Min, Q1, median, Q3, max: A box-and-whisker plot shows the five-number summary: minimum, lower quartile Q₁, median, upper quartile Q₃, maximum. The box spans Q₁ to Q₃ with the median inside; the whiskers reach the min and max.
[Diagram: math-box-plot] - Available in full study mode
IB-style question — read the plot
A box plot of exam scores shows minimum 14, Q₁ = 30, median 38, Q₃ = 47, maximum 62. Write down the median and find the range.
Step by step
- Median = the line inside the box.
- Range = max − min.
Final answer
Median = 38; range = 48.
Range vs IQR: Range = max − min (uses the whiskers). IQR = Q₃ − Q₁ (the box width). Don't mix them up.
Each section holds about a quarter of the data: The four sections (min→Q₁, Q₁→median, median→Q₃, Q₃→max) each contain about 25% of the data. The IQR = Q₃ − Q₁ holds the middle 50%. To compare two distributions, compare their medians (centre) and IQRs/ranges (spread).
IB-style question — IQR and a region
For the scores with Q₁ = 30, median 38, Q₃ = 47, find the IQR and state what percentage of students scored between 38 and 47.
Step by step
- IQR = Q₃ − Q₁.
- Median to Q₃ is one of the four quarters.
Final answer
IQR = 17; about 25% scored between the median (38) and Q₃ (47).
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Fences at 1.5 IQRs beyond the quartiles: A value is an outlier if it lies more than 1.5 × IQR beyond a quartile: below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR. Work out the fences, then compare.
IB-style question — is it an outlier?
A data set has Q₁ = 20, Q₃ = 32. Determine whether a value of 54 is an outlier.
Step by step
- IQR and the upper fence.
- Compare 54 with the upper fence.
Final answer
Upper fence = 50, and 54 > 50, so 54 is an outlier.
Smallest non-outlier: The smallest value that is not a low outlier is exactly the lower fence Q₁ − 1.5·IQR (here 20 − 18 = 2).