Interpreting Bivariate Data and Drawing Conclusions

Reading a table as evidence

Numerical data help geographers answer questions about patterns and relationships. Before interpreting a table, read its title, headings and units. Establish what each row represents: a place, a date, a person or another observation. Then compare values that measure the same thing in the same units.

The following constructed practice data represent the distances and elapsed times of six separate fictional walks. They are not fieldwork observations from real routes.

Each row contains a pair: the 2 km walk, for example, took 21 minutes. Distance ranges from 1 to 6 km, while time ranges from 13 to 60 minutes. The overall pattern is that longer walks take more time, although the 4 km entry does not follow the pattern closely. A useful interpretation includes both the general pattern and any exception, rather than listing every number. Tables give exact values, but a graph can make their overall relationship easier to see.

Seeing relationships in bivariate data

These are bivariate data because each observation contains two variables. A variable is a quantity that can take different values, such as distance or journey time. A scatter plot makes their relationship easier to see: distance goes on the horizontal axis and time on the vertical axis. Each row becomes one point, such as $(2,21)$. The points are not joined in sequence.

Data for Distance and time: interpreting a constructed scatter plot

A pattern rising from left to right shows positive correlation: as one variable increases, the other tends to increase. A falling pattern shows negative correlation: as one increases, the other tends to decrease. If there is no clear rising or falling pattern, there is no clear correlation.

Direction and strength are different. Strength concerns how closely the points follow a common trend, not how steep that trend is. Points tightly grouped around a trend show a stronger relationship than widely scattered points. Here, most points follow a fairly close upward pattern, but the 4 km, 55 minute point lies noticeably above it.

Correlation describes an association; it does not alone establish its cause. Other factors may affect the observations. For these fictional walks, the graph cannot tell us whether the unusual time resulted from a pause, a recording mistake or something else.

Sketching a trend and drawing a best-fit line

A trend line summarises the overall direction of the cloud of points. Where the pattern is approximately straight, an estimated line of best fit can be drawn by eye using a ruler.

Position the line through the middle of the main pattern and extend it across the data range. Aim for roughly balanced numbers of points above and below it, with their distances from the line also reasonably balanced. Moving a ruler across the graph helps you judge this balance. The line represents the whole relationship, rather than a connection between selected observations.

The line shown is one reasonable estimate for the practice data. The unusual 4 km point should not pull it away from the main pattern. Keep that point visible and discuss it: recognising an outlier is not evidence that the observation is wrong. Different sensible estimated lines can give slightly different predictions.

Predicting within and beyond the data

To estimate time from distance, start at the distance on the horizontal axis, move vertically to the best-fit line, then horizontally to the time axis. To estimate distance from time, reverse the process: move horizontally from the time axis to the line, then vertically to the distance axis.

For 3.5 km, the illustrated line gives approximately 37 minutes. This is interpolation, because 3.5 km lies within the available distance range of 1–6 km. It estimates the likely value from the overall pattern; it is not an observed journey time.

If the same line were extended to 8 km, it would suggest approximately 82 minutes. This is extrapolation, because 8 km lies outside the data range. It is less dependable: the trend might change beyond 6 km, for example if longer walks involved more rest stops. Interpolation is generally more dependable than extrapolation, but even an interpolated estimate is uncertain when points are widely scattered.

Spotting weaknesses in statistical presentations

A presentation can make a pattern difficult to judge, or make it appear more convincing than the evidence allows. Check what the display lets you establish:

• Labels, units and keys: without these, values or different data series may have no clear meaning.
• Scales: check the starting values and intervals. Unequal intervals on an ordinary numerical axis distort positions. A shortened vertical scale can exaggerate differences, especially in a bar chart. A non-zero start is not automatically wrong, but it must be clear.
• Selection of data: omitted observations or a narrow selection of places or dates may hide exceptions and limit how representative the pattern is.
• Comparability: values collected over different time periods or using different units cannot necessarily be compared directly.
• Summary information: an average alone can conceal variation and unusual values. A graph showing the individual observations may reveal more.
• Proportions and totals: percentages allow comparisons of proportions, but do not establish which group has the greater number unless the group totals are known.

For example, imagine two bars showing pedestrian counts of 100 and 120. If the vertical axis starts at 90, the visible bar heights are 10 and 30 units. The second bar looks three times as tall, even though the count is only 20% higher. Reading the scale reveals the difference between the visual impression and the numerical evidence.

Always connect the weakness to its consequence. For example, an unlabelled time axis prevents the reader from knowing whether a journey took seconds, minutes or hours. This is a more precise criticism than saying that the graph is poor.

Drawing and justifying a conclusion

A conclusion answers the original enquiry question using the evidence. State the overall finding, support it with relevant figures, then consider exceptions and limits. Describing identifies what the data show; explaining offers reasons for that pattern. Any proposed reason needs geographical understanding or further evidence, rather than being treated as something the graph proves.

For the practice data, a justified conclusion would be: “Longer distances are generally associated with longer elapsed journey times. Time rises from 13 minutes for the 1 km walk to 60 minutes for the 6 km walk, and most points follow an upward trend. However, the 55 minute value for the 4 km walk lies above the main pattern. These six constructed observations support a relationship in this example, not a prediction for every walking route.”

The qualification matters. A pattern can support a conclusion without proving it universally. When using real geographical data, consider whether the observations cover enough places, times and conditions to justify the breadth of the claim. For example, pedestrian counts collected on one weekday morning may support a conclusion about that period, but not about weekend shopping activity. Measurement errors or a limited sample also reduce confidence in the conclusion. Explain the particular limitation and how it affects the claim you can make.

Read and describe

• Check table headings, units and what each observation represents.
• Identify the overall pattern, support it with values and note exceptions.
• Bivariate data contain paired values for two variables.
• Positive correlation: both tend to increase together. Negative: one tends to decrease as the other increases.
• Strength depends on how closely points follow the trend, not its steepness.

Fit and predict

• Sketch the overall trend; for an approximately straight relationship, draw an estimated best-fit line through the middle of the points.
• Balance points and their distances on either side. Consider outliers without assuming they are errors.
• Read from an axis to the line, then across to the other axis.
• Interpolation is within the data range; extrapolation extends beyond it and is usually less dependable.

Evaluate and conclude

Check labels, units, keys, scales, omitted data, comparability and concealed variation. Percentages alone do not establish numbers without group totals.

Conclusion = answer to the enquiry + numerical evidence + exceptions and limits.

Correlation alone does not prove causation; conclusions should not extend beyond the places, times and conditions the data support.