Mathematics 9709 · Probability & Statistics 1

Types of data & stem-and-leaf

Raw numbers are hard to read until you turn them into a clear display.
This chapter is about choosing the right display and reading it without being tricked by scale, class width, or rounding.
Back-to-back stem-and-leaf diagrams are the most common exam question because they keep every raw value while showing shape.

First ask: counted or measured?

Before you draw anything, ask: can you count it, or do you measure it?
Discrete data is counted: shoe sizes, goals, number of siblings. The values are separate steps.
Continuous data is measured: heights, times, masses. The values can land anywhere in a range, then get rounded.
Qualitative data is words or categories, like favourite colour.
Discrete does not have to mean whole-number. Money and shoe sizes can be discrete because they move in fixed steps.

Counted → discrete. Measured → continuous. The type decides which display is legal.

Situation	Display	Why
Discrete & small	stem-and-leaf	keeps every raw value
Discrete & large	bar chart / line graph	height = frequency, bars gapped
Continuous	histogram or cumulative-frequency graph	area / running total
Comparing two sets	back-to-back stem-and-leaf	the most common exam question

Stem-and-leaf: a sorted list that is secretly a bar chart

A stem-and-leaf keeps every original value. That is why it is stronger than a normal bar chart for small data sets.
Split each value into a stem and a leaf. In 53, the stem can be 5 and the leaf can be 3.
Stems run down the side. Leaves must be ordered, and the diagram must have a key with units.

5|3 5 8 8

6|1 1 2 4 7 9

7|2 7 9

9|2 7

Key: 5 | 3 represents a score of 53%

15 Physics scores. Each leaf is one student; the longest row (the 60s) is the most common band — the shape pops out like a sideways bar chart.

Worked example

Read a single stem-and-leaf (the simplest rung)

For the scores above: $58$ and $61$ tie as most common (each appears twice); below 70 is $4+6=10$ ; 80 or more is $0+2=2$ . Count check $4+6+3+0+2=15$ ✓.

Answer

\text{below 70: } 10,\quad \text{80 or more: } 2

Build a back-to-back that scores all four marks

One shared central stem; list every stem lowest to highest, no gaps even for empty rows.
Right team: leaves outward left-to-right, ascending.
Left team: leaves outward right-to-left — smallest leaf nearest the stem.
No commas, leaves lined up vertically so each row is a bar.
A key with units that names BOTH sides.
Count-check: leaves on each side = the count you were given.

Worked example

Back-to-back, fully worked (the marquee exam shape)

The number of days it rained each month in a town: 2016 17, 20, 13, 12, 10, 8, 0, 1, 5, 11, 16, 9; 2017 9, 13, 11, 8, 6, 3, 1, 2, 2, 4, 8, 7. Display this back-to-back and compare the two years.

Classes (the stem): $0\text{–}9,\ 10\text{–}19,\ 20\text{–}29$ , so stems 0, 1, 2 — equal width, no gaps.
2017 (right) ascends outward; 2016 (left) ascends outward — smallest leaf next to the stem on each side.
Count-check: 12 leaves on each side ✓. Then write the key naming both years with units.
Compare with TOTALS, not just shape: $2016=122$ days vs $2017=74$ days.

2016 (leaves)stem2017 (leaves)

9 8 5 1 001 2 2 3 4 6 7 8 8 9

7 6 3 2 1 011 3

Key: 1 | 0 | 1 means 10 days (2016) and 11 days (2017)

Rain-days back-to-back. The left (2016) leaves ascend OUTWARD from the stem; the key names both years with units.

More rainy days fell in 2016 (122 vs 74). But nothing was said about amounts, so it would be wrong to claim more rain fell in 2016.

Answer

\text{2016: }122\text{ days},\ \ \text{2017: }74\text{ days}

Back-to-back is THE exam shape: shared stem, both sides ascend outward, key names both sides with units. Compare by total/centre, and never over-claim beyond what the count measures.

Worked example

Read median, IQR, and judge the average

Ten monthly salaries ($1000s), already ordered: 28, 29, 30, 31, 32, 33, 34, 35, 36, 82. Median $=\tfrac{32+33}{2}=32.5$ ; $Q_1=30,\ Q_3=35$ so $\text{IQR}=5$ . But the mean is $\tfrac{370}{10}=37$ . (9709/52/M/J/24 Q4) (9709/52/M/J/23 Q3)

Answer

\text{median }\$32{,}500\text{ — the lone }\$82{,}000\text{ inflates the mean}

The mean got dragged because one salary sits far out. Drag that outlier and watch the effect — the mean chases it while the median holds firm:

outlier

mean = 22.0·median = 20·mean > median → positive skew

The median resists outliers; the mean does not. When one value sits far out, quote the median and name the outlier in your comment.

Examiner note

Four B marks: (1) correct shared stem, (2) left leaves correct + ordered outward + no punctuation, (3) right leaves the same, (4) a key with units naming both teams — write the key first, it is the free mark most often dropped. The average comment scores only if you name the value: “the median, because the $82k anomaly distorts the mean.” A spread comparison must compare IQR/range in context, not the centre. (9709/52/O/N/24 Q6)

Common mistake

Drawing two separate plots — a back-to-back wants ONE shared central stem; two side-by-side diagrams cap your marks. And on the left side the leaves must ascend outward (smallest next to the stem), not small-to-big left-to-right.

Common mistake

Forgetting the key (or its units / one side), leaving leaves unsorted, using unequal stem rows, or claiming “more rain” when the data only counts days.

