Statistics: construct and interpret diagrams for grouped discrete data and continuous data, ie histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use (Higher tier only)

The idea

Grouped data are summarised in classes. Discrete data take exact values such as counts, while continuous data are measured on a scale such as height. For grouped discrete data, use suitable class boundaries when constructing the diagram so adjacent classes are represented correctly. A histogram represents grouped continuous data with bars whose area is proportional to frequency, so frequency density equals frequency divided by class width. With equal class intervals the bars have equal width and height can represent frequency, but with unequal intervals you must use frequency density so area stays correct. A cumulative frequency graph plots the running total against the upper class boundary, and from it you estimate the median, quartiles and interquartile range. Use histograms to show distribution shape and cumulative frequency graphs to estimate positional statistics.

What earns the marks

• Distinguish grouped discrete data from continuous data and select a suitable diagram for each.
• Construct a histogram for continuous data by calculating frequency density as frequency divided by class width.
• Draw histogram bars with correct widths and heights so that the area of each bar represents its class frequency.
• Construct a cumulative frequency graph by plotting cumulative frequency against the upper class boundary and joining points with a suitable curve or polygon.
• Interpret a histogram by comparing areas or frequency densities to describe the distribution.
• Interpret a cumulative frequency graph to estimate the median, lower and upper quartiles and the interquartile range.
• Explain when a histogram is more appropriate than a cumulative frequency graph and vice versa.

Common mistakes and how to put them right

• Using frequency as the bar height when class intervals are unequal.

  Put it right: Use frequency density, calculated as frequency divided by class width, so that bar area equals frequency.

• Plotting cumulative frequency against the midpoint of each class instead of the upper class boundary.

  Put it right: Cumulative frequency is plotted at the upper class boundary of each class.

• Assuming a single joining style is mandatory for a cumulative frequency graph.

  Put it right: Plot cumulative frequencies at the correct class boundaries and use the board-accepted curve or polygon style consistently.

• Reading the median from the vertical axis instead of the horizontal axis.

  Put it right: Find half the total frequency on the vertical axis, move across to the curve, then read the value on the horizontal axis.

Exam tips

• Always add a frequency density column to the table before drawing a histogram with unequal class intervals.
• Label the vertical axis of a histogram as frequency density and include the correct units where given.
• When estimating from a cumulative frequency graph, show your construction lines so the method is clear.