One Quantitative Variable: Histograms, Stems, and Dots — Worked Solutions
Worked Solutions
Compare your reasoning with the worked solutions after attempting the questions. A different valid representation or explanation can earn credit when it satisfies the stated criteria.
Original generated practice for the Fall 2026 course. Return to the lesson and independent checks.
Place the endpoint once: worked answer
A histogram uses intervals 0≤x<10, 10≤x<20, and 20≤x<30 for wait times in minutes.
Part a
Identify the bin for 10, 19, and 20 minutes and explain why the endpoint rule is needed.
10 and 19 go in 10≤x<20; 20 goes in 20≤x<30. Each observation enters exactly one bin, preventing overlap or double counting.
| Evidence for credit | Points |
|---|---|
| Correct three placements and single-count rule. | 1 |
Build an ordered stem: worked answer
Five values in grams are 23, 21, 28, 21, and 20.
Part a
Construct the stem-and-leaf row and key. Explain the repeated leaf.
Stem 2 has ordered leaves 0 1 1 3 8; key 2|3=23 grams. Two leaves 1 represent two 21-gram observations.
| Evidence for credit | Points |
|---|---|
| Ordered row and correct key. | 1 |
| Repeated observations retained. | 1 |
One dot for each observation: worked answer
Recorded scores are 4, 4, 4, 5, and 7.
Part a
Construct a dotplot on paper or as an ordered text plot with one symbol per observation. State the axis and stack heights.
On a score axis, place three stacked dots at 4, one at 5, none at 6, and one at 7. This accounts for all five observations and preserves numerical order.
| Evidence for credit | Points |
|---|---|
| Five dots and correct stacks/empty position. | 1 |
| Ordered labeled quantitative axis. | 1 |
Same data, different grouping: worked answer
Waits are 3,4,4,6,7,8,9,10,10,12,13,17 minutes.
Part a
Construct bin frequencies for 0≤x<5, 5≤x<10, 10≤x<15, 15≤x<20, then for 0≤x<10 and 10≤x<20. Explain the appearance change and a horizontal-bar alternative.
Narrow frequencies are 3,4,4,1; wide frequencies are 7,5. Wider intervals merge distinct parts of the distribution and hide some detail. A valid horizontal histogram places the ordered bins on the vertical axis and frequency lengths on the horizontal axis.
| Evidence for credit | Points |
|---|---|
| Both complete frequency sets. | 1 |
| Bin-width detail explanation and axis-swapped alternative. | 1 |
Represent a new waiting-time sample: worked answer
Twenty customer waits in minutes: 4, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 27, 29.
Part a
Construct a frequency histogram with intervals 0≤x<10, 10≤x<20, and 20≤x<30. Label intervals, units, baseline, and bar heights; check the total.
Heights are 5,10,5 customers; total 20. Draw ordered touching equal-width bins on the minutes axis, frequency on the other axis, and a zero baseline.
| Evidence for credit | Points |
|---|---|
| Correct three heights and total. | 1 |
| Complete ordered quantitative histogram with labeled axes/intervals. | 1 |
Part b
Construct a relative-frequency version and a valid horizontal version. Explain what stays the same.
Relative heights are 0.25,0.50,0.25, summing to 1. A horizontal version places the same ordered intervals on the vertical axis and proportion lengths on the horizontal axis. The observations, intervals, and shares stay the same.
| Evidence for credit | Points |
|---|---|
| All shares and ordered horizontal-bin construction. | 1 |
| Explains invariant underlying data and proportions. | 1 |
Part c
Construct a stem-and-leaf plot with tens stems and an explicit minutes key.
0|4 6 7 8 9; 1|0 0 1 2 3 4 5 6 7 8; 2|0 2 4 7 9. Key 1|2=12 minutes. Stems and leaves are ordered and all 20 observations are retained.
| Evidence for credit | Points |
|---|---|
| Complete ordered stems/leaves with duplicate 10s. | 1 |
| Correct key and observation total. | 1 |
Part d
Construct a dotplot. Label the axis, retain repeated waits, and state what individual detail it shows that your histogram hides.
Place one dot at every listed value, with two stacked at 10 minutes. The axis is wait in minutes in natural order. It shows exact individual waits such as the two 10s and one 18, which the 10-to-less-than-20 histogram bar combines.
| Evidence for credit | Points |
|---|---|
| All observations and correct 10 stack. | 1 |
| Ordered units and exact-value versus interval distinction. | 1 |
Part e
Construct 5-minute bin counts using [0,5), [5,10), [10,15), [15,20), [20,25), [25,30). Explain how this changes visible detail without changing the data.
Counts are 1,4,6,4,3,2; total 20. These narrower bins distinguish the concentration from 10 to less than 15 and separate it from 15 to less than 20, which the wider histogram merged. The same observations are regrouped, not newly measured.
| Evidence for credit | Points |
|---|---|
| All six counts and total. | 1 |
| Specific detail comparison tied to unchanged data. | 1 |