with identical statistics but dissimilar graphs.
Contemporary Research
Recent work does not replace the box plot. It tests it against modern data and proposes supplements. The strongest evidence is a systematic review of what researchers actually publish.
Do Bar Graphs Hide the Data?
- Aim: Weissgerber et al. (2015) asked how continuous data are presented in small-sample studies, and whether readers can see the underlying distribution.
- Method: They carried out a systematic review of research articles (n = 703) published in top physiology journals and recorded the type of graph used for continuous data.
- Results: Papers rarely included scatterplots, box plots or histograms, which let readers evaluate the data. Most used bar and line graphs, where many different distributions can produce the same graph.
- Conclusion: The authors argued for training investigators in data presentation, fuller presentation of data and changes to journal editorial policy. They supplied Excel templates for scatterplots of small samples.
The review describes physiology rather than psychology. It counts display types without testing whether readers misjudge them.
For very small samples, the tool the authors supplied is a scatterplot of the individual points, not a box plot.
Box Plots for Large Data
Hofmann et al. (2017) pointed out that the box plot was built for small batches drawn by hand.
Large data sets allow precise estimates of tail behavior. Yet a box plot shows many extreme but unsurprising scores as individual points.
Their fix was the letter-value plot. It displays further quantiles beyond the quartiles, as far as the sample can estimate them reliably. They illustrated it on real data and released code in an R package.
Showing the Raw Data Alongside the Summary
Allen et al. (2019) described raincloud plots. They combine the raw data points, a half-violin density curve and summary statistics such as the median and a confidence interval. The aim is to keep the at-a-glance quality of a bar graph while exposing the distribution.
The paper is a tutorial. It offers open-source code in R, Python and Matlab. It reports no test of whether readers interpret the plots more accurately.
Taken together, these papers show that the box plot’s weaknesses are well understood. They lie mainly in what it leaves out.
Treat the box plot as a compact summary layer. Add the raw points when the sample is small, and state the quartile rule and whisker convention.
Strengths
- Quick Summary of Key Statistics: The median, IQR and outliers show a dataset’s center and spread at a glance.
- Simple, Side-by-Side Comparisons: Aligned boxes make group differences stand out. Compare medians, overall spread and outlier prevalence.
- Outlier Detection: Because boxplots explicitly plot values beyond 1.5 × IQR (or another chosen threshold) as separate points, unusual or extreme values are immediately visible.
- Resistance and Economy: The summary does not depend on the mean, standard deviation or any assumption about the shape of the distribution. It is compact: a dozen boxes fit on one figure.
- A Fairer Comparison Than a Bar Graph: A bar graph of means shows one number per group. A box plot shows five, plus the outliers.
Key Takeaways
- Five Numbers: A box plot shows the minimum, first quartile (Q1), median, third quartile (Q3) and maximum of a data set.
- The Box: It spans Q1 to Q3 (the IQR) and holds the middle 50% of scores. The line inside is the median, not the mean.
- Whiskers: They usually end at the most extreme scores inside the 1.5 × IQR fences. Points beyond the fences are plotted as potential outliers.
- Skew: A median off-center in the box and unequal whiskers signal skew. The longer side is the side of the tail.
- Comparison: Side-by-side boxes compare medians, spread and outliers across groups, but they cannot show whether a distribution has one peak or several.
- Software Differences: Programs calculate quartiles differently, so the same data can give slightly different boxes and outlier flags.
- Sample Size: With very small samples, show the points themselves. With very large ones, a box plot flags many ordinary scores as outliers.
References
Allen, M., Poggiali, D., Whitaker, K., Marshall, T. R., & Kievit, R. A. (2019). Raincloud plots: A multi-platform tool for robust data visualization. Wellcome Open Research, 4, 63. https://doi.org/10.12688/wellcomeopenres.15191.1
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BSc (Hons) Psychology, MSc Psychology of Education
Associate Editor for Simply Psychology
Olivia Guy-Evans is a writer and associate editor for Simply Psychology, where she contributes accessible content on psychological topics. She is also an autistic PhD student at the University of Birmingham, researching autistic camouflaging in higher education.
Chartered Psychologist (CPsychol)
BSc (Hons) Psychology, MRes, PhD, University of Manchester
Saul McLeod, PhD, is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.