Interpreting and representing data
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给教育者： 面向 Interpreting and representing data（Mathematics，Statistics）的即用型课程幻灯片, 复习笔记——用在你的课堂上，或将该知识点作为学习者可实时游玩的互动课堂活动来运行。
课程笔记
Choosing the Right Chart
- Categorical data (e.g. favourite colour) → bar charts, pie charts, or pictograms.
- Ungrouped discrete numerical data (e.g. shoe size) → vertical line charts (also called bar-line graphs).
- Time series data (e.g. temperature each month) → line graphs or time series graphs.
- Always include a title, labels on axes, and a key for pictograms or multiple bars.
- For bar charts, use equal bar widths and a gap between bars; the height shows frequency.
- Pie charts show proportions of a whole, not frequencies – the total must be known.
Constructing Pie Charts
- Each category's angle = (frequency ÷ total) × 360°.
- Check that all angles add up to 360°.
- Draw sector angles accurately with a protractor. Starting at 12 o’clock and working clockwise is convenient, but other starting positions are valid.
- Label each sector with the category and percentage or angle.
- Pie charts are best when comparing parts to a whole with a small number of categories.
Favourite colours of 60 students
Bar Charts and Pictograms
- Bar charts have separate bars for each category; the height (or length) represents frequency.
- Bars must have equal width and can be vertical or horizontal.
- Pictograms use symbols to represent a set number of items (e.g. one symbol = 5 people).
- Always include a key showing what one symbol represents.
- When drawing pictograms, use half or quarter symbols if needed, but keep them consistent.
Vertical Line Charts
- Used for ungrouped discrete data (e.g. number of siblings).
- Draw a vertical line (or dot) above each value, with height equal to frequency.
- The x-axis shows the data values, the y-axis shows frequency.
- Use a numerical horizontal scale: vertical lines sit at the actual data values, so unequal value gaps have unequal spacing.
- This chart is useful for showing the shape of the distribution.
Time Series Graphs
- Plot data over time (e.g. months, years) on the x-axis.
- Join the points with straight lines to show trends.
- Look for trends (overall direction), seasonal patterns (regular fluctuations), and outliers.
- Use a suitable scale so the graph is not misleading.
- Time series graphs help compare changes over time for one or more variables.
Histograms (Higher Tier)
- Histograms are for continuous data, often with unequal class widths.
- The area of each bar represents frequency, not the height.
- Calculate frequency density = frequency ÷ class width.
- Plot frequency density on the y-axis and class boundaries on the x-axis.
- Bars are drawn touching (no gaps) because data is continuous.
- To find frequency from a bar: frequency = frequency density × class width.
Unequal-width classes
Cumulative Frequency Graphs (Higher Tier)
- Cumulative frequency is the running total of frequencies up to each upper class boundary.
- Plot cumulative frequency against the upper class boundary (not the midpoint).
- Join points with a smooth curve (or straight lines).
- To find the median, locate the value at ½ of the total frequency on the y-axis, then read across and down.
- To find the lower quartile, use ¼ of total frequency; for upper quartile, use ¾.
- The interquartile range = upper quartile − lower quartile.
Cumulative frequency
Box Plots (Higher Tier)
- A box plot shows the minimum, lower quartile, median, upper quartile, and maximum.
- Draw a box from lower to upper quartile, with a line at the median.
- Extend whiskers to the minimum and maximum values (unless outliers are shown).
- Box plots allow quick comparison of two or more distributions.
- The length of the box shows the interquartile range (middle 50% of data).
A five-number summary
Comparing Distributions
- Always compare both a measure of centre (mean, median, mode) and a measure of spread (range, interquartile range).
- For example: 'The median for group A is higher, but the interquartile range is larger, so group A has more variability.'
- When comparing, use the same type of chart for both sets of data.
- Be careful with misleading graphs: non-zero axes, incorrect scales, or omitted keys can distort comparisons.
Common Mistakes to Avoid
- In histograms, do not use frequency as the height when class widths are unequal – use frequency density.
- Plot cumulative frequency at the upper class boundary, not the midpoint.
- Do not read a pie chart as a frequency without knowing the total.
- Do not compare distributions by centre alone – always include spread.
- Always include labels, titles, and keys; check that axes start at zero unless there is a good reason not to.
幻灯片
练习题
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1.Which type of chart is most suitable for displaying categorical data?
Easy- ABar chart
- BHistogram
- CLine graph
- DCumulative frequency graph
2.A pictogram uses symbols to represent frequencies.
EasyTrue or false?
3.A pie chart is drawn for 90 students. The angle for the 'Football' category is 120°. How many students chose football?
Medium- A30
- B40
- C45
- D60
4.Which displays can show the frequency of each separate, ungrouped discrete value? Select all that apply.
Medium- AVertical line chart
- BBar chart
- CHistogram
- DPictogram
- ECumulative frequency graph
5.When drawing a pie chart, the angle for each category is calculated as (frequency / total) × 360°.
EasyTrue or false?
6.Match each graphical representation with the type of data it is most suitable for.
Medium- Line graph
- Histogram
- Pie chart
- Time series data
- Continuous data with unequal class widths
- Categorical data
7.A histogram has a 0–10 class of frequency density 5 and a 10–20 class of frequency density 3. What is the ratio of their frequencies?
Medium- A5:3
- B3:5
- C1:1
- D10:9
8.A box plot shows a median of 30, lower quartile 20, upper quartile 45, and whiskers extending to 10 and 60. Which statement is correct?
Hard- AThe interquartile range is 25.
- BThe range is 40.
- CThe median is 45.
- DThe lower quartile is 30.
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