Standard Deviation, Quartiles & Percentiles • Topic 4 of 4

Boxplots

A boxplot (box-and-whisker plot) is a compact picture of the five-number summary: minimum, Q1, median, Q3 and maximum. The box spans Q1 to Q3, so its width is the IQR; a line inside the box marks the median; and the whiskers reach out to the minimum and maximum. Reading one is a GRE skill in itself. The position of the median line inside the box signals skew — a median hugging the left of the box means the upper half is more spread out (right skew), and vice versa. The whiskers show the reach of the extremes but say nothing about how many values sit out there. A frequent trap: each of the four regions of a boxplot — below Q1, Q1 to median, median to Q3, above Q3 — contains about 25% of the data, so a wide section means those 25% are spread out, not that more values live there. Boxplots are ideal for comparing two data sets side by side, which is exactly how the GRE tends to use them.

A labeled boxplot on a number line, each of its four regions holding about 25 percent of the dataEach region holds about 25% of the data25%25%25%25%min 12Q1 20median 28Q3 34max 50

✅ Solved examples

1. A boxplot has min 12, Q1 20, median 28, Q3 34, max 50. Find the IQR and the range.
IQR = Q3 − Q1 = 34 − 20 = 14. Range = max − min = 50 − 12 = 38.
2. In the same boxplot, roughly what percent of the data lies between 20 and 34?
That is Q1 to Q3, the middle 50% of the data. About 50%.
3. A boxplot's median line sits much closer to Q1 than to Q3. What does that indicate?
The upper half (median to Q3 and beyond) is more spread out than the lower half, indicating a right (positive) skew.
4. Quantitative Comparison. Two boxplots share the same median but Set A's box is wider than Set B's. Quantity A: the IQR of Set A. Quantity B: the IQR of Set B. Which is greater?
The box width IS the IQR (Q1 to Q3). A wider box means a larger IQR. Quantity A is greater.

✏️ Practice — try these, take hints as needed

1. A boxplot shows Q1 = 15 and Q3 = 27. Find the IQR.
IQR is the box width.
Q3 − Q1.
27 − 15.
12
2. A boxplot has min 5 and max 45. Find the range.
Range spans whisker to whisker.
max − min.
45 − 5.
40
3. About what percent of the data lies above Q3 in any boxplot?
The four regions each hold about a quarter.
Above Q3 is the top region.
One quarter.
about 25%
4. If a boxplot's median line is nearer Q3 than Q1, which half is more spread out?
A short median-to-Q3 gap means the upper values bunch.
The longer side is more spread.
Compare the two gaps.
The lower half (left skew)
5. Two boxplots have equal IQRs but different whisker lengths. Which measure differs — the IQR or the range?
IQR is the box width and is equal here.
Whiskers reach the extremes.
The extremes set the range.
The range

📝 Topic test — 8 questions

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