Tables, graphs and correlations

Turning research results into a readable pattern

Psychologists often collect many numerical results, such as recall scores or reaction times. A table organises those results; a graph makes their pattern easier to see. Choosing a display depends on what you want to show: how often scores occur, differences between categories, or a relationship between two variables.

Always begin by identifying what each number measures. A recall score is a count of words, whereas a reaction time is a measurement on a continuous scale. This difference affects how the results should be displayed.

Building and reading a frequency table

A frequency is the number of times a value or category occurs. To construct a frequency table, list the possible values or categories, tally each observation in the appropriate row, then count the tallies. Include values with a frequency of zero when they belong within the range being displayed.

Suppose ten participants recall these numbers of words: 0, 1, 1, 2, 2, 2, 3, 3, 4 and 4. Their results can be organised as follows:

The row for two words means that three participants each recalled two words. It does not mean that two participants recalled three words. Adding the frequencies gives ten, so every participant has been counted once.

A frequency table preserves how often each score occurs, but not which participant obtained it. It also provides the numbers needed to draw a frequency diagram.

Bar charts and discrete frequency diagrams

A bar chart compares separate categories or discrete values. The bars have equal widths and gaps between them, because the horizontal positions do not form continuous intervals.

For the recall table, put words recalled on the horizontal axis and frequency on the vertical axis. Draw bars with heights 1, 2, 3, 2 and 2 above the scores 0 to 4. This is a frequency diagram: the height of each bar shows how many participants obtained that score.

Bar charts can also compare categories. Suppose six participants prefer studying in silence and four prefer studying with music. Two separated bars, labelled Silence and Music, would show frequencies of six and four.

The vertical axis need not always show frequency. A chart comparing mean recall in two experimental conditions would instead show mean words recalled. Read the axis label before deciding what a bar means.

Histograms show grouped continuous data

A histogram displays continuous measurements grouped into intervals. Reaction time is continuous: a participant could respond in 0.42 seconds, 0.425 seconds or another value within the range. Rather than drawing a bar for every possible time, group the measurements.

For example, suppose ten reaction times are summarised like this:

The interval notation prevents overlap: a time of exactly 0.4 seconds belongs in the second interval, not the first. Each observation must belong in just one interval.

Put reaction time on the horizontal axis, with boundaries at 0.2, 0.4, 0.6 and 0.8 seconds. Because these intervals have equal widths, the bar heights can show frequencies of two, five and three. The bars touch because the intervals join along a continuous scale. In a histogram, bar area represents frequency; with equal-width intervals, taller bars also represent larger frequencies.

The middle reaction-time interval contains five participants, compared with two in the first interval. However, the grouped table and histogram cannot tell you the exact times of those five participants. Changing from raw numbers to grouped data makes the pattern clearer but loses some detail.

Scatter diagrams keep two measurements together

A scatter diagram shows whether two variables are related. Each point represents a pair of measurements from the same participant or observation. It does not show frequencies as bar heights.

Suppose six participants report their hours of sleep and complete a recall task:

To plot these results, label the horizontal axis Sleep and the vertical axis Words recalled. Choose evenly spaced scales covering all the values. Participant B becomes the point $(5, 6)$: move across to five hours, then up to six words. Participants C and D have the same horizontal position but different vertical positions. Plot all six points without joining them together.

Data for Sleep and recall: a constructed example

These points rise overall from bottom-left to top-right. This indicates a positive correlation: participants reporting more sleep tend to recall more words. It is a tendency, not a rule applying perfectly to every pair of participants.

Interpreting relationships and experimental graphs

A positive correlation rises from bottom-left to top-right. A negative correlation falls from top-left to bottom-right: as one variable increases, the other tends to decrease. With no correlation, the points show no clear upward or downward pattern.

Strength concerns how closely the points follow the overall pattern. Points clustered closely around a straight-line pattern show a stronger correlation than widely scattered points. A steep slope does not itself mean a strong correlation.

Correlation alone cannot establish cause and effect. In the sleep example, another factor could influence both sleep and recall. The graph therefore supports an association, not the conclusion that extra sleep caused better recall.

When plotting experimental results, usually place the independent variable—the variable the researcher changes—on the horizontal axis, and the dependent variable—the outcome measured—on the vertical axis. For example, a numerical presentation time could be plotted against the number of words recalled. If the independent variable consists of separate conditions, such as silence and music, a bar chart is more suitable.

To interpret any graph, read its title, axes, units and scale first. Describe the main pattern in terms of the actual variables, then support it with values. To translate a plotted point back into numbers, read its horizontal and vertical coordinates and record them together in a table. This reverses the plotting process while preserving the paired measurements.

Choose the display

• Frequency table: counts observations in each value, category or interval.
• Bar chart: compares separate categories or discrete values; bars have gaps.
• Frequency diagram: shows how often values occur.
• Histogram: groups into intervals; bars touch. Area represents ; for equal-width intervals, height can show frequency.