Last Updated on October 11, 2026 by Karl Thompson
Bivariate Analysis for Quantitative Social Research
Bivariate analysis involves analysing two variables at a time in order to find out whether the two variables are related.
Exploring relationships between variables means searching for evidence that the variation in one variable coincides with variation in another variable.
There are a variety of techniques you can use to conduct bivariate analysis, but which one you use depends on the types of variable being analysed. The table below shows which technique fits which combination of variables. (If you need a reminder of the different types, see Variables in Quantitative Research.)
| Type of variable | Nominal | Ordinal | Interval/ratio | Dichotomous |
|---|---|---|---|---|
| Nominal | Contingency table + chi-square + Cramér’s V | Contingency table + chi-square + Cramér’s V | Contingency table + chi-square + Cramér’s V; compare means and eta | Contingency table + chi-square + Cramér’s V |
| Ordinal | Contingency table + chi-square + Cramér’s V | Spearman’s rho | Spearman’s rho | Spearman’s rho |
| Interval/ratio | Contingency table + chi-square + Cramér’s V; compare means and eta | Spearman’s rho | Pearson’s r | Spearman’s rho |
| Dichotomous | Contingency table + chi-square + Cramér’s V | Spearman’s rho | Spearman’s rho | Phi |
Bivariate analysis for different types of variable (based on Bryman 2016)
Bivariate Analysis: Relationships, not Causality
If there is a relationship between two variables, this does not necessarily mean one causes the other.
Even if there is a causal relationship, we need to take care to make sure the direction of causality is the right way round. Researchers must be careful not to let their own assumptions decide which variable is causing which.
For example, Sutton and Rafaeli (1988) conducted bivariate analysis on the relationship between the display of positive emotions by convenience store staff and levels of store sales.
Common sense might tell you that friendly staff sell more. However, Sutton and Rafaeli found a negative relationship: stores with higher sales had staff who displayed fewer positive emotions. Their explanation was that the causality ran the other way. High sales mean busy stores, and in busy stores, with queues and time pressure, staff stick to neutral, efficient behaviour rather than being chatty and friendly. This was unexpected, but it also makes sense!
Sometimes you can infer the direction of causality with complete certainty. Take the relationship between age and voting: younger people are less likely to vote, and age must be the independent variable here. There is no way that voting can influence how old you are.
Contingency Tables
A contingency table is like a frequency table, but it allows two variables to be analysed at the same time so that relationships between them can be examined.
Contingency tables usually contain percentages, since these make the relationships easier to see.
| Male: number | Male: percent | Female: number | Female: percent | |
|---|---|---|---|---|
| Sociology | 60 | 30 | 120 | 40 |
| Maths | 20 | 10 | 60 | 20 |
| English | 20 | 10 | 60 | 20 |
| Dance | 100 | 50 | 60 | 20 |
| Total | 200 | 100 | 300 | 100 |
Students studying subjects in one college, by gender (hypothetical data)
The table above contains both the number of students in each cell and that number as a percentage of the column total.
These are column percentages: each number is calculated as a percentage of the total for its column, which is why each percent column adds up to 100.
In the table above, we can see that there are more female students than male students, and that a higher percentage of female students than male students study every subject except dance, which is much more popular among male students. (It’s quite an unusual college!)
Contingency tables can be used with all types of variable, but they are not always the most efficient method. They are usually reported alongside a chi-square test, which tells you how confident you can be that the relationship in the table isn’t just down to chance in the sample.
Pearson’s r
Pearson’s r is a method for examining relationships between interval/ratio variables. The main features of this method of analysis are:
- The coefficient will lie between −1 and +1.
- Its size, ignoring the sign, indicates the strength of the relationship: 0 means no relationship, and 1 (or −1) means a perfect relationship. The closer the coefficient is to +1 or −1, the stronger the relationship; the closer to 0, the weaker the relationship.
- Whether the coefficient is positive or negative indicates the direction of the relationship. A positive correlation means that as one variable goes up, so does the other. A negative correlation means that as one goes up, the other goes down.
Examples of Pearson’s r correlations
The table below shows the relationship between age and four other variables. (Note that this data is hypothetical, or made up, and for illustrative purposes only!)
| Age group | Happiness score | Wealth (£) | Hours watching TV per week | Average number of friends |
|---|---|---|---|---|
| 20 | 10 | £10,000 | 15 | 5 |
| 30 | 8 | £20,000 | 10 | 8 |
| 40 | 6 | £30,000 | 33 | 11 |
| 50 | 4 | £40,000 | 22 | 10 |
| 60–69 | 2 | £50,000 | 9 | 16 |
| Pearson’s r | −1 | 1 | 0 | 0.93 |
The correlations are as follows:
- between age and happiness: a perfect negative correlation
- between age and wealth: a perfect positive correlation
- between age and watching TV: no correlation
- between age and number of friends: a strong positive correlation
The scatter plots for the above data are as follows:
Age and happiness
Age and wealth
Age and TV
Age and friends
Spearman’s rho
Spearman’s rho is often represented by the Greek letter ρ (rho) and is designed for use with ordinal variables. It can also be used when one variable is ordinal and the other is interval/ratio.
It works in the same way as Pearson’s r, in that the computed value will be between −1 and +1, with the sign showing the direction of the relationship.
The difference is that Pearson’s r can only be used when both variables are interval/ratio, whereas Spearman’s rho can be used when one of the variables is ordinal.
Phi and Cramér’s V
The phi coefficient is used to analyse the relationship between two dichotomous variables. Like Pearson’s r, it produces a statistic which can be positive or negative and varies between −1 and +1.
Cramér’s V can be used with nominal variables. It varies between 0 and 1, and it can only show the strength of the relationship between two variables, not the direction.
Cramér’s V is usually reported along with a contingency table and a chi-square test.
Comparing Means and Eta
Sometimes you need to examine the relationship between an interval/ratio variable and a nominal variable. If the nominal variable can be clearly identified as the independent variable, it can be useful to compare the means of the interval/ratio variable for each subgroup of the nominal variable. For example, you could compare the average income of people in different occupational groups.
This procedure is often accompanied by a test of association between variables called eta. This statistic expresses the level of association between the two variables, and it is always positive.
Eta-squared expresses the amount of variation in the interval/ratio variable that is due to the nominal variable.
Signposting and Sources
This material should be of interest to anyone studying quantitative social research methods. It goes beyond what you need for A-level sociology, so treat it as extension material. It is closer to what you would cover on a first-year undergraduate research methods course.
For analysing one variable at a time, see Univariate Analysis in Quantitative Social Research.
For the difference between correlation and causation as it applies to positivist research, see Scientific Quantitative Methodology in Sociology.
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Sources
Bryman, A. (2016) Social Research Methods. 5th edition. Oxford: Oxford University Press.
Sutton, R. I. and Rafaeli, A. (1988) “Untangling the relationship between displayed emotions and organizational sales: the case of convenience stores”, Academy of Management Journal, 31(3), pp. 461–487.