Displaying Data
A picture is worth a thousand numbers. The same data set can tell completely different stories depending on how you display it. A poorly chosen graph can hide patterns, exaggerate differences, or mislead viewers. A well-chosen display reveals structure, shows relationships, and communicates findings clearly.
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
Twenty students took a test. Their marks were: 45, 52, 55, 58, 60, 62, 65, 65, 68, 70, 72, 75, 78, 80, 82, 85, 88, 90, 92, 95.
Before reading on how would you group these to show the distribution clearly? Which type of display would you use?
Three core displays are tested in HSC Maths Standard. Each serves a different purpose and reveals different aspects of a distribution.
Frequency table: organises raw data into classes with a count for each group. Use equal class widths when comparing frequency counts across groups; if widths differ, use frequency density for a histogram.
Histogram: a display for grouped continuous data with adjacent bars; use frequency density for unequal class widths so bar area represents frequency.
Stem-and-leaf plot: shows all individual values while revealing the shape of the distribution. Back-to-back version compares two groups on the same stem.
Key facts
- Types of data displays
- When to use each display
- Class width and class boundaries
Concepts
- How display choice affects perception
- Why bars touch in histograms
- How back-to-back plots compare groups
Skills
- Create frequency tables with equal class widths
- Draw histograms and stem-and-leaf plots
- Describe distribution shape and critique displays
A frequency table groups data into classes and counts how many values fall in each class.
Example: Test marks for 30 students, grouped into unequal-width bands
| Mark range | Frequency | Cumulative frequency |
|---|---|---|
| 0–49 | 2 | 2 |
| 50–59 | 4 | 6 |
| 60–69 | 8 | 14 |
| 70–79 | 9 | 23 |
| 80–89 | 5 | 28 |
| 90–100 | 2 | 30 |
Frequency tables group data into classes. Class width = upper − lower boundary. Relative frequency = frequency ÷ total. Cumulative frequency adds up as you go down. Use midpoints to calculate mean from a frequency table.
Pause, copy the frequency table structure (class, frequency, relative frequency = freq ÷ total, cumulative frequency) and the midpoint method for estimating the mean from grouped data (use class midpoint as the representative value) into your book.
Quick check: A frequency table has classes 20–29, 30–39, 40–49, 50–59. The class width is:
We just saw frequency tables grouping data into classes with class width, relative frequency, and cumulative frequency columns. That raises a question: a table shows the numbers, but how do you show the overall shape of the distribution visually? This card answers it → a histogram plots class boundaries on the x-axis and frequency (or frequency density for unequal-width classes) on the y-axis, with adjacent bars; it can reveal whether the data is symmetric, positively skewed, or negatively skewed.
A histogram is a bar graph for grouped numerical data where:
- Each bar represents one class (group)
- With equal-width classes, bar height can show frequency (or relative frequency); with unequal-width classes, use frequency density so bar area represents frequency.
- Bars touch each other, no gaps, because the data is continuous
- Equal-width classes allow direct frequency heights; unequal-width classes need frequency density as the height.
Key features to identify when reading a histogram:
- Symmetry: Are both halves roughly the same shape?
- Skew: Positive skew = long tail to the right; negative skew = long tail to the left.
- Centre: Where do most values cluster (mode class)?
- Spread: How wide is the distribution across the x-axis?
- Modality: Unimodal (one peak) or bimodal (two peaks)?
- Outliers: Isolated bars far from the main body?
Histograms show grouped numerical data: x-axis = continuous scale (class boundaries), bars touch (no gaps). With equal class widths, y-axis may show frequency; with unequal widths, use frequency density so bar area represents frequency. Shape reveals distribution: symmetric, positively skewed (longer right tail), or negatively skewed (longer left tail).
Pause, copy the histogram rules (x-axis = continuous class boundaries; adjacent bars; frequency heights for equal-width classes or frequency-density heights for unequal-width classes) and the three shape types (symmetric: roughly mirror image; positively skewed: longer right tail; negatively skewed: longer left tail) into your book.
True or false: In a histogram, the bars have gaps between them to distinguish each class.
We just saw histograms revealing distribution shape, but grouping data into classes loses the individual values. That raises a question: is there a display that shows the shape of the distribution while keeping every individual value visible and readable? This card answers it → a stem-and-leaf plot preserves every value (stem = leading digit, leaf = final digit), lets you read the median and mode directly, and can compare two data sets back-to-back.
A stem-and-leaf plot shows all data values while displaying the distribution's shape. Each value is split into a stem (leading digit/s) and a leaf (last digit).
Example: Ages: 21, 23, 25, 28, 31, 32, 35, 38, 41, 42
2 | 1 3 5 8
3 | 1 2 5 8
4 | 1 2
Key: 2 | 1 = 21
Advantages:
- Preserves all raw data values (unlike a histogram)
- Median and quartiles can be read directly
- Distribution shape is immediately visible
Back-to-back stem-and-leaf: Two groups share the same stem, with leaves extending in opposite directions. Used to compare distributions side by side.
Stem-and-leaf plot: stem = leading digit(s), leaf = final digit. Back-to-back plots compare two data sets sharing one stem column. Every individual value is preserved. Sorted leaves let you read median and quartiles directly.
Pause, copy the stem-and-leaf construction rules (stem = leading digit, leaves sorted in order, each leaf is one data point) and the back-to-back format (two data sets sharing one stem column, leaves going outward in opposite directions) into your book.
