Read values and percentages from a boxplot

Which feature of this boxplot shows its median?

Move the median and maximum. The two edges of the box stay at 5 and 14 minutes. Compare position, IQR and range

Does moving the median necessarily change the IQR?

Read the landmarks first

For an ordinary boxplot, locate the two whisker ends, two box edges and the median line

1. 1Units All five values use the numerical axis units
2. 2Differences Range is maximum minus minimum. IQR is Q3 minus Q1.
3. 3Modified plots If individual outlier points are shown, the full data range includes them

Five-number summary (minutes)

\(\displaystyle (2,5,9,14,18)\)

Use the five-number summary to identify the median

Five-number summary (minutes)

\(\displaystyle (2,5,9,14,18)\)

Read the lower quartile from that summary

Five-number summary (minutes)

\(\displaystyle (2,5,9,14,18)\)

Read the upper quartile from that summary

Five-number summary (minutes)

\(\displaystyle (2,5,9,14,18)\)

Calculate the IQR from box edges 5 and 14.

Five-number summary (minutes)

\(\displaystyle (2,5,9,14,18)\)

Calculate the range from extremes 2 and 18.

Quartiles describe ordered positions

Use approximate proportions when interpreting a boxplot. Exact counts need more information

1. 1Typical reading About 25% below Q1, about 50% below the median, about 75% below Q3.
2. 2Box The box represents roughly the middle half of the ordered observations
3. 3Limits Ties, small samples and conventions can prevent exact percentage claims. Width shows numerical spread, not a bar-chart frequency

Using the usual approximate interpretation, about what percentage lies above Q3?

Using the usual approximate interpretation, about what percentage lies between Q1 and Q3?

One whisker is much longer than the other. What does this establish?

Why exact percentages need care

Consider eight observations: 1, 1, 1, 1, 1, 1, 1, 9. Under medians of halves, Q1, median and Q3 all equal 1.

1. 1Strictly below Q1 There are no values below 1, so the exact percentage strictly below Q1 is 0%
2. 2At or below Q1 Seven of eight values are at or below 1, which is 87.5%
3. 3Lesson A five-number summary cannot generally certify an exact strict-inequality percentage

Seven observations are 1. Move the eighth observation from 9 to 2, then 1. Count the dots strictly above Q3 = 1. The actual records change while all three quartiles remain 1.

What does the experiment show?

Recorded values

\(\displaystyle 1,\ 1,\ 1,\ 1,\ 1,\ 1,\ 1,\ 9\)

For the original records 1, 1, 1, 1, 1, 1, 1, 9, what exact percentage is strictly above Q3 = 1?

From a boxplot alone, can you always find the exact number of observations strictly above Q3?

Can a boxplot alone reveal the mean or sample size?

Check an interpretation

For a plot with Q1 = 10, median = 16 and Q3 = 22 minutes, write one precise spread statement and one qualified proportion statement

1. 1Spread IQR = 22 − 10 = 12 minutes
2. 2Proportion Approximately half the observations lie between the quartiles under the usual interpretation
3. 3Limits Do not invent the exact sample count, mean or individual values. Written explanations need self or teacher review

Non-outlier whisker endpoints (metres)

\(\displaystyle 1.2,\ 4.4\)

Q1, median, Q3 (metres)

\(\displaystyle 2.1,\ 2.8,\ 3.6\)

Separate outlier (metres)

\(\displaystyle 6.5\)

Read the maximum of the entire data set, including the separate point

Non-outlier whisker endpoints (metres)

\(\displaystyle 1.2,\ 4.4\)

Q1, median, Q3 (metres)

\(\displaystyle 2.1,\ 2.8,\ 3.6\)

Separate outlier (metres)

\(\displaystyle 6.5\)

Calculate the full range of these recorded lengths

Non-outlier whisker endpoints (metres)

\(\displaystyle 1.2,\ 4.4\)

Q1, median, Q3 (metres)

\(\displaystyle 2.1,\ 2.8,\ 3.6\)

Separate outlier (metres)

\(\displaystyle 6.5\)

Calculate the IQR of these recorded lengths

Non-outlier whisker endpoints (metres)

\(\displaystyle 1.2,\ 4.4\)

Q1, median, Q3 (metres)

\(\displaystyle 2.1,\ 2.8,\ 3.6\)

Separate outlier (metres)

\(\displaystyle 6.5\)

What is the upper whisker endpoint of this modified boxplot?

Compare the evidence each display retains

First use eight durations recorded to the nearest second: 2, 2, 2, 2, 4, 6, 8, 10 seconds. Draw a dot plot and an ordinary boxplot on a common 0–12-second scale. Use medians of halves. Calculate the exact percentages strictly below Q1 and at or below Q1, and explain what the boxplot alone does not establish. Then repeat the retained investigation with eight continuous measurements of your own

1. 1Count from the raw display Choose strictly below or at-or-below a quartile. Count records and calculate the exact percentage, including ties
2. 2Read the summary Describe the usual approximate quartile interpretation separately. Explain why the exact count may differ
3. 3Retain context State units, sample size, recording precision and the medians-of-halves convention. Your drawings and explanation need self or teacher review
4. 4Choose and verify a tool Record your selected plotting and counting functions, the quartile convention and both inequality criteria. Check every dot, summary feature and count directly against the raw records, and explain why a default approximate quartile share is a different statement
5. 5Complete before continuing Save both plots, exact counts, explanation and tool record before opening the bounded model. Your own measurements will vary. Complete writing, drawing and external-tool execution are not automatically marked

Model response: retain ties and name the inequality

The sorted records are 2,2,2,2,4,6,8,10 seconds. The ordinary summary is (2,2,3,7,10) seconds, giving IQR 5 seconds and full range 8 seconds. Q1 equals 2 because the lower four records all equal 2.

1. 1Complete both drawings Place four stacked dots at 2 and single dots at 4,6,8,10 on the labelled 0–12-second axis. Draw the box from 2 to 7 with median 3. The lower ordinary whisker has zero length because the minimum and Q1 coincide; the upper whisker reaches 10. Keep all eight records represented
2. 2Exact percentages from the records No record is strictly below the lower quartile of 2, so the exact share is 0%. Four of the eight records are at or below 2, so the exact share is 50%. Ties and the inequality matter. These are observed counts, not the usual approximate 25% interpretation at the lower quartile
3. 3One reproducible workflow Enter the sorted data in A2:A9. Calculate Q1 with MEDIAN(A2:A5), median with MEDIAN(A2:A9), and Q3 with MEDIAN(A6:A9). Use COUNTIF with <Q1 and <=Q1 separately, each divided by COUNT. Check the counts against the four repeated 2s and the other four records. A frequency table with values 2,4,6,8,10 and counts 4,1,1,1,1 helps verify the dot stacks
4. 4Explain the information limit A boxplot by itself does not supply the original sample size, individual records or exact strict-inequality counts. Equal geometric widths do not imply equal observed counts, and a zero-length whisker does not imply there are no observations there. The separate raw display supports these exact claims; preserve the recording precision and sample context

For the recorded durations 2, 2, 2, 2, 4, 6, 8 and 10 seconds, which statement correctly handles Q1 = 2 seconds?