Interpreting the Interquartile Range on a Box Plot

A box plot summarizes a distribution using quartiles. The box begins at the first quartile (Q1) and ends at the third quartile (Q3); the median is shown inside it. The distance from Q1 to Q3 is the interquartile range, a measure of spread that is less sensitive to extreme values than the full range.

Calculate the IQR

Use IQR = Q3 − Q1. If Q1 is 12 and Q3 is 27, the IQR is 15. The subtraction order matters: subtract the lower quartile from the upper quartile so the spread is nonnegative. Read the scale carefully and use the marked quartile values, not the whisker endpoints.

What the IQR tells you

The IQR covers the middle half of ordered observations, from the 25th percentile through the 75th percentile. It does not tell you the exact values of all data points or how the data are distributed within the box. Two data sets can have the same IQR but different medians, shapes, and tails.

Outliers and box plots

A common convention flags a value as a possible outlier if it falls below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. Some box plots show whiskers extending only to the most extreme observations within those fences and plot farther values as individual points. Other plots use whiskers differently. Read the graph’s key or instructions before assuming what its whiskers represent.

Common mistakes

• Using the whisker endpoints instead of Q1 and Q3.
• Subtracting Q3 from Q1 and getting a negative spread.
• Calling the IQR the range of the entire data set.
• Assuming the IQR reveals how values are distributed inside the box.

Read a box plot from its quartile marks

A box plot places the median inside a box whose ends mark Q1 and Q3. The IQR is the horizontal or vertical distance between those edges, not the length of the whiskers. If the scale shows Q1 = 12 and Q3 = 27, the middle half spans 15 units. Read the axis increments: if each grid interval represents 5, a box two intervals wide has IQR 10, not 2.

Understand what the middle 50 percent means

Q1 marks the point below which about one quarter of ordered observations fall; Q3 marks the point below which about three quarters fall. The interval from Q1 to Q3 contains the middle half by rank. It does not mean that observations are evenly spaced within the box. A short box indicates that the middle half is concentrated in a narrower numerical range; it does not reveal the distribution's shape by itself.

Compare spread and center separately

Two groups can have the same median but different IQRs, indicating different spreads among their middle observations. They can also have the same IQR but different medians. Compare the median for center and the IQR for middle spread. Whisker lengths and individual points may describe tails or possible outliers, depending on how the graph is defined. Do not infer a group's average from the median line.

Interpret outlier fences cautiously

Under the common 1.5 × IQR rule, calculate the lower fence Q1 − 1.5(IQR) and upper fence Q3 + 1.5(IQR). Points beyond these cutoffs are flagged as potential outliers; the rule does not prove that a measurement is wrong or should be removed. For Q1 = 12 and Q3 = 27, IQR = 15, so the fences are −10.5 and 49.5. A point beyond them deserves investigation in context, not automatic deletion.

Check graph conventions and quartile methods

Some box plots draw whiskers to the minimum and maximum; others stop at the most extreme values inside the fences and plot farther observations individually. Read the key before interpreting whiskers. For a small raw data set, textbooks may use different procedures for locating quartiles when the number of observations is even or odd. Follow the method the problem specifies, and when Q1 and Q3 are already labeled, calculate directly from those values.

Key takeaway

Read Q1 and Q3 from the box edges, subtract Q1 from Q3, and interpret the result as the spread of the middle half. Use the graph’s stated whisker convention when discussing outliers.

Calculate the interquartile range

The interquartile range (IQR) is the distance between the third and first quartiles: IQR = Q3 − Q1. It measures the spread of the middle 50% of ordered observations. If Q1 = 14 and Q3 = 26, then IQR = 12. It has the same units as the data and is less sensitive to extreme values than the full range.

Quartiles divide ordered data into four parts. Different conventions can place Q1 and Q3 slightly differently, especially with small samples. Follow the method specified by the problem or use quartiles read from the provided plot. Do not assume that each quartile contains exactly one quarter of the observations when tied values occur at boundaries.

Use the IQR rule to flag possible outliers

A common rule flags values below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR) for investigation. With Q1 = 14 and Q3 = 26, IQR = 12, so the lower fence is 14 − 18 = −4 and the upper fence is 26 + 18 = 44. A value of 50 is flagged by this rule. It is not automatically an error; it is a point far from the middle spread that merits examination.

Some box-and-whisker plots draw whiskers to the most extreme values within these fences and show flagged points separately. Other introductory plots extend whiskers to the minimum and maximum. Check the plot’s convention. The IQR rule describes a statistical flag, not a universal definition of an outlier.

Compare distributions robustly

When comparing groups, use median for center and IQR for spread of the middle half. A higher median indicates a higher central value; a wider IQR indicates more variability among central observations. Similar IQRs do not mean the full distributions are identical, because tails and sample sizes may differ. A box plot does not reveal every individual value.

Because the IQR is resistant to extremes, it can complement the mean and standard deviation. For skewed data, median and IQR often describe typical location and spread more faithfully than mean and standard deviation. Choose measures based on the distribution and question.

• Compute IQR as Q3 − Q1.
• Use the 1.5×IQR fences to flag values for review, not automatic removal.
• Check whether whiskers include all extrema or exclude flagged outliers.
• Compare median for center and IQR for middle spread.
• Follow the stated quartile convention for small samples.