Construct and interpret diagrams for grouped discrete data and continuous…

The idea

A histogram shows frequency by area, not height. For unequal class intervals the vertical axis must be frequency density, calculated as frequency ÷ class width. The area of each bar (width × frequency density) then equals the frequency. For grouped discrete data, e.g. ages 1–10, 11–20, continuous class boundaries are needed so bars touch; use halfway points such as 0.5–10.5, 10.5–20.5. For continuous data grouped as 0 ≤ x < 10, 10 ≤ x < 20, the boundaries are simply 0, 10, 20. A cumulative frequency graph plots a running total of frequencies against the upper class boundary of each interval, and the points are joined by a smooth curve. This graph is used to estimate the median (at the n/2 value), quartiles (n/4 and 3n/4) and other percentiles. Use a histogram to see the shape of a distribution; use a cumulative frequency graph to estimate medians and quartiles.

What earns the marks

• Calculate at least one correct frequency density where the method (frequency ÷ class width) is clear.
• Draw histogram bars with correct widths corresponding to the continuous class boundaries, with no gaps between them.
• Draw bar heights corresponding to the calculated frequency densities.
• Plot cumulative frequency points accurately against the upper class boundary for each interval.
• Join the plotted cumulative frequency points with a smooth curve, starting from the lowest boundary on the horizontal axis.
• Clearly show reading lines on a cumulative frequency graph when estimating the median, quartiles or percentiles.
• Choose an appropriate diagram for a given purpose, e.g. a histogram to compare frequencies across classes or a cumulative frequency graph to estimate the median.

Common mistakes and how to put them right

• Plotting frequency on the vertical axis of a histogram when class widths are unequal.

  Put it right: The vertical axis must be frequency density (frequency ÷ class width).

• Plotting cumulative frequency against class midpoints instead of upper class boundaries.

  Put it right: Cumulative frequency is a running total up to the end of an interval, so it must be plotted at the upper boundary.

• For grouped discrete data (e.g. 1–10, 11–20), drawing bars with gaps between them.

  Put it right: A histogram's bars must touch; use continuous class boundaries halfway between the discrete intervals, e.g. 0.5–10.5, 10.5–20.5.

• Joining cumulative frequency points with straight line segments.

  Put it right: Join them with a smooth curve, as required for Edexcel GCSE.

Exam tips

• Before drawing a histogram, add two columns to the table: one for class width and one for frequency density. This organises your calculations.
• Always draw the lines on the cumulative frequency graph that you used to find an estimate. This allows you to be awarded method marks even if your reading is slightly inaccurate.