Mathematics • Year 10 • Unit 4 • Lesson 4

Stem-and-Leaf Plots in the Real World

Apply ordered and back-to-back stem-and-leaf plots to real contexts: HSC trial results, junior vs senior athletics, BOM temperature records, and exam pass-rate comparisons across classes. Read off medians, count students above a cut-off, and use Lesson 4's "data values are preserved" advantage.

Apply · Real-World Maths

1. Word problems

Each problem uses Lesson 4 ideas: build a plot, read the median by counting to the middle, or compare two data sets with a back-to-back plot. Show your working — a final answer with no working only earns half marks.

1.1 — HSC trial maths marks. A class of 15 students sits a trial HSC maths paper. Marks (out of 100):
58, 67, 72, 49, 88, 73, 91, 65, 70, 82, 76, 60, 84, 79, 95.

(a) Build an ordered stem-and-leaf plot with a key.
(b) How many students scored 70 or above? 3 marks

Stuck? "70 or above" includes any leaf in the 7, 8 or 9 row.

1.2 — BOM monthly maximum temperatures. The daily maximum temperatures (°C) in Sydney for the first 14 days of January are:
28, 31, 26, 35, 29, 32, 30, 27, 38, 33, 29, 31, 36, 34.

(a) Build an ordered stem-and-leaf plot with a key.
(b) Use the plot to find the median temperature for those 14 days.
(c) On how many days was the maximum 32 °C or above? 3 marks

Stuck? Median of 14 values = average of the 7th and 8th values when ordered.

1.3 — Athletics carnival 100 m times. Times (in seconds) for the Year 10 girls 100 m race were recorded as:
15.2, 14.8, 16.1, 13.9, 14.5, 17.3, 15.0, 16.5, 14.2, 15.8, 13.7, 16.0.

(a) Using stems = whole seconds and leaves = tenths digit (so 14.5 → 14 | 5), build an ordered stem-and-leaf plot with a key.
(b) How many girls ran in under 15.0 s? 3 marks

Stuck? Decimal data works the same way — pick what counts as stem (the whole number) and what counts as leaf (the tenths digit), and state it in your key.

1.4 — Boys vs girls heights (back-to-back). Year 10 PE class records the heights (cm) of 12 girls and 12 boys.
Girls: 152, 158, 161, 155, 163, 168, 159, 165, 154, 160, 162, 157.
Boys: 165, 172, 168, 175, 170, 178, 162, 173, 169, 180, 174, 167.

(a) Build a back-to-back ordered stem-and-leaf plot (girls on left, boys on right). Use stems of 15, 16, 17, 18.
(b) Identify the shortest girl and the tallest boy.
(c) Write one sentence comparing the medians visually. 3 marks

1.5 — Reading a given plot. A stem-and-leaf plot of weekly study hours for 18 students shows:
0 | 5 8 8 9
1 | 0 2 2 3 5 6 8
2 | 0 1 4 5 8
3 | 2 5 Key: 1 | 2 = 12 hours.

(a) State the range (largest − smallest).
(b) Find the median.
(c) The lesson advantage card says stem-and-leaf plots "preserve the original data values". Explain in one sentence what this means in terms of being able to compute the mean. 3 marks

2. Explain your thinking

This question is about communication, not just drawing. Use full sentences. 4 marks

2.1 A friend says: "A histogram is better than a stem-and-leaf plot because it looks cleaner." Using Lesson 4 (especially the lesson's stated advantage that stem-and-leaf preserves the original data), write a four-sentence reply that (i) gives one situation where the friend is right, (ii) gives one situation where the stem-and-leaf is more useful, (iii) explains specifically what is lost when continuous data is shown only as a histogram, and (iv) names one calculation that is easier from a stem-and-leaf plot than from a histogram.

Stuck? Histograms only show frequency per class — you cannot read off the individual values. Stem-and-leaf plots show every original value, so the median (and even the mean) can be calculated directly.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — HSC trial marks

(a) Ordered:
4 | 9
5 | 8
6 | 0 5 7
7 | 0 2 3 6 9
8 | 2 4 8
9 | 1 5 Key: 7 | 2 = 72.

(b) "70 or above" = leaves in the 7, 8 and 9 rows = 5 + 3 + 2 = 10 students.

1.2 — January max temps

(a) Ordered:
2 | 6 7 8 9 9
3 | 0 1 1 2 3 4 5 6 8 Key: 2 | 8 = 28 °C.

(b) n = 14, median = average of 7th and 8th values = (31 + 31) / 2 = 31 °C.
(c) "32 °C or above" = leaves 2, 3, 4, 5, 6, 8 in the 3 row = 6 days.

1.3 — 100 m race times

(a) Ordered:
13 | 7 9
14 | 2 5 8
15 | 0 2 8
16 | 0 1 5
17 | 3 Key: 14 | 5 = 14.5 s.

(b) "Under 15.0 s" = leaves in 13 and 14 rows = 2 + 3 = 5 girls.

1.4 — Boys vs girls heights (back-to-back)

(Girls leaves outward) | stem | (Boys leaves)

4 2 | 15 |
8 7 5 3 2 1 0 9 8 | 16 | 2 5 7 8 9
| 17 | 0 2 3 4 5 8
| 18 | 0 Key: 15 | 4 = 154 cm.

(Girls leaves in row 16 in descending order out from the stem: 0, 1, 2, 3, 5, 8 — written 8 5 3 2 1 0.)

(b) Shortest girl = 152 cm; tallest boy = 180 cm.
(c) Visually the boys distribution sits about 10 cm higher on the stems than the girls — boys median around 171 cm, girls median around 160 cm.

1.5 — Reading a study-hours plot

Smallest = 5, largest = 35.
(a) Range = 35 − 5 = 30 hours.
(b) n = 18, median = average of 9th and 10th values = (13 + 15) / 2 = 14 hours.
(c) Because the plot shows every original value (e.g. 5, 8, 8, 9, ...), we can simply add all 18 values and divide by 18 to compute the mean exactly — no estimation needed.

2.1 — Histogram vs stem-and-leaf (sample response)

(i) A histogram is better when there are hundreds or thousands of data values and we only need to see the overall shape — listing every leaf would be unreadable. (ii) A stem-and-leaf is more useful for smaller data sets (say, 10-50 values) because it preserves every original value, which Lesson 4 explicitly calls out as the main advantage. (iii) A histogram loses the individual values — once data is grouped into bars, you no longer know whether a 60-69 bar held five 60s or five 69s. (iv) The median can be read directly from an ordered stem-and-leaf plot by counting to the middle value, but on a histogram you can only estimate it using the median class.

Marking: 1 mark each for (i), (ii), (iii), (iv).