L: Data presentation and interpretation: Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency. Connect to probability distributions.

The idea

Single-variable data can be displayed in diagrams such as histograms, bar charts, box plots and cumulative frequency graphs. In a histogram, bars are drawn so that area, not height, represents frequency; this matters when class widths are unequal. Frequency density is calculated as frequency divided by class width, and the area of each bar equals frequency. Interpreting these diagrams involves reading frequencies, identifying modal classes and estimating probabilities. Connecting to probability distributions means recognising that relative frequencies from data can estimate probabilities, and that the shape of a histogram can suggest an underlying distribution. For example, a histogram of waiting times with unequal classes: a taller bar does not necessarily mean a higher frequency unless width is considered. Always check class width before comparing bars.

What earns the marks

• Identifies that in a histogram the area of each bar represents the frequency of the class, not the height alone.
• Calculates frequency density using frequency divided by class width, and uses it to draw or interpret histogram bars correctly.
• Reads frequencies from a histogram by multiplying frequency density by class width for each bar.
• Interprets other single-variable diagrams such as bar charts, box plots or cumulative frequency graphs, extracting relevant values such as median, quartiles or frequencies.
• Connects relative frequency from a diagram to probability, for example estimating the probability that a randomly chosen item falls in a given class.
• Recognises that the overall shape of a histogram or other diagram can suggest features of a probability distribution, such as skewness or the location of the mode.

Common mistakes and how to put them right

• Assuming the height of a histogram bar is the frequency.

  Put it right: Frequency is represented by area; height is frequency density, so frequency = frequency density × class width.

• Comparing histogram bars of different widths by height alone.

  Put it right: Compare areas, or calculate frequencies, because a wider class with the same height contains more data.

• Treating a bar chart and a histogram as interchangeable.

  Put it right: Bar charts are for discrete categories with gaps and height represents frequency; histograms are for continuous data with no gaps and area represents frequency.

Exam tips

• Before answering a histogram question, check whether the vertical axis is labelled frequency or frequency density; if it is frequency density, use area to find frequencies.
• When estimating a probability from a diagram, write the relative frequency as a fraction or decimal and relate it to the total frequency, not to the number of bars.
• If asked to connect a diagram to a probability distribution, comment on the shape, centre and spread, and explain how relative frequencies approximate probabilities for large samples.