Histograms and frequency polygons (Edexcel GCSE Statistics): Revision Notes
Histograms and frequency polygons
What is a histogram?
A histogram is a special type of graph used to display grouped continuous data. Unlike bar charts, histograms have some distinctive features that make them perfect for showing continuous information.
The most important characteristic of a histogram is that there are no gaps between the bars (unless one of the class intervals has zero frequency). This is because the data is continuous, meaning the values flow seamlessly from one group to the next.
Key features of histograms:
- Used for grouped continuous data
- Frequency is always plotted on the vertical axis
- No spaces between bars
- When class intervals are equal, all bars must have equal width
The continuous nature of the data is what distinguishes histograms from bar charts. In a bar chart, gaps between bars represent distinct categories, but in histograms, the lack of gaps shows that the data flows continuously from one interval to the next.
What is a frequency polygon?
A frequency polygon provides another way to visualise the same data shown in a histogram. It's created by joining the midpoints of the tops of histogram bars using straight lines. This creates a polygon shape that clearly shows the distribution pattern of your data.
Frequency polygons are particularly useful because they:
- Show the overall shape of a distribution clearly
- Allow easy comparison between different data sets
- Provide a smooth visual representation of the data pattern
Interpreting histograms
When reading a histogram, you can extract valuable information about your data:
Reading frequencies directly: Look at the height of each bar to find how many data points fall within that class interval.
Calculating cumulative frequencies: You can add up frequencies from different bars to find totals. For example, if you want to know how many people threw a rock up to 6 metres, you would add the frequencies for all bars representing distances from 0 to 6 metres.
Worked Example: Reading a Histogram
Looking at a histogram showing rock-throwing distances, you might see that 5 people threw between 4m and 5m, and 8 people threw between 5m and 6m. This means 13 people threw up to 6m in total.
Drawing histograms step-by-step
When constructing a histogram from data, follow these essential steps:
-
Set up your axes: Always put frequency on the vertical (y) axis and your continuous variable on the horizontal (x) axis.
-
Mark your scale: Ensure your scale can accommodate all your data values and frequencies.
-
Draw the bars: Each bar represents a class interval. The height corresponds to the frequency for that interval.
-
Use proper tools: Always use a ruler for straight edges and a sharp pencil for precision, especially in exams.
-
Check bar widths: For equal class intervals, all bars must be the same width with no gaps between them.
Remember that the precision of your histogram affects how easily others can read and interpret your data. Taking time to draw neat, accurate bars will make your work much clearer.
Drawing frequency polygons step-by-step
To create a frequency polygon from histogram data:
-
Start with your histogram: Either draw the histogram first or visualise where the bars would be.
-
Find the midpoints: Calculate the midpoint of each class interval. For example, for the interval 110 ≤ h < 120, the midpoint is 115.
-
Plot the points: At each midpoint position on the x-axis, plot a point at the height of the corresponding frequency.
-
Connect with straight lines: Join all the plotted points using straight lines to form your polygon.
-
Complete the shape: The polygon should start and end at the x-axis to create a closed shape.
Worked example: Children's heights
Worked Example: Creating Histogram and Frequency Polygon
Let's work through a complete example using data about children's heights in a club:
Given data:
- 110 ≤ h < 120: 8 children
- 120 ≤ h < 130: 13 children
- 130 ≤ h < 140: 16 children
- 140 ≤ h < 150: 10 children
- 150 ≤ h < 160: 7 children
Drawing the histogram:
- Set up axes with Height (cm) on x-axis and Frequency on y-axis
- Mark intervals: 110, 120, 130, 140, 150, 160 on x-axis
- Mark frequency scale: 0, 5, 10, 15, 20 on y-axis
- Draw bars with heights 8, 13, 16, 10, 7 respectively
- Ensure no gaps between bars
Drawing the frequency polygon:
- Calculate midpoints: 115, 125, 135, 145, 155
- Plot points at (115, 8), (125, 13), (135, 16), (145, 10), (155, 7)
- Connect these points with straight lines
- The resulting shape shows the distribution pattern clearly
Common exam tips and traps
Key exam tips:
- Always use a ruler and sharp pencil for neat, accurate graphs
- Check that frequency is on the y-axis
- Ensure no gaps between histogram bars
- For frequency polygons, join midpoints with straight lines only
Common traps to avoid:
- Don't leave gaps between histogram bars (this is the most common mistake)
- Don't confuse histograms with bar charts - histograms are for continuous data
- Remember that equal class intervals require equal bar widths
- Don't forget to join points with straight lines for frequency polygons
Remember!
Key Points to Remember:
- Histograms display grouped continuous data with no gaps between bars
- Frequency polygons are created by joining the midpoints of histogram bar tops
- Always put frequency on the vertical axis
- Use proper drawing tools (ruler and sharp pencil) in exams
- Equal class intervals mean equal bar widths in histograms