Bivariate Data and Evaluating Statistical Presentation

Seeing a relationship between two variables

Geographers often investigate whether two measurements are related. For example, a river investigation might compare channel width with channel depth. At each site, the two measurements form a pair: the width and depth must come from the same site. This is bivariate data, meaning data about two variables.

A scatter graph makes the relationship visible. Put one variable on the horizontal axis and the other on the vertical axis, label both with their units, and choose suitable numerical scales with equal intervals. Plot one point for each pair of measurements. Unlike a line graph showing a sequence, a scatter graph does not join successive observations: the purpose is to reveal the overall pattern between the variables.

The following graph demonstrates the method.

Data for Using river width to estimate river depth

Describing direction, strength and unusual results

A positive correlation means that as one variable increases, the other tends to increase. The points generally rise from bottom left to top right. In the graph, wider river channels tend to be deeper.

A negative correlation means that as one variable increases, the other tends to decrease. The points generally slope from top left to bottom right. With no clear correlation, the points show no consistent rising or falling pattern.

Direction is different from strength. A strong relationship has points clustered closely around a clear trend; a weak relationship has more scatter. A steep line does not necessarily mean a strong correlation: steepness describes how quickly one variable changes relative to the other, whereas strength concerns how closely the points follow the pattern.

A useful description of the graph is: ‘There is a strong positive correlation: as channel width increases from 2 m to 12 m, depth generally increases from about 0.35 m to 1.25 m.’ This gives the direction, strength, geographical meaning and supporting values.

An outlier lies well away from the main pattern. It may reflect a recording error or a genuine difference at a particular site. It should be recognised and investigated, rather than automatically treated as wrong.

Sketching a trend and drawing a line of best fit

A trend line summarises the general direction of the points. First look across the whole scatter graph: is there a clear rising or falling pattern, or does the pattern curve? Sketch through the centre of that pattern rather than tracing a route between individual points.

Where the pattern is approximately straight, use a ruler to draw an estimated line of best fit. Position it across the full observed range, with roughly as many points above it as below it. The distances between the points and the line should also be reasonably balanced. The aim is to represent the whole pattern, not simply connect the first and last points.

The line need not pass through every point, or through the origin. An isolated outlier should not pull it away from the main cluster, but this is different from deleting evidence because it does not support your conclusion. If there is no clear relationship, a sloping line would provide a poor basis for prediction.

Making predictions within and beyond the data

A line of best fit can estimate one variable from the other. To predict depth from width, find the chosen width on the horizontal axis, move vertically to the line, then horizontally to the depth axis. To estimate width from depth, reverse the process.

In the graph, a width of 7 m gives an estimated depth of about 0.8 m. The observed widths run from 2 m to 12 m, so this estimate lies within their range. This is interpolation. The estimate comes from the line, even though no observation was plotted at exactly 7 m.

The line has also been extended beyond the observations. At a width of 16 m, it suggests a depth of about 1.7 m. This is extrapolation, because 16 m lies beyond the observed width range. It assumes that the same relationship continues where no measurements are available. A wider channel might behave differently, so this prediction is less secure.

Even interpolation is an estimate, not a guaranteed result. Predictions are more convincing when there are enough observations, the relationship is strong and the chosen value is close to the measured data. On a real fieldwork graph, use its actual scales and measurements rather than the values in this illustration.

Why correlation does not prove a cause

A scatter graph shows an association, but it does not explain why it occurs. Two variables may both respond to a third factor. For example, sunglasses sales and ice-cream sales may rise together because hot, sunny weather increases demand for both. Buying sunglasses does not necessarily cause people to buy ice cream.

Likewise, a relationship between river width and depth does not by itself prove that greater width causes greater depth. Explaining a geographical process needs additional evidence, not just a matching statistical pattern.

Recognising selective or misleading presentation

Statistics can be accurate individually yet create a misleading overall impression. Evaluate what has been shown, what has been left out and how the display shapes the reader’s interpretation.

Axis scales can exaggerate differences. For example, in a bar chart comparing traffic counts of 100 and 110 vehicles, a vertical axis starting at 90 makes the second bar twice the displayed height of the first. The actual increase is only 10%. Bars are normally compared by their length from zero, so a cut-off baseline can distort that comparison. A non-zero axis on a scatter or line graph is not automatically wrong, but its scale must be clear and interpreted carefully.

Selected dates or sites can hide the wider pattern. Showing only two months when a value rose may conceal a longer decline. Similarly, showing only river sites that fit a proposed relationship could make the correlation appear stronger. Ask whether the omitted results would alter the conclusion.

The sample may not represent the population. Asking only people at a swimming pool whether they enjoy swimming would probably overstate enthusiasm among the wider public. A small sample may also be unusually influenced by a few responses. Consider both how many observations were collected and how they were selected; there is no single sample size that guarantees a fair result.

A chosen average can hide variation. A few very high incomes can pull the mean upwards while most people earn much less. Reporting only that mean may give an unrealistic impression of a typical income. Check which average is used and whether the spread or unusual values have been concealed.

Pictures can magnify change. Doubling both the height and width of an icon makes its area four times larger. If the value has only doubled, the image exaggerates the increase. Read the key and check what the size or number of symbols actually represents.

A strong evaluation connects the weakness to its effect: ‘Only the sites supporting the trend are shown, so the graph may overstate the strength of the relationship.’ This is more useful than simply calling the graph misleading.

Describe the relationship

• Positive: both variables tend to increase together.
• Negative: one tends to decrease as the other increases.
• No clear correlation: no consistent pattern.
• Strength: judge closeness to the trend, not steepness.
• Name both variables, support the pattern with values and identify outliers.

Draw and predict

• Keep scatter points separate.
• Sketch the overall trend; draw a straight estimated best-fit line where appropriate.
• Balance points and distances on either side across the observed range.
• Read from an axis to the line, then across to the other axis; include units.
• Interpolation: within the data range.
• Extrapolation: beyond the data range; less reliable because the trend may not continue.
• Correlation does not prove causation.

Evaluate the presentation

Check for distorted scales, selected dates or sites, small or biased samples, a misleading choice of average and exaggerated pictogram sizes.

Explain how the specific weakness changes the impression, and what missing information would help test the conclusion.