C9.4; E9.4 - Stem-and-leaf and frequency distributions

C9.4; E9.4 - Stem-and-leaf and frequency distributions

A good display can organise observations while keeping their exact values. You will choose a clear stem-and-leaf key, order and retrieve the observations, and build simple frequency distributions without confusing a value with how often it occurs.

Choosing stems, leaves and a key

Could exactly the same rows of digits represent lengths ten times as large? Yes: a stem-and-leaf diagram needs a key before its digits have a definite meaning.

A stem-and-leaf diagram separates each value into a stem and a single final digit called a leaf. For whole numbers such as 1717, use stem 11 for the tens and leaf 77 for the units. Each occurrence has its own leaf, so repeated values stay repeated. The stem goes to the left of a dividing line and the leaves go to its right.

Stem-and-leaf key

A key states how to reconstruct one full value, including its unit if applicable. For example, the key “1∣71\mid7 represents 17 cm17\,\mathrm{cm}” assigns tens and units. Changing it to “1∣71\mid7 represents 1.7 cm1.7\,\mathrm{cm}” makes every value one tenth as large.

Choose stems that leave one digit per leaf at the required precision. For measurements recorded to one decimal place, the whole-number part can be the stem and the tenths digit the leaf. Do not delete zeros: a leaf of 00 is part of a value, not an empty place.

Worked example 1: whole-number measurements

Question. Draw a stem-and-leaf diagram for these lengths in centimetres:

12, 17, 20, 20, 24, 4112,\ 17,\ 20,\ 20,\ 24,\ 41

Plan. Use tens as stems and units as leaves, keeping both occurrences of twenty.

12⟶1∣2one ten and two units20⟶2∣0two tens and no units41⟶4∣1four tens and one unit\begin{aligned} 12&\longrightarrow1\mid2 && \text{one ten and two units}\\ 20&\longrightarrow2\mid0 && \text{two tens and no units}\\ 41&\longrightarrow4\mid1 && \text{four tens and one unit} \end{aligned}
StemLeaves
11272\quad7
220040\quad0\quad4
33
4411

Key: 1∣21\mid2 represents 12 cm12\,\mathrm{cm}.

Answer. The ordered diagram above includes the empty thirties row so the gap in the data is visible.

Check. Read it back: 12,17,20,20,24,4112,17,20,20,24,41. There are six leaves for six observations, including two zeros in the twenties row.

Now divide every measurement by ten. The digits can stay in exactly the same places; the key must change.

Worked example 2: decimal measurements

Question. Draw a stem-and-leaf diagram for these lengths in centimetres:

1.2, 1.7, 2.0, 2.0, 2.4, 4.11.2,\ 1.7,\ 2.0,\ 2.0,\ 2.4,\ 4.1

Plan. Use whole centimetres as stems and tenths as leaves.

1.2⟶1∣2one and two tenths2.0⟶2∣0keep the zero tenths4.1⟶4∣1four and one tenth\begin{aligned} 1.2&\longrightarrow1\mid2 && \text{one and two tenths}\\ 2.0&\longrightarrow2\mid0 && \text{keep the zero tenths}\\ 4.1&\longrightarrow4\mid1 && \text{four and one tenth} \end{aligned}
StemLeaves
11272\quad7
220040\quad0\quad4
33
4411

Key: 1∣21\mid2 represents 1.2 cm1.2\,\mathrm{cm}.

Answer. The diagram's digits match Example 1, but the new key makes the measurements one tenth as large.

Check. Its largest entry is 4.1 cm4.1\,\mathrm{cm}, agreeing with the original decimal list rather than 41 cm41\,\mathrm{cm}.

Watch out

A diagram without a key leaves the place value uncertain. A zero leaf also matters: 2∣02\mid0 is an observation of 2020 or 2.02.0, according to the key; it is not a blank row.

Use a new decimal key to reconstruct the observations.

Ordering and reading the data

An ordered stem-and-leaf diagram has stems increasing down the page and leaves increasing across each row. This lets you retrieve the smallest, largest or any specified position without sorting the whole list again. Count leaves, not stems: a stem can hold several observations or none.

When constructing a diagram, first place every leaf on the correct row, then sort the leaves in that row. Keep a repeated digit each time it occurs. Finally count the leaves and read back a few entries, checking against the raw list.

Worked example 3: ordering an unsorted list

Question. Draw an ordered stem-and-leaf diagram of these scores and find the third smallest score:

26, 18, 31, 24, 18, 42, 30, 2726,\ 18,\ 31,\ 24,\ 18,\ 42,\ 30,\ 27

Plan. Separate tens and units, sort each row, then count from the smallest leaf.

StemLeaves
11888\quad8
224674\quad6\quad7
33010\quad1
4422

Key: 2∣62\mid6 represents a score of 2626.

Number of leaves=2+3+2+1=8check all observationsFirst three scores=18,18,24read in increasing order\begin{aligned} \text{Number of leaves}&=2+3+2+1=8 && \text{check all observations}\\ \text{First three scores}&=18,18,24 && \text{read in increasing order} \end{aligned}

Answer. The third smallest score is 2424.

Check. Exactly two scores in the original list are less than 2424, both equal to 1818. The repeated smallest score occupies two positions.

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