Constructing Histograms, Dispersion Graphs and Scatter Graphs

Choosing a graph for the enquiry

Geographers choose graphs according to the question they want to investigate. A histogram shows how frequently values occur within numerical groups. A dispersion graph keeps the individual values visible, showing how closely grouped or widely spread they are. Both can represent one variable, such as pebble size. A scatter graph has a different purpose: it investigates a relationship between two variables, such as traffic counts and noise levels.

Before drawing any of these graphs, identify the variable or variables, their units and the smallest and largest values. Choose scales that accommodate the data and make differences easy to read. Equal distances along a numerical axis must represent equal changes in value.

Histograms: grouping continuous measurements

Continuous data can take any value within a range. For example, a pebble's length might be 4 cm, 4.3 cm or 4.35 cm, depending on the precision of the measurement. A histogram groups these measurements into class intervals and shows the , or number of observations, in each interval.

For this technique, use class intervals of equal width. Suppose a sample of 20 pebbles has the following lengths, grouped into intervals 2 cm wide:

The boundaries prevent double counting: a pebble exactly 4 cm long belongs in the 4 to less than 6 cm interval, not the preceding one.

To construct the histogram, label the horizontal axis Pebble length (cm) and mark the boundaries 0, 2, 4, 6, 8 and 10 at equal spacings. Label the vertical axis Frequency and use a scale starting at zero that reaches at least 7. Draw each bar across its entire class interval, with its height equal to the frequency. The bars touch because the length scale is continuous; these are not separate categories.

The tallest bar identifies the modal class: 4 to less than 6 cm, containing seven pebbles. There are 12 pebbles between 4 cm and less than 8 cm, so this is where most of the sample is concentrated. However, the histogram does not reveal their exact individual lengths.

The overall shape also matters. An approximately bell-shaped distribution, balanced around its centre, suggests a normal distribution. An uneven distribution with a longer tail towards one end is skewed. Describe the whole pattern rather than judging its shape from the tallest bar alone.

Dispersion graphs: keeping individual values visible

A dispersion graph plots every observation along a numerical scale. It is useful when an enquiry needs to compare variation within samples, such as pebble sizes at different points on a beach.

Suppose a sample contains pebble sizes of 18, 19, 20, 22, 24, 25, 26, 28, 31, 33 and 34 mm. Draw a vertical axis labelled Pebble size (mm), with a scale covering all these values. Above a sample label, plot one small cross at the height of each measurement. All the crosses belong to the same sample column; horizontal position is not a second measured variable. If a value occurs more than once, place the marks beside one another at the same height so that no observations disappear.

The lowest point is 18 mm and the highest is 34 mm. Their difference gives the range:

$$\text{range}=\text{maximum}-\text{minimum}=34-18=16\text{ mm}$$

This means the sample spans 16 mm from its smallest to its largest pebble. The gaps and clusters also show how the values are distributed within that span.

Ordering the values helps identify the middle of the sample. Here, the sixth of the 11 values is the median, 25 mm. The lower and upper quartiles are 20 mm and 31 mm, giving an interquartile range of 11 mm. Unlike the full range, this describes the spread of the middle half of the data.

To compare samples, draw a separate column for each on the same vertical scale. A more tightly grouped column suggests more similar pebble sizes; a wider spread suggests greater variation. This can help investigate sediment sorting, although the graph alone does not explain what caused the differences.

Scatter graphs: plotting paired observations

A scatter graph needs two measurements for each observation. For example, an enquiry might record both distance from an urban centre and noise level at each site. Each point represents one site's pair of measurements.

Put the independent variable, used to investigate possible changes in the other variable, on the horizontal axis. Put the dependent variable, whose variation is being investigated, on the vertical axis. Here, distance goes horizontally and noise level vertically. In an enquiry comparing countries' gross national income, or GNI, with infant mortality, GNI would go horizontally and infant mortality vertically.

Suppose five sites have these distance and noise-level pairs: (0 m, 64 dB), (100 m, 57 dB), (200 m, 43 dB), (300 m, 37 dB) and (400 m, 24 dB). Choose a distance scale covering 0–400 m and a noise scale covering 0–70 dB. For the point (200, 43), locate 200 m horizontally, then 43 dB vertically, and mark their intersection. Repeat for every pair without joining the points.

Data for Plotting distance and noise level

Read the pattern from left to right:

• Positive correlation: as the horizontal variable increases, the vertical variable generally increases.
• Negative correlation: as the horizontal variable increases, the vertical variable generally decreases.
• No correlation: there is no clear overall relationship.

This example shows negative correlation: noise levels tend to be lower farther from the centre. A line of best fit represents that overall trend. Draw it through the middle of the point pattern, aiming for approximately equal numbers of points above and below it rather than connecting individual points.

Points close to the line suggest a strong relationship; widely scattered points suggest a weaker one. A scatter graph gives a visual judgement, not an exact numerical measure of strength. Its appearance also depends on the axis scales and proportions. Finally, an association does not by itself establish cause: explaining a geographical relationship requires evidence about the processes involved.

Choose the right graph

• Histogram: frequencies of continuous data grouped into equal-width class intervals.
• Dispersion graph: individual values and their spread within a sample.
• Scatter graph: relationships between two paired variables.