Summarising and Representing Data

Turning measurements into a useful summary

A scientific investigation can produce many readings. A list preserves individual measurements, but a summary helps us compare results and spot patterns. An average describes a typical value; a frequency table shows how often values occur; a graph makes the pattern easier to see.

Quantitative data are numerical data, such as a plant’s height, a reaction time or the number of organisms counted. The kind of data and the question being investigated determine which summary is useful.

Mean, median and mode describe different things

The arithmetic mean uses every value. Add all the readings, then divide by the number of readings:

$$\text{mean}=\frac{\text{sum of the values}}{\text{number of values}}$$

Suppose five seedlings have heights of 2, 3, 3, 4 and 8 cm. Their total height is 20 cm, so their mean height is $20\div5=4\text{ cm}$. The mean has the same unit as the original measurements. It need not be a value that occurs in the data.

The median is the middle value after the readings have been put in numerical order. These five heights are already ordered, so the third value, 3 cm, is the median: two readings lie on either side of it. With an even number of readings, there are two middle values. For example, the median of 2, 3, 5 and 8 cm is $(3+5)\div2=4\text{ cm}$.

The mode is the most frequent value. In the five-seedling example, 3 cm occurs twice, while every other height occurs once, so the mode is 3 cm. A data set can have more than one mode if values tie for the highest frequency, or no mode if no value occurs more often than the others.

These averages answer slightly different questions. The mean takes account of every measurement, but an unusually large or small reading can pull it towards that value. The median depends on the position of the middle readings, so it is less affected by extreme values. The mode identifies the most common result. Here, the 8 cm seedling helps explain why the mean is higher than the median and mode.

Building and reading a frequency table

Frequency means how often something occurs. To construct a frequency table, list the possible values or categories, count each observation once, and record the totals. Tally marks can help: group them in fives so that they are easy to count.

For the five seedling heights, the table is:

The frequency column tells us that two seedlings are 3 cm tall, not that their combined height is 2 cm. Adding the frequencies gives the total number of seedlings. The largest frequency identifies the modal height.

Continuous measurements, such as height or time, can take values between whole numbers. If there are many different readings, group them into class intervals rather than making a separate row for every measurement. Make the intervals non-overlapping so that a reading cannot belong to two groups.

For example, the same heights can be grouped as follows:

The interval $2\leq h<4$ includes 2 cm and 3 cm but excludes 4 cm. Grouping makes the distribution clearer, although the exact measurements within each interval are no longer visible.

Choosing between a bar chart and a histogram

A bar chart compares separate categories. For example, suppose a survey records 4 daisies, 7 dandelions and 3 buttercups. Put the species on the horizontal axis and the number counted on the vertical axis. Draw bars of equal width with equal gaps between them. Their heights show the frequencies. Dandelions are the most common category in this example.

A histogram shows continuous measurements grouped into numerical intervals. Put the measurement and its unit on the horizontal axis, and keep the intervals in numerical order. The bars touch because neighbouring intervals belong to one continuous scale. For equal-width intervals, as in the seedling table, bar height can show frequency directly.

To construct either graph, choose a useful title, label both axes and select a numerical scale that accommodates the data. Draw each bar to the correct height. For the seedling histogram, retain the empty intervals: leaving their positions visible shows where no measurements occurred.

When interpreting a graph, first read the axes and scale. Then identify the most common category or interval, compare frequencies and describe where measurements are concentrated. Here, three of the five seedlings are between 2 cm and just below 4 cm tall.

Scatter diagrams reveal relationships

A scatter diagram displays paired measurements of two variables. Each point represents one pair, not a total or an average. Put one variable on the horizontal axis and the other on the vertical axis, label their units, and plot a cross where each pair of values meets. Do not join the points one after another.

Suppose temperature and gas volume are recorded for a fixed amount of gas at constant pressure. The following example shows how to plot and interpret such pairs.

Data for Identifying a positive correlation

A point at $(20,104)$ represents a temperature of 20 °C paired with a gas volume of 104 cm³. Across the graph, higher temperatures tend to occur with higher volumes: this is a positive correlation.

A negative correlation has the opposite pattern: as one variable increases, the other tends to decrease. With no correlation, the points show no clear overall relationship. For a roughly straight-line relationship, points clustered closely around an upward or downward trend indicate a stronger correlation than widely scattered points.

Describe the overall pattern rather than relying on one pair of points. A correlation is evidence of an association, but it does not by itself establish that changing one variable causes the other to change. The investigation’s conditions and other possible influences must also be considered.

Averages

• Mean: sum of values ÷ number of values.
• Median: middle of the ordered data; average the two middle values if necessary.
• Mode: most frequent value.
• Extreme readings affect the mean more than the median.

Frequencies and graphs

• A frequency table counts occurrences; total frequency = number of observations.
• Class intervals must not overlap.
• Bar chart: separate categories, equal-width bars and gaps.
• Histogram: continuous numerical intervals in order; bars touch. For equal-width intervals, height can show frequency.
• Read titles, axis labels, units and scales before comparing values.

Scatter diagrams

Each point represents one paired measurement.

• Positive correlation: both variables tend to increase together.
• Negative correlation: one tends to decrease as the other increases.
• No correlation: no clear overall pattern.
• Tighter clustering around a trend indicates a stronger relationship.
• Correlation alone does not establish causation.