Statistical summaries and normal distributions

Summarising psychological data

A researcher might collect the number of words each participant recalls in a memory task. A list of individual scores contains useful detail, but it can be difficult to see the overall pattern. Descriptive statistics summarise features of the data.

Two features matter here: a typical score and how spread out the scores are. The mean, median and mode are measures of central tendency: different ways of identifying the centre or typical value. The range is a measure of dispersion, which means spread. Using both helps a researcher describe results more clearly than using an average alone.

Mean: sharing the total equally

The mean is the arithmetic average. Add all the scores, then divide by the number of scores:

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

Suppose four participants recall 4, 6, 7 and 9 words. Their total is $4 + 6 + 7 + 9 = 26$ words. Dividing by four gives a mean of $26 \div 4 = 6.5$ words. Think of this as sharing their total equally between the four participants. Nobody needs to have recalled exactly 6.5 words for this to be a meaningful average.

The mean uses every score, so a change in any score changes the mean. This is useful, but it also makes the mean sensitive to outliers: unusually high or low scores. If most participants have similar scores but one has a very high score, the mean is pulled upwards and may describe the typical participant poorly.

Median: finding the middle position

The median is the middle score after the data have been arranged in numerical order. It depends on position rather than adding every score.

For an odd number of scores, there is one middle value. Suppose five participants recall 9, 4, 7, 6 and 7 words. In order, these are 4, 6, 7, 7, 9. Two scores lie on each side of the middle score, so the median is 7 words.

For an even number of scores, there are two middle values. Find their mean. For 4, 6, 7, 9, the middle values are 6 and 7, so the median is $(6 + 7) \div 2 = 6.5$ words.

The median is less affected by extreme scores than the mean. In 4, 6, 7, 7, 20, the median is still 7 words, whereas the mean is $44 \div 5 = 8.8$ words. Here, the median better represents the cluster of lower scores. Its limitation is that it does not use the actual size of every score.

Mode: finding what occurs most often

The mode is the most frequently occurring score or category. In 4, 6, 7, 7, 9, the mode is 7 words, because 7 occurs twice and every other score occurs once.

The mode can also summarise categories. Suppose participants report their preferred revision method as flashcards, reading, flashcards, practice questions and flashcards. The modal category is flashcards. A mean cannot meaningfully be calculated from these category names.

A dataset can have two or more modes when values tie for the highest frequency. If every value occurs equally often, there is no single most frequent value. The mode is simple to identify, but it does not take every score's numerical value into account and may give a limited picture of the results.

Range: describing how far scores extend

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

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

For memory scores of 4, 6, 7, 7, 9 words, the range is $9 - 4 = 5$ words. A larger range means the scores extend across a wider interval; a smaller range means the highest and lowest scores are closer together.

The range adds information that an average misses. Suppose one group scores 5, 6, 7 words and another scores 2, 6, 10 words. Both means are 6 words, but their ranges are 2 words and 8 words respectively. The second group's scores are more spread out.

The range is quick to calculate, but uses only the two extreme scores. One outlier can greatly increase it, and it tells us nothing about how scores are distributed between the lowest and highest values.

Normal distributions: recognising the overall pattern

A distribution describes how scores are spread across possible values. A normal distribution has a symmetrical, bell-shaped curve. Most scores cluster around the centre, and progressively fewer occur towards either extreme.

Read the horizontal axis as the score and the vertical axis as how common scores are. The highest point shows the most common score. The matching shape on either side of the centre shows symmetry: equally distant scores below and above the centre are equally common.

In a perfect normal distribution, the mean, median and mode are equal and lie at the centre. Scores to the left are below the mean; scores to the right are above it. Half the distribution lies on each side. The tails of the ideal curve approach the horizontal axis without touching it: very extreme values are increasingly uncommon.

Real research results may only approximate this shape. Do not assume that every psychological dataset is normally distributed; examine the pattern of scores as well as its numerical summaries.

Significant figures: reporting sensible precision

Significant figures express the precision with which a number is written. Count from the first non-zero digit. For example, 6.50 has three significant figures: its final zero communicates precision. 0.0065 has two significant figures: the zeros before 6 only locate the decimal point.

To round to a chosen number of significant figures, find the last digit to keep and inspect the next digit. If the next digit is 5 or more, round up; if it is below 5, leave the retained digit unchanged. Remove the remaining digits, keeping any zeros needed to preserve place value.

For example, 6.478 becomes 6.5 to two significant figures and 6.48 to three significant figures. 14,489 becomes 14,000 to two significant figures: the first two significant digits are 1 and 4, and the next digit is 4.

Significant figures are not the same as decimal places. Decimal places count digits after the decimal point; significant figures start at the first non-zero digit, wherever it occurs.

Choose a sensible precision for statistical results. Follow a stated instruction; otherwise consider the precision of the original measurements and avoid reporting a long calculator display as though every digit were meaningful. Keep full precision while calculating, then round the final result.

Central tendency

• Mean: add all scores and divide by the number of scores. Uses every score; sensitive to outliers.
• Median: order the scores and find the middle. With an even number, average the two middle values. Less affected by outliers.
• Mode: most frequent score or category. Can summarise categories; there may be more than one mode.