Strand 1 — Statistics & Probability

Representing Data: Stem-and-Leaf Plots and Histograms

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to construct a stem-and-leaf plot from a given set of data.
✓By the end of this lesson students will be able to interpret information presented in a stem-and-leaf plot, including back-to-back plots.
✓By the end of this lesson students will be able to construct a back-to-back stem-and-leaf plot to compare two data sets.
✓By the end of this lesson students will be able to construct a histogram from a frequency table with equal class intervals.
✓By the end of this lesson students will be able to interpret information presented in a histogram.

Key concepts

Stem-and-Leaf Plot

A stem-and-leaf plot is a way of organising numerical data to show its shape and distribution. Each data value is split into a 'stem' (the leading digit(s)) and a 'leaf' (the trailing digit). The leaves are ordered numerically from smallest to largest for each stem. A key must always be included to explain how to read the plot.

Back-to-Back Stem-and-Leaf Plot

A back-to-back stem-and-leaf plot is used to compare two related sets of data. It shares a common stem in the centre, with one set of leaves extending to the left and the other set of leaves extending to the right. The leaves on both sides are ordered numerically, usually from the stem outwards. A key for both sides is essential.

Histogram

A histogram is a graphical representation of the distribution of numerical data. It is similar to a bar chart but is used for continuous data, where the bars represent class intervals and are drawn without gaps (unless a class interval has zero frequency). For Junior Cycle, histograms typically use equal class intervals, and the height of each bar represents the frequency of data within that interval. The horizontal axis represents the data values (class intervals), and the vertical axis represents the frequency.

Key facts to remember

1A stem-and-leaf plot displays data in a way that preserves the original data values.
2Every stem-and-leaf plot must have a clear key to indicate how to read the data values.
3Leaves in a stem-and-leaf plot should always be ordered numerically from smallest to largest for each stem.
4Back-to-back stem-and-leaf plots are excellent for comparing the distributions of two related data sets.
5Histograms are used for continuous numerical data, showing the frequency distribution within class intervals.
6In a histogram, the bars must touch each other (unless a class interval has zero frequency), unlike a bar chart.
7For Junior Cycle, the height of a bar in a histogram typically represents the frequency of the class interval.
8The horizontal axis of a histogram represents the data values (class intervals), and the vertical axis represents the frequency.

Worked examples

Example 1

The following data shows the number of minutes 15 students spent on their maths homework last night: 25, 32, 18, 40, 28, 35, 22, 30, 15, 38, 25, 42, 30, 20, 33. Construct a stem-and-leaf plot for this data.

I1. Identify the smallest and largest values to determine the stems. The smallest value is 15, and the largest is 42. So, the stems will be 1, 2, 3, and 4.

II2. Draw two columns, one for the stem and one for the leaves.

III3. Go through each data value and place its leaf (the units digit) next to its stem (the tens digit).

IV Stem | Leaves

V -----|-------

VI 1 | 8 5

VII 2 | 5 8 2 5 0

VIII 3 | 2 5 0 8 0 3

9 4 | 0 2

104. Order the leaves numerically for each stem.

11 Stem | Leaves

12 -----|-------

13 1 | 5 8

14 2 | 0 2 5 5 8

15 3 | 0 0 2 3 5 8

16 4 | 0 2

175. Add a key to explain how to read the plot.

Answer

Stem | Leaves -----|------- 1 | 5 8 2 | 0 2 5 5 8 3 | 0 0 2 3 5 8 4 | 0 2 Key: 1 | 5 = 15 minutes

Always remember to include a key with your stem-and-leaf plot. This is crucial for interpreting the data correctly.

Example 2

A class of students took a science test. The scores for the boys and girls are given below. Boys' scores: 65, 72, 58, 80, 75, 62, 68, 70, 55, 78 Girls' scores: 60, 70, 85, 68, 72, 75, 63, 82, 70, 65 Construct a back-to-back stem-and-leaf plot to compare their scores.

I1. Identify the smallest and largest scores for both groups to determine the common stems. The smallest score is 55 (boys) and the largest is 85 (girls). So, the stems will be 5, 6, 7, and 8.

II2. Draw a central stem column with two leaf columns, one for boys (left) and one for girls (right).

III3. For each boy's score, place the leaf (units digit) to the left of the stem (tens digit). For each girl's score, place the leaf to the right of the stem.

IV Boys' Scores | Stem | Girls' Scores

V -------------|------|--------------

VI | 5 | 0

VII 8 5 | 6 | 0 8 3 5

VIII 2 5 0 8 | 7 | 0 2 5 0

9 0 | 8 | 5 2

104. Order the leaves numerically, from the stem outwards, for both sides.

11 Boys' Scores | Stem | Girls' Scores

12 -------------|------|--------------

13 8 5 | 5 | 0

14 8 5 2 | 6 | 0 3 5 8

15 8 5 2 0 | 7 | 0 0 2 5

16 0 | 8 | 2 5

175. Add a key for both sides.

Answer

Boys' Scores | Stem | Girls' Scores -------------|------|-------------- 8 5 | 5 | 0 8 5 2 | 6 | 0 3 5 8 8 5 2 0 | 7 | 0 0 2 5 0 | 8 | 2 5 Key: 5 | 0 = 50 (Girls), 5 | 8 = 58 (Boys)

When ordering leaves on the left side of a back-to-back plot, they should increase in value as you move away from the stem (e.g., 8 5 means 58, not 85).

Example 3

The table below shows the heights (in cm) of 30 students in a class. Height (cm) | Frequency ------------|---------- 140 - 149 | 4 150 - 159 | 10 160 - 169 | 12 170 - 179 | 4 Construct a histogram to represent this data.

I1. Determine the class intervals and frequencies from the table.

II2. Draw and label the horizontal axis (x-axis) for 'Height (cm)' and the vertical axis (y-axis) for 'Frequency'.

III3. Mark the boundaries of the class intervals on the horizontal axis. Since the data is continuous, the bars should touch. To achieve this, we adjust the class boundaries to be continuous. For 140-149, the true boundaries are 139.5 to 149.5. For Junior Cycle, it's common to use the given intervals as labels and draw bars touching.

IV4. For each class interval, draw a bar whose height corresponds to its frequency. Ensure the bars are of equal width.

V5. Title the histogram.

Answer

A histogram with: - Horizontal axis labelled 'Height (cm)' with intervals 140-149, 150-159, 160-169, 170-179. - Vertical axis labelled 'Frequency' scaled from 0 to at least 12. - Bar for 140-149 with height 4. - Bar for 150-159 with height 10. - Bar for 160-169 with height 12. - Bar for 170-179 with height 4. - All bars touching each other. - Title: 'Heights of Students'.

Remember that for a histogram, the bars must touch each other, as the data is continuous. This is a key difference from a bar chart.

Common mistakes

✗Forgetting to include a key for stem-and-leaf plots, making them unreadable.
✗Not ordering the leaves numerically in a stem-and-leaf plot.
✗Drawing gaps between bars in a histogram, confusing it with a bar chart.
✗Incorrectly labelling axes or forgetting to label them entirely on histograms.
✗Misinterpreting the ordering of leaves on the left side of a back-to-back stem-and-leaf plot (e.g., reading '8 5 | 6' as 85 instead of 58).

Exam tips

★Always use a ruler and pencil to draw your plots neatly and accurately in exams.
★Double-check that your key is clear and correct for all stem-and-leaf plots, especially back-to-back ones.
★Ensure all axes on your histograms are clearly labelled with units where appropriate, and the bars touch.
★Read the question carefully to determine if you need to construct a plot or interpret an existing one, and what specific information is being asked for.

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