Univariate and Bivariate Analysis

Univariate analysis means analysing one variable on its own (the prefix "uni" means one). You are describing the distribution of a single set of values: what they look like when you line them all up. The tools that suit one variable include frequency tables, dot plots, histograms and bar charts. To summarise the variable you describe its centre using a mean, median or mode, and its spread using the range, exactly the descriptive statistics you practised in Lesson 12.

For example, "What is the typical shoe size in Year 9?" is a univariate question. You collect one number from each student, build a dot plot, and report that the most common size is 8 and the values range from 5 to 11. You are not comparing shoe size against anything else, you are simply describing one variable well.

Bivariate analysis means analysing two variables together to see whether there is a relationship between them (the prefix "bi" means two). Instead of describing one column of data, you now have two measurements for each record and you ask: as one changes, what happens to the other? The right tool depends on the kind of variables. For two numerical variables you use a scatter plot. For two category variables you use a two-way table (a contingency table). When one variable is time, a line graph shows how the other changes over that time.

For example, "Do taller students have bigger hand spans?" is bivariate. Each student gives two numbers, height and hand span, so each becomes one point on a scatter plot. If the points rise from lower left to upper right, taller students do tend to have bigger hand spans, and we say the two variables move together.

Deciding which analysis a question needs is simpler than it sounds: count the variables in the question. If the question is about one variable, "how tall is our class?", "what is the typical test score?", you do univariate analysis and describe its distribution. If the question links two variables, "do taller students have bigger hand spans?", "does temperature affect sales?", you do bivariate analysis and look for a relationship.

A useful clue: bivariate questions usually contain a joining word such as "affect", "relate to", "depend on", or "compared with". Univariate questions usually ask "what is", "how many" or "how typical" about a single thing. Once you know the count, the right tool follows: one variable points to a dot plot or histogram, two numerical variables point to a scatter plot, and two categories point to a two-way table.

A scatter plot is the main tool for bivariate analysis of two numerical variables. Each point is one record with two values, one read off the horizontal axis and one off the vertical axis. To read a scatter plot, describe two things. First the direction of the trend: a positive relationship means the points rise from left to right (as one variable goes up, so does the other); a negative relationship means they fall from left to right (as one goes up, the other goes down); and if the points are scattered with no slope, there is no relationship. Second, describe roughly how strong the trend is: points hugging a clear line show a strong relationship, while a loose cloud that only leans a little shows a weak one.

Look at the scatter plot below of height against arm span for ten students. Each dot is one student. The points rise from the lower left to the upper right, so the trend is positive: taller students tend to have a longer arm span. Because the points sit close to an upward line, the relationship looks fairly strong. Important: a clear trend like this still does not prove that being tall causes a long arm span, that careful distinction between a relationship and a cause is exactly what you will tackle in Lesson 14.

Univariate worked example. A class records shoe sizes: 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11. That is one variable, so this is univariate. Build a frequency table, draw the dot plot (shown earlier), then summarise: the mode is 8 (it appears six times), the values range from 5 to 11, and the distribution is fairly symmetric around 8. The whole job is describing one variable clearly.

Bivariate worked example. The same class now records, for ten students, both height and arm span: (142, 141), (150, 149), (155, 153), (158, 158), (162, 164), (166, 169), (170, 173), (174, 178), (178, 182), (182, 187) in centimetres. That is two measurements per student, so this is bivariate. Plot each pair as one point (the scatter plot above), then describe the trend: the points rise from lower left to upper right, so taller students tend to have a longer arm span, a positive and fairly strong relationship. Notice how the same class data answered a one-variable question one way and a two-variable question a completely different way.