When a row gets too long, and the combined-stats twist

If one row would hold ~30 leaves, split into narrower classes (e.g. 0–4, 5–9 instead of 0–9), giving repeated stems. And when a back-to-back question also hands you $\sum x,\ \sum x^2,\ \sum y,\ \sum y^2$ and asks for the combined mean and SD of all the values: combined mean $=\tfrac{\sum x+\sum y}{n_x+n_y}$ , combined variance $=\tfrac{\sum x^2+\sum y^2}{n_x+n_y}-(\text{combined mean})^2$ (a Ch3 bridge).

Your turn— tap to reveal the worked answer (9709-style)

Top scores. Girls: 84, 88, 86, 86, 88, 90, 90, 91, 93, 93, 94, 94. Boys: 82, 85, 86, 86, 89, 91, 93, 93, 94, 95, 96, 99. Who scored higher on the whole?

Girls total $=1077$ , boys total $=1089$ — the boys, by total and by median.

Answer

\text{boys } (1089 \text{ vs } 1077)

NextHistograms

Types of data & stem-and-leaf

Raw numbers are hard to read until you turn them into a clear display.
This chapter is about choosing the right display and reading it without being tricked by scale, class width, or rounding.
Back-to-back stem-and-leaf diagrams are the most common exam question because they keep every raw value while showing shape.

First ask: counted or measured?

Before you draw anything, ask: can you count it, or do you measure it?
Discrete data is counted: shoe sizes, goals, number of siblings. The values are separate steps.
Continuous data is measured: heights, times, masses. The values can land anywhere in a range, then get rounded.
Qualitative data is words or categories, like favourite colour.
Discrete does not have to mean whole-number. Money and shoe sizes can be discrete because they move in fixed steps.

Counted → discrete. Measured → continuous. The type decides which display is legal.

Situation	Display	Why
Discrete & small	stem-and-leaf	keeps every raw value
Discrete & large	bar chart / line graph	height = frequency, bars gapped
Continuous	histogram or cumulative-frequency graph	area / running total
Comparing two sets	back-to-back stem-and-leaf	the most common exam question

Stem-and-leaf: a sorted list that is secretly a bar chart

A stem-and-leaf keeps every original value. That is why it is stronger than a normal bar chart for small data sets.
Split each value into a stem and a leaf. In 53, the stem can be 5 and the leaf can be 3.
Stems run down the side. Leaves must be ordered, and the diagram must have a key with units.

5|3 5 8 8

6|1 1 2 4 7 9

7|2 7 9

9|2 7

Key: 5 | 3 represents a score of 53%

15 Physics scores. Each leaf is one student; the longest row (the 60s) is the most common band — the shape pops out like a sideways bar chart.

Worked example

Read a single stem-and-leaf (the simplest rung)

For the scores above: $58$ and $61$ tie as most common (each appears twice); below 70 is $4+6=10$ ; 80 or more is $0+2=2$ . Count check $4+6+3+0+2=15$ ✓.

Answer

\text{below 70: } 10,\quad \text{80 or more: } 2

Build a back-to-back that scores all four marks

One shared central stem; list every stem lowest to highest, no gaps even for empty rows.
Right team: leaves outward left-to-right, ascending.
Left team: leaves outward right-to-left — smallest leaf nearest the stem.
No commas, leaves lined up vertically so each row is a bar.
A key with units that names BOTH sides.
Count-check: leaves on each side = the count you were given.

Worked example

Back-to-back, fully worked (the marquee exam shape)

Classes (the stem): $0\text{–}9,\ 10\text{–}19,\ 20\text{–}29$ , so stems 0, 1, 2 — equal width, no gaps.
2017 (right) ascends outward; 2016 (left) ascends outward — smallest leaf next to the stem on each side.
Count-check: 12 leaves on each side ✓. Then write the key naming both years with units.
Compare with TOTALS, not just shape: $2016=122$ days vs $2017=74$ days.

2016 (leaves)stem2017 (leaves)

9 8 5 1 001 2 2 3 4 6 7 8 8 9

7 6 3 2 1 011 3

Key: 1 | 0 | 1 means 10 days (2016) and 11 days (2017)

Rain-days back-to-back. The left (2016) leaves ascend OUTWARD from the stem; the key names both years with units.

More rainy days fell in 2016 (122 vs 74). But nothing was said about amounts, so it would be wrong to claim more rain fell in 2016.

Answer

\text{2016: }122\text{ days},\ \ \text{2017: }74\text{ days}

Back-to-back is THE exam shape: shared stem, both sides ascend outward, key names both sides with units. Compare by total/centre, and never over-claim beyond what the count measures.

Worked example

Read median, IQR, and judge the average

Answer

\text{median }\$32{,}500\text{ — the lone }\$82{,}000\text{ inflates the mean}

The mean got dragged because one salary sits far out. Drag that outlier and watch the effect — the mean chases it while the median holds firm:

outlier

mean = 22.0·median = 20·mean > median → positive skew

The median resists outliers; the mean does not. When one value sits far out, quote the median and name the outlier in your comment.

Examiner note

Common mistake

Forgetting the key (or its units / one side), leaving leaves unsorted, using unequal stem rows, or claiming “more rain” when the data only counts days.

When a row gets too long, and the combined-stats twist

Your turn— tap to reveal the worked answer (9709-style)

Top scores. Girls: 84, 88, 86, 86, 88, 90, 90, 91, 93, 93, 94, 94. Boys: 82, 85, 86, 86, 89, 91, 93, 93, 94, 95, 96, 99. Who scored higher on the whole?

Girls total $=1077$ , boys total $=1089$ — the boys, by total and by median.

Answer

\text{boys } (1089 \text{ vs } 1077)