4.2.3.2 - Descriptive statistics

What descriptive statistics do

Psychologists rarely hand an examiner a list of every raw score and stop there. Raw data need to be summarised so that the reader can see what is typical, how varied the scores are, and whether two measured variables appear to move together.

In AQA research methods, descriptive statistics do not prove a hypothesis by themselves. They describe the data so that the researcher can make cautious, evidence-based interpretations. For example, a memory researcher might report:

The exam skill is choosing and interpreting the statistic, not just naming it. A strong answer links the statistic to the data in the stem: "The median is more suitable because one participant's unusually high score would distort the mean" is much better than "use the median because it is good".

Descriptive summaries must be read in context. A higher mean reaction time indicates slower responses, whereas a higher mean recall score indicates more words remembered. Neither difference alone establishes statistical significance.

Measures of central tendency

An average describes a typical score, but different averages preserve different information.

Mean

For the scores:

5, 7, 7, 9, 10, 11, 12, 14, 15, 17

the total is 107. There are 10 scores, so:

Here $\sum x$ means add all scores, $n$ is the number of scores and $\bar{x}$ is the mean.

The mean is often the most useful average for interval-style numerical data, such as reaction time in milliseconds or number of words recalled. Its main strength is that it uses every score. This makes it sensitive to small changes across the whole dataset.

That strength is also its weakness. If the final score of 17 were replaced with 98, the mean would become 18.8. That would not describe most participants very well because one extreme score has pulled the average upwards.

Median

If there is an odd number of scores, the median is the central score. If there is an even number of scores, it is halfway between the two middle scores.

Always order the data before finding the median. For ten scores, the middle positions are the fifth and sixth; do not average the lowest and highest scores. For the same dataset:

5, 7, 7, 9, 10, 11, 12, 14, 15, 17

the two middle scores are 10 and 11, so:

The median is useful when data contain extreme scores because it is based on position in the ordered list rather than the size of every value. Whether the highest score is 17 or 98, the median remains 10.5.

The limitation is that the median ignores the exact values of most scores. If scores below and above the middle change a lot, the median may stay the same even though the dataset has changed substantially.

Mode

In the dataset above, the mode is 7 because 7 occurs twice and every other score occurs once. A dataset can have more than one mode, such as two equally common anxiety ratings. It can also have no useful mode if every score occurs once.

Do not average category codes such as 1 = flashcards and 2 = videos: changing the arbitrary labels would change the calculated mean without changing anyone's preference. The mode is the only average that works naturally with nominal data: categories with names rather than scores. For example, if students choose their preferred revision method from flashcards, mind maps or practice questions, the modal category is the most popular method.

The limitation is that the mode can be crude. In the dataset above, the mode of 7 is quite far from the mean of 10.7 and the median of 10.5, so it may not represent the overall pattern well.

Worked example

Dr Patel records the number of words recalled by ten participants. Nine participants recall between 12 and 16 words, but one participant recalls 45 words. The median would give a fairer description of typical recall than the mean because the single score of 45 would inflate the mean.

Measures of dispersion

Two groups can have the same average while their individual scores vary very differently.

Measures of central tendency tell us what is typical. Measures of dispersion tell us whether scores are clustered close together or scattered widely. This matters because two groups can have the same mean but very different patterns of scores.

Imagine two memory conditions:

Both conditions have the same mean, but the noisy-room scores are much more spread out. A researcher who reports only the mean hides an important difference.

Range

For the scores 5, 7, 7, 9, 10, 11, 12, 14, 15, 17:

The range is quick and easy to calculate. It gives a basic sense of the distance between the smallest and largest values.

The weakness is that it uses only the two most extreme scores. If most students score between 47 and 57 on a test but one student scores 0 and another scores 100, the range looks huge even though most scores are tightly grouped. This means the range can exaggerate variation when outliers are present.

Standard deviation

Standard deviation is more informative than range because it is based on every score, not just the highest and lowest. In psychological research, it helps the researcher judge whether participants responded in a consistent way.

For example:

The silence condition has the higher mean, so participants recalled more words on average. Its low standard deviation also suggests the scores were consistent. The music condition has a lower mean and a higher standard deviation, suggesting participants' recall varied more when music was playing.

Calculating standard deviation

First find the mean, subtract it from each score, square each deviation, add the squared deviations, divide by the specified denominator and take the square root. Squaring prevents positive and negative deviations cancelling.

Squaring the deviations matters: adding signed deviations alone would give zero even for widely spread data. For four recall scores 2, 4, 4, 6 words, the mean is $16/4=4$ words:

Using the sample standard deviation formula:

Here $s$ is sample standard deviation, $n=4$, and the answer is rounded to two decimal places. A population formula divides by $n$, giving $\sqrt{8/4}=1.41$ words for these same scores. Use the formula or calculator setting requested; never switch denominator silently. Standard deviation has the same units as the original scores. Keep unrounded intermediate values.

Choosing a measure of dispersion

Standard deviation is usually a more informative summary of numerical spread around the mean because it uses all scores. It is sensitive to changes across the whole dataset, but extreme scores can still affect it because they increase the deviations from the mean. It is also harder to calculate than the range.