Chance & Data: Statistics Bivariate Data By the end of this lesson you will be able to explain/calculate the following: Independent & Dependant Variables Correlations

Bivariate Data Univariate data is where only one variable was considered for each piece of data Sets of data where each piece is represented by two variables are called bivariate We will discuss the ways of measuring the relationship between the following pairs of variables: a numerical variable and a categorical variable (for example, weight and nationality) two categorical variables (for example, gender and religious denomination) two numerical variables (for example, height and weight).

Dependant & Independent Variables In a relationship involving two variables, if the values of one variable ‘depend’ on the values of another variable, then the 1st is referred to as the dependent variable and the 2nd is referred to as the independent variable It is useful to identify the independent and dependent variables where possible When displaying data on a graph: the independent variable is on the horizontal axis and the dependent variable is on the vertical axis.

Scatterplots Bivariate data are best represented using a scatterplot. Each piece of data on a scatterplot is shown by a point. The x-coordinate of this point is the value of the independent variable and the y-coordinate is the corresponding value of the dependent variable.

Worked Example

Correlation When analysing bivariate data we are often interested to see whether any relationship exists between the two variables and, if it does, what type of relationship it is. The relationship between the two variables is called correlation. If correlation exists, it can be classified according to its: form — whether it is linear or non-linear direction — whether it is positive or negative strength — whether it is strong, moderate or weak The scatterplot is an excellent tool that assists in classifying the relationship between the two variables.

Linear and Non-linear Relationships (Form) If a scatterplot is in the shape of a ‘corridor’ and fitting a straight line to it seems reasonable, then the relationship between the two variables can be called linear. Otherwise the relationship is non-linear. Non-linear relations can be classified further as being quadratic, exponential and so on, but further classification is not required at Year 10

Positive and Negative Correlation (Direction) If one variable tends to increase as the other variable increases, the correlation between the two variables is said to be positive. If one variable tends to decrease with the increase of the other, the correlation is said to be negative.

The Strength of the Correlation (Strength) The narrower the path, the stronger the correlation between the two variables. Sometimes the points on the scatterplot form a straight line.

The Strength of the Correlation (Strength) Sometimes the points on the scatterplot appear to be in no particular order

Worked Example The points on the scatterplot form a narrow path that resembles a straight ‘corridor’ (that is, it would be reasonable to fit a straight line to it). Therefore the relationship is linear. The path is directed from the bottom left corner to the top right corner and the value of y increases as x increases. Therefore the correlation is positive. Furthermore the points are quite tight; that is, they form a thin corridor. So the correlation can be classified as being strong. There is a strong, positive, linear relationship between x and y.

Worked Example