Produce a five-number summary and construct a boxplot

Which pair determines the two edges of the box?

Move the median, then the maximum of this ordinary boxplot. Follow the line and whisker on a fixed scale. The model changes a summary; it does not recalculate quartiles from raw observations

Which quantity is unchanged in this model?

From five numbers to a drawing

Use medians of halves, as in the earlier quartile lesson. Construct an ordinary boxplot first; a modified plot requires the outlier rule and observed non-outlier endpoints

1. 1Ordinary boxplot Here the whiskers reach the minimum and maximum. A modified boxplot can show unusual observations separately
2. 2Five features Minimum, Q1, median, Q3 and maximum are placed on one numerical scale

Build a small example

Seven times are 14, 2, 9, 18, 6, 5 and 11 minutes

1. 1Order 2, 5, 6, 9, 11, 14, 18.
2. 2Halves The median is 9. Lower half: 2, 5, 6. Upper half: 11, 14, 18.
3. 3Summary Minimum 2, Q1 5, median 9, Q3 14, maximum 18.

Ordered times (minutes)

\(\displaystyle 2,5,6,9,11,14,18\)

What is the median of those seven ordered times?

Ordered times (minutes)

\(\displaystyle 2,5,6,9,11,14,18\)

What is Q1?

Ordered times (minutes)

\(\displaystyle 2,5,6,9,11,14,18\)

What is Q3?

The completed boxplot is shown. What does the line inside the box represent?

The box edges are 5 and 14 minutes. What is the IQR?

The whiskers end at 2 and 18 minutes. What is the range?

Even-sized lists

For 4, 6, 8, 10, 14, 18, 20, 24, split into two halves of four. Do not remove either central observation

1. 1Centre Average the two central observations \(\displaystyle \mathrm{median}=\frac{10+14}{2}=12\)
2. 2Lower quartile Average 6 and 8. \(\displaystyle Q_1=(6+8)/2=7\)
3. 3Upper quartile Average 18 and 20. \(\displaystyle Q_3=(18+20)/2=19\)
4. 4Summary 4, 7, 12, 19, 24.

Which is a valid summary in increasing order?

Ordered lengths (metres)

\(\displaystyle \begin{gathered}1.8,\ 2.1,\ 2.4,\ 2.6\\2.9,\ 3.1,\ 3.6\end{gathered}\)

What is the median of these seven recorded lengths?

Ordered lengths (metres)

\(\displaystyle \begin{gathered}1.8,\ 2.1,\ 2.4,\ 2.6\\2.9,\ 3.1,\ 3.6\end{gathered}\)

Find the lower box edge, Q1.

Ordered lengths (metres)

\(\displaystyle \begin{gathered}1.8,\ 2.1,\ 2.4,\ 2.6\\2.9,\ 3.1,\ 3.6\end{gathered}\)

Find the upper box edge, Q3.

For lengths 1.8, 2.1, 2.4, 2.6, 2.9, 3.1, 3.6 metres, move the box edges to the quartiles you calculated. Then try moving both to 2.6. The controls draw a proposed summary; they do not change the recorded data

Which edge settings represent the recorded lengths?

Ordered lengths (metres)

\(\displaystyle \begin{gathered}1.8,\ 2.1,\ 2.4,\ 2.6\\2.9,\ 3.1,\ 3.6\end{gathered}\)

For the correctly constructed boxplot, what is the IQR?

Does a five-number summary tell you exactly how many measurements were recorded?

An extreme observation is flagged as an outlier. How do ordinary and modified boxplots differ?

All observations equal 8. What happens to the boxplot?

Construct and check your own plot

Draw a boxplot on a 0–30 minute scale from the five-number summary (4, 7, 12, 19, 24)

1. 1Place Mark all five values on a linear scale. Draw the box from 7 to 19 and median at 12.
2. 2Whiskers Draw from 4 to 7 and from 19 to 24. The median must not be forced to the box centre
3. 3Check Range 20 minutes; IQR 12 minutes. Your drawing needs self or teacher review

Construct a continuous-data example

The recorded lengths are 1.8, 2.1, 2.4, 2.6, 2.9, 3.1 and 3.6 metres. Draw an ordinary boxplot for these lengths on a labelled 1.5–4 metre scale. This activity needs your own drawing, not just a matched option

1. 1Check the features Whiskers reach 1.8 and 3.6; box edges are 2.1 and 3.1; median is 2.6
2. 2Check the scale Equal distances represent equal changes in length. Include the measurement unit
3. 3Check the limits The summary does not retain all seven values or the sample size. Compare with a dot plot of the same observations. Your drawing and explanation need self or teacher review
4. 4Choose and verify a tool Choose a spreadsheet or statistical tool. Enter all seven recorded lengths, state the medians-of-halves convention and verify the five features against sorted records. Record the formulas/settings and explain why an ordinary plot is required. Check its dot-plot counterpart and the proposed numerical scale
5. 5Complete before continuing Save both drawings, the tool record and your explanation before the model. Complete plots, writing and external-tool execution are not automatically marked

Model response: preserve the measurement scale

The seven recorded lengths are 1.8,2.1,2.4,2.6,2.9,3.1,3.6 metres. Their ordinary five-number summary is (1.8,2.1,2.6,3.1,3.6) metres. The IQR is 1 metre and full range 1.8 metres

1. 1Complete drawing On the requested linear 1.5–4-metre axis, put box edges at 2.1 and 3.1, the median at 2.6 and whisker endpoints at 1.8 and 3.6. Label metres. Plot the seven original values as seven dots on the same scale. The two drawings summarise the same sample; the boxplot does not display every observation or encode its count
2. 2One reproducible workflow Enter the sorted lengths in A2:A8. Use MIN, MAX and MEDIAN for the full range of cells. Exclude the central record A5 when finding quartiles: MEDIAN(A2:A4)=2.1 and MEDIAN(A6:A8)=3.1. Record the convention instead of assuming every QUARTILE function uses it. Place those five numerical features in an ordinary-boxplot tool or draw them from the table
3. 3Independent verification Check the fourth ordered observation and the medians of both three-record halves manually. Verify seven dots, equal numerical intervals and both differences. The whiskers here represent the full minimum and maximum. Review drawing accuracy and tool operation yourself or with a teacher

The lower half of the ordered lengths is 1.8, 2.1 and 2.4 metres. A tool puts the lower box edge at the minimum 1.8 metres. What is the appropriate check?