Summarising Numerical Data

Summarising scores: typical values and spread

Psychologists often collect numerical scores, such as reaction times or the number of words recalled in a memory task. A long list of scores can be difficult to interpret, so descriptive statistics summarise important features of the data.

Two features matter here. A measure of central tendency describes the centre or typical score: the mean, median and mode do this in different ways. A measure of dispersion describes how spread out the scores are. The range is one measure of dispersion. An average and a range therefore answer different questions about the same data.

The mean: sharing the total equally

The mean, also called the arithmetic mean, is found by adding all the scores and dividing by the number of scores. Imagine sharing the total equally between all the participants: the amount each would receive is the mean.

$$\text{Mean} = \frac{\text{sum of all scores}}{\text{number of scores}}$$

Suppose five participants recall 4, 6, 6, 9 and 10 words. The total is $4 + 6 + 6 + 9 + 10 = 35$ words. Dividing by the five participants gives a mean of $35 \div 5 = 7$ words recalled.

Every score contributes to the calculation, including repeated scores. The mean does not have to be a score that anyone actually obtained: nobody in this example recalled exactly seven words. It can also be a decimal, even when individual scores are whole numbers.

The median: finding the middle

The median is the middle score once the scores are arranged from lowest to highest. Ordering the data is essential because the middle of an unordered list need not represent its centre.

For the ordered scores 4, 6, 6, 9, 10, the third score is in the middle, with two scores on either side. The median is therefore 6 words.

When there is an even number of scores, there is no single middle score. Find the two middle scores and calculate their mean. Suppose four participants recall 4, 6, 9 and 10 words. The middle scores are 6 and 9, so the median is $(6 + 9) \div 2 = 7.5$ words.

The mode: finding the most frequent value

The mode is the score that occurs most often. In 4, 6, 6, 9, 10, the mode is 6 words, because 6 occurs twice and every other value occurs once.

The mode is the value itself, not the number of times it appears. A dataset can have more than one mode if values tie for the highest frequency. If every value occurs only once, there is no mode.

These three measures need not give the same answer. For the five recall scores, the mean is 7 words, while the median and mode are both 6 words. Each summarises a different feature of the scores.

The range: measuring how far scores extend

The range is the difference between the highest and lowest scores.

$$\text{Range} = \text{highest score} - \text{lowest score}$$

For 4, 6, 6, 9, 10 words, the range is $10 - 4 = 6$ words. A larger range means the scores extend across a wider interval; a smaller range means they lie within a narrower interval. Compare ranges using the same measurement and units.

An average alone does not show this spread. Suppose one group recalls 6, 7 and 8 words, while another recalls 2, 7 and 12 words. Both groups have a mean of 7 words, but their ranges are 2 words and 10 words respectively. The second group's scores are much more spread out.

The range uses only the two extreme scores, so an unusually high or low score can change it substantially. Such a score also affects the mean because the mean includes every value. The median is less affected by how extreme that score is, because it depends on the middle position.

Significant figures: reporting numbers at a suitable precision

Significant figures are the digits retained to express a number's precision. Count from the first non-zero digit. For example, 3.40 seconds has three significant figures: the final zero shows that the value is expressed to hundredths of a second. In 0.00486 seconds, the leading zeros locate the decimal point; the three significant figures are 4, 8 and 6.

To round to a given number of significant figures, identify the last digit to keep and look at the next digit. If the next digit is 5 or more, increase the last retained digit by 1. If it is 4 or less, leave that digit unchanged. Remove the remaining digits, keeping any zeros needed to preserve the number's place value.

For example, 43,600 becomes 44,000 to two significant figures. The retained digits are initially 4 and 3; the next digit is 6, so the 3 rounds up to 4. Similarly, 0.00486 becomes 0.0049 to two significant figures.

Significant figures are not the same as decimal places, which count positions to the right of the decimal point. For example, 0.0472 has three significant figures but four decimal places. To two significant figures it becomes 0.047; to two decimal places it becomes 0.05.

An appropriate answer follows any stated rounding instruction and does not imply more measurement precision than is justified. Extra digits do not make a measurement more accurate. When calculating a mean, retain the available digits during the calculation and round the final answer rather than rounding each step.

Central tendency

• Mean: add all scores and divide by the number of scores.
• Median: order the scores and find the middle; with an even number, average the two middle scores.
• Mode: the most frequent value; there may be multiple modes or no mode.

$$\text{Mean} = \frac{\text{sum of scores}}{\text{number of scores}}$$

Dispersion

$$\text{Range} = \text{highest score} - \text{lowest score}$$

A larger range indicates a wider spread. Equal means do not imply equal ranges. Extreme scores can strongly affect the mean and range.

Significant figures

Count from the first non-zero digit. Leading zeros do not count; zeros between significant digits and trailing decimal zeros do.

Check the next digit: 5–9 → round up; 0–4 → leave unchanged. Preserve place value.

Significant figures count retained significant digits; decimal places count positions after the decimal point. Round the final answer as requested.