Overview – Descriptive statistics

Descriptive statistics are ways of summarising and describing data collected by researchers. They allow researchers to identify patterns in their data and present their findings clearly.

This page covers the key concepts in descriptive statistics for psychology:

Measures of central tendency

Measures of central tendency are ways of identifying the typical or central value in a set of data. They are used to summarise a large set of scores using a single value.

For example, let’s say we had 9 participants complete a memory test and they got the following scores:

Participant 1 2 3 4 5 6 7 8 9
Score 6 8 7 5 9 7 4 6 8

There are 3 main measures of central tendency:

  • Mean – the average of all the scores.
  • Median – the middle score when the data is arranged in order.
  • Mode – the most frequently occurring score.

Mean

The mean is the average of a set of scores.

It is calculated by adding all the scores together and then dividing by the number of scores. So, using the data set above, we would calculate the mean like this:

(6 + 8 + 7 + 5 + 9 + 7 + 4 + 6 + 8) ÷ 9 = 60 ÷ 9 = 6.67

So the mean score for the data set above is 6.67.

The mean uses every score in the dataset, but it can be affected by extreme scores or outliers.

Median

The median is the middle score when the data is arranged in order from lowest to highest.

So, using the same data above, we would put the scores in order and pick the one in the middle like this:

4, 5, 6, 6, 7, 7, 8, 8, 9

There are nine scores, so the middle score is the fifth score. So the median is 7.

The median is less affected by extreme scores than the mean, making it useful when data is skewed.

Mode

The mode is the score that occurs most frequently in a set of data.

Again, using the same data above:

4, 5, 6, 6, 7, 7, 8, 8, 9

The scores 6, 7, and 8 each occur twice. So this dataset has three modes: 6, 7, and 8.

A dataset can have more than one mode or it may have no mode if every value occurs only once.

Measures of dispersion

Measures of dispersion are ways of describing how spread out the scores in a data set are.

For example, let’s use the same memory test scores as above:

Participant 1 2 3 4 5 6 7 8 9
Score 6 8 7 5 9 7 4 6 8

There are 3 main measures of dispersion:

  • Range – the difference between the highest and lowest scores.
  • Variance – a measure of how far the scores spread out from the mean.
  • Standard deviation – a measure of how much the scores typically differ from the mean.

Range

The range is the difference between the highest and lowest scores in a data set.

Using the data above:

Highest score − lowest score = 9 − 4 = 5

So the range is 5.

The range is a simple way of showing how spread out the scores are, but it can be strongly affected by extreme scores or outliers.

Variance

Variance is a measure of how far the scores in a data set spread out from the mean.

To calculate the variance, we:

  • Work out the mean.
  • Work out how far each score is from the mean.
  • Square each of these differences.
  • Add the squared differences together.
  • Divide the total by the number of scores.

Using the data above, the mean is 6.67.

We can then work out how far each score is from the mean and square the differences:

Score Difference from mean Squared difference
6 −0.67 0.44
8 1.33 1.78
7 0.33 0.11
5 −1.67 2.78
9 2.33 5.44
7 0.33 0.11
4 −2.67 7.11
6 −0.67 0.44
8 1.33 1.78

The squared differences add up to 20 (0.44 + 1.78 + 0.11 + 2.78 + 5.44 + 0.11 + 7.11 + 0.44 + 1.78 = 20).

We then divide this by the number of scores:

20 ÷ 9 = 2.22

So the variance is 2.22.

A larger variance means the scores are more spread out, while a smaller variance means the scores are closer together.

Standard deviation

Standard deviation is another measure of how spread out the scores are. It tells us approximately how much the scores typically differ from the mean.

The standard deviation is calculated by taking the square root of the variance.

Using the variance calculated above:

√2.22 = 1.49

So the standard deviation is approximately 1.49.

A larger standard deviation means the scores are more spread out, while a smaller standard deviation means the scores are closer to the mean.

Standard deviation is often more useful than the range because it takes all of the scores into account rather than just the highest and lowest scores.

Ratios

Ratios are used to compare the relative size of two or more quantities.

A ratio shows how much of one quantity there is compared with another. Ratios are often written using a colon –  for example 2:1 means that for every 2 of one thing, there is 1 of another.

For example, let’s say our memory test with 9 participants consisted of 6 female 3 male participants:

Participant Female Male
Number 6 3

The ratio of female to male participants is:

6:3

Ratios can be simplified by dividing both numbers by the same number. In this example, both numbers can be divided by 3:

6 ÷ 3 : 3 ÷ 3 = 2:1

So the ratio of female to male participants is 2:1.

Ratios can also be used to compare different quantities in psychological research, such as the number of participants in two experimental conditions.

Percentages

Percentages are used to express a value as a proportion of 100. They are particularly useful for describing how many participants in a study fall into a particular category.

For example, in our memory study, let’s say 6 out of the 9 participants correctly remembered a particular item. To calculate the percentage of participants who remembered that item, we divide the number of participants who gave the particular response by the total number of participants and multiply by 100:

(6 ÷ 9) × 100 = 66.67%

So 66.67% of the participants correctly remembered the item.

The general formula for calculating a percentage is:

(number in the category ÷ total number) × 100

Percentages make it easier to compare results between groups, particularly when the groups contain different numbers of participants.

Fractions

Fractions are used to express a quantity as a part of a whole. A fraction has two parts:

  • the numerator, which is the number on top,
  • the denominator, which is the number on the bottom.

For example, in our memory study, 6 out of the 9 participants correctly remembered a particular item. This can be written as the fraction:

6/9

The numerator (6) tells us how many participants correctly remembered the item, while the denominator (9) tells us the total number of participants.

Fractions can be simplified by dividing the numerator and denominator by the same number:

6/9 = 2/3

So 2/3 of the participants correctly remembered the item.

Fractions can also be converted into percentages. To do this, divide the numerator by the denominator and multiply by 100:

(2 ÷ 3) × 100 = 66.67%

So 2/3 is equivalent to 66.67%.

Fractions are useful in psychology when describing the proportion of participants who show a particular behaviour or response.

Frequency

Frequency tallies are used to record how often each value or category occurs in a data set.

A tally is a simple mark used to keep track of each occurrence. Every fifth tally is usually drawn across the first four tallies, making it easier to count groups of five.

For example, in our memory study above, the 9 participants received the following scores:

6, 8, 7, 5, 9, 7, 4, 6, 8

We can use a frequency table to record how often each score occurred:

Memory score Tally Frequency
4 | 1
5 | 1
6 || 2
7 || 2
8 || 2
9 | 1

The frequency is the number of times each score occurs. For example, a score of 6 has a frequency of 2 because two participants achieved this score.

Frequency tallies are useful because they provide a quick way of organising raw data and identifying which scores or categories occur most often (i.e. the mode).