In simple terms
A friendly intro before the formal notes — no formulas yet.
Decoding Data Displays
We turn long lists of numbers into clear pictures like histograms and box plots. These visuals help us quickly grasp the data's story by showing its centre, spread, and shape.
Imagine you've just bought a massive box of assorted chocolates. To understand what you have, you wouldn't just stare at the pile. You'd group them by type (caramel, nut, dark, milk), count how many of each you have, and maybe find the 'average' chocolate. Representing data is the same: we group, count, and summarise numbers to make sense of them, finding the average value (mean) or the middle value (median).
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First, group raw data into classes. For a histogram, calculate frequency density (frequency divided by class width) to determine the bar height, ensuring area represents frequency.
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To estimate the mean from grouped data, use the midpoint of each class as a representative value. The formula is the sum of (midpoint × frequency) divided by the total frequency.
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Plot a cumulative frequency curve (ogive) to find the median. Locate the position of the median at n/2 on the vertical axis, then read across to the curve and down to the horizontal axis.
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Finally, measure the data's spread. Calculate the interquartile range (IQR) from the ogive or use your calculator for standard deviation to compare how consistent or varied different datasets are.
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Key formulas
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$Frequency Density = \frac{\text{Frequency}}{\text{Class Width}}$
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Full topic notes
Formal explanation with the rigour you need for the exam.
Stem-and-Leaf Diagrams
A stem-and-leaf diagram is a simple but effective way to display quantitative data while retaining the original data values. It separates each data point into a 'stem' (the leading digit or digits) and a 'leaf' (the final digit). The stems are listed vertically, and the leaves are listed horizontally next to their corresponding stem. It's crucial to include a key to explain how to read the diagram. For comparing two datasets, a back-to-back stem-and-leaf diagram can be used, with a central stem and leaves for each dataset branching out on either side.
Shows the shape of the distribution (e.g., symmetry, skewness).
Retains all original data values.
Data is ordered, which makes finding the median and quartiles straightforward.
Always include a key (e.g., 5 | 2 represents 52).
Histograms for Grouped Continuous Data
When data is continuous and has been grouped into class intervals, a histogram is the appropriate diagram. Unlike a bar chart, there are no gaps between the bars. A crucial feature of a histogram is that the area of each bar is proportional to the frequency of that class. If the class widths are all equal, the heights of the bars are proportional to the frequencies. However, if the class widths are unequal, you must calculate and plot frequency density on the vertical axis.
$Frequency Density = \frac{\text{Frequency}}{\text{Class Width}}$
To estimate the mean from a grouped frequency table, we assume that all the data values within a class are evenly distributed and can be represented by the mid-point of that class. We then calculate a weighted average.
Cumulative Frequency, Quartiles, and Box Plots
A cumulative frequency graph, or ogive, is a running total of the frequencies. It is plotted using the upper class boundary of each interval against the cumulative frequency. These graphs are extremely useful for estimating values that relate to position in the dataset, most notably the median (), the lower quartile (), and the upper quartile (). The median is the value of the middle data point, while the quartiles divide the data into four equal parts.
The difference between the upper and lower quartiles is the Interquartile Range (IQR), a robust measure of spread that is not affected by extreme outliers. A box-and-whisker plot provides a concise visual summary of the data using five key values: the minimum value, , the median, , and the maximum value. They are excellent for comparing the distributions of two or more datasets side-by-side.
Worked examples
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The times, minutes, taken by 80 people to complete a crossword puzzle are summarised in the table below.
| Time (t mins) | Frequency |
|---|---|
| 10 ≤ t < 15 | 10 |
| --- | --- |
| 15 ≤ t < 25 | 30 |
| 25 ≤ t < 35 | 25 |
| 35 ≤ t < 50 | 15 |
(i) Draw a histogram to represent this information. (ii) Calculate an estimate for the mean time taken.
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(i) First, we need to calculate the class widths and frequency densities for the unequal class intervals.
The cumulative frequency graph shows the heights, in cm, of 120 plants.
(A cumulative frequency graph is shown, starting at (0,0), passing through (10, 20), (20, 70), (30, 105), (40, 120). The x-axis is 'Height (cm)' and the y-axis is 'Cumulative Frequency'.)
(i) Use the graph to estimate the median height. (ii) Estimate the interquartile range of the heights. (iii) It was found that the shortest plant was 5 cm and the tallest was 38 cm. Draw a box-and-whisker plot to represent the data.
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(i) The total number of plants is . The median is the value of the th plant, which is the 60th plant. Reading from 60 on the cumulative frequency axis across to the curve and down to the height axis gives an estimated median. Median () ≈ 18 cm.
How it all connects
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Glossary
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Quick check
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Revision flashcards
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What is a key difference between a histogram and a bar chart?
A histogram is for continuous data with no gaps between bars, and the area of each bar is proportional to frequency. A bar chart is for discrete or categorical data and has gaps between bars.
Key takeaways
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Shows the shape of the distribution (e.g., symmetry, skewness).
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Retains all original data values.
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Data is ordered, which makes finding the median and quartiles straightforward.
- ✓
Always include a key (e.g., 5 | 2 represents 52).
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Practice Questions: Representation of Data
Practice Questions: Representation of Data
Extra simulations & links
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Checkpoint
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