Fill the gap: A stem-and-leaf plot for 23, 25, 28, 31, 32, 35 would have stems and . The leaves for stem 3 are: .
Worked examples · reveal each step
Heights (cm) of 20 students: 152, 155, 158, 160, 162, 163, 165, 166, 168, 170, 171, 172, 175, 178, 180, 182, 185, 188, 190, 195. Create a frequency table (class width 10) and describe the distribution.
Create a stem-and-leaf plot for: 23, 25, 28, 31, 32, 32, 35, 38, 41, 42, 45, 48
Common errors · the 3 traps that cost marks
Match each display to its best use case:
- Histogram
- Stem-and-leaf plot
- Frequency table
- Back-to-back stem-and-leaf
- Count how many values fall in each class
- Compare two distributions side by side
- Show shape of a large continuous data set
- Retain all raw values while showing shape
Quick-fire practice · 2 activities
Create a frequency table (class width 5) for: 12, 15, 18, 22, 25, 28, 30, 32, 35, 38, 40, 42, 45. Draw a stem-and-leaf plot for: 56, 58, 62, 65, 68, 70, 72, 75, 78, 80, 82, 85, 88, 90, 92.
A histogram uses class widths of 1, 5 and 10 in different sections. Explain why this is problematic. When would a stem-and-leaf plot be better than a histogram?
Top 3 list: Name THREE ways a graph could be drawn to deliberately mislead the reader. For each, describe the technique and explain what it makes the data appear to show.
For the data in this activity, class-width-10 frequencies are 40–49 (1), 50–59 (3), 60–69 (5), 70–79 (4), 80–89 (4), 90–99 (3), for a total of 20. The distribution is unimodal, peaking in 60–69; a histogram with equal-width classes can show this shape. A stem-and-leaf plot would preserve all 20 individual values while showing the same shape, and lets you read the median and quartiles directly, an advantage for HSC calculation questions.
What has changed in your understanding? What did you get right? What surprised you?
Pick your answer, then rate your confidence. That tells the system what to drill next.
Q1. Which feature distinguishes a histogram from a bar chart?
Q2. A frequency table uses classes 30–39, 40–49, 50–59. How many of these values belong to the class 40–49? Values: 35, 42, 45, 48, 51, 39, 44.
Q3. A distribution has mean = 75, median = 70, mode = 65. This distribution is most likely:
Q4. A histogram uses classes 0–30, 30–50, 50–60, 60–70, 70–100 and uses frequency as bar height. What is wrong with this display?
Q5. Which display is best for comparing the test scores of two classes while retaining all individual values?
SA 1. Create a frequency table with class width 5 for: 22, 24, 28, 31, 33, 35, 36, 38, 40, 42, 45, 48, 50, 52, 55. Then draw a stem-and-leaf plot for the same data and describe the distribution shape. (2 marks)
SA 2. A histogram of exam marks uses intervals [0, 30), [30, 50), [50, 60), [60, 70), [70, 100] and uses raw frequency as bar height. Explain the problem this causes and how the display could be corrected. Suggest equal class widths if using frequency as height. (2 marks)
SA 3. (a) A newspaper presents the same unemployment data in two ways: Graph A uses a y-axis from 0–10% with monthly data; Graph B uses a y-axis from 4–6% with thick bars. How does each shape the reader's perception? (b) Discuss the ethical responsibility of data presenters when choosing graphical displays, using specific examples of how misleading displays can affect public opinion or policy. (3 marks)
Comprehensive answers (click to reveal)
MC 1, C: Histograms represent continuous data; bars are adjacent with no gaps. Bar charts for categorical data have gaps.
MC 2, B: Values in 40–49: 42, 45, 48, 44 = 4 values.
MC 3, A: When mean > median > mode, the distribution is right (positively) skewed, a long tail pulls the mean up.
MC 4, D: The class widths are 30, 20, 10, 10 and 30. With raw frequency as the height, bar areas over-represent wider classes; use frequency density so each area represents frequency, or use equal-width classes with frequency heights.
MC 5, B: Back-to-back stem-and-leaf retains all individual values and displays both distributions on the same scale for direct comparison.
SA 1 (2 marks): Freq. table: 20–24: 2, 25–29: 1, 30–34: 2, 35–39: 3, 40–44: 2, 45–49: 2, 50–54: 2, 55–59: 1 [0.5]. Stem-and-leaf: 2|2 4 8; 3|1 3 5 6 8; 4|0 2 5 8; 5|0 2 5; Key: 2|2 = 22 [0.5]. Description: roughly symmetric, slight right skew, unimodal around 35–40 [1].
SA 2 (2 marks): Raw frequency heights with unequal-width classes make bar areas misrepresent the counts [1]. Use frequency density as bar height, or keep frequency heights and use equal-width classes, e.g., [0,20), [20,40), [40,60), [60,80), [80,100] [1]. The given half-open intervals assign boundary values unambiguously.
SA 3 (3 marks): (a) Graph A (0–10%): shows full context; changes appear small, reader perceives stability. Graph B (4–6%): zooms in; tiny changes look dramatic, reader perceives crisis or volatility [1]. (b) Data presenters have an ethical duty not to mislead through truncated axes, unequal scales or cherry-picked time frames. Examples: climate data with selective date ranges to deny trends; COVID graphs using log vs linear scale creating different impressions of growth; political polling presented without sampling methodology. These manipulations affect public health compliance, policy support and voting behaviour [2].
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