Chapter 25: Review

The chapter problem, solved

The ordered travel times have 20 values. Their median is the mean of positions 10 and 11:

Using the median-of-halves convention, $Q_1=(17+18)/2=17.5$ and $Q_3=(26+27)/2=26.5$.

The five-number summary is $12,17.5,21.5,26.5,42$. The upper fence is $26.5+1.5\times9=40$, so 42 is a possible outlier. The data are not grouped, so you can make a box plot directly. Grouped data would give estimates from a cumulative frequency graph or histogram.

Mixed practice — choose the method first

For each question, name the method before working.

1. A list has 24 values. At what cumulative frequencies would you estimate the lower quartile, median and upper quartile?
2. A class has width 8 and frequency 28. Find its histogram density.
3. Two groups have medians 16 and 20. Which has the higher typical value?
4. The lower quartile is 10 and upper quartile is 22. Find the IQR.
5. A scatter graph rises from left to right. Describe its correlation.
6. A data set has $Q_1=12$, $Q_3=20$. Find its upper IQR fence.
7. A bar has width 5 and frequency density 3. Find its frequency.
8. A line of best fit predicts 18 at $x=0$ and 38 at $x=10$. Estimate at $x=6$.
9. Group A has IQR 4; Group B has IQR 9. Which middle half is more spread out?
10. A cumulative frequency graph uses upper class boundaries. Explain why its curve cannot go down.

What you can now do

• find quartiles, the IQR and a five-number summary;
• draw and interpret cumulative frequency graphs and histograms;
• describe correlation and use a line of best fit for estimates;
• compare distributions and identify possible outliers.

Your relevance reflection

Return to your example from the opener. Which measure or graph could help compare the values? Write one sentence explaining your choice. Has this chapter changed what you notice about variation in that situation?