 # Interpreting diagrams for OCR A-level Maths 1. Correlation
2. Interpolation
3. Extrapolation

Scatter graphs can be used to investigate the correlation between the variables they represent. The variables can have a positive, negative or no correlation, depending on the steepness of the pattern the scatter points follow. The two variables that are positively correlated increase together and a variable that is negatively correlated decreases while the other one increases. Although two variables can be correlated, it does not necessarily mean that the one variable causes the other, as there may be a third one that affects both. Thus, it can be said that correlation does not imply causation. The line of best fit of a scatter plot can be used for interpolation. Interpolation is the process used to estimate the value of the dependent variable from the independent one without a scatter point for that value. For interpolation to work, the value must lie within the range of the values of the graph. Extrapolation is the process used to estimate the value of the dependent variable from the independent one without a scatter point for that value for values which are outside the recorded range. Since the values are not within the range, estimates made by extrapolation are less accurate than those made by interpolation. # 1

Describe what you would expect the relationship between the number of ice creams sold and the number of flipflops sold to be.

It is expected that the two variables are positively correlated, however this may not be because the increase in sales in the one causes the sales to increase for the other. The positive correlation between the two variables is better explained by the fact that in summer, the demand for both ice cream and swimsuits increases. Thus, although the two variables are positively correlated, they do not cause one another, as the third variable of the season may be the one affecting both of them and causing them to be correlated. # 2

What values is interpolation carried out for?

Interpolation can be used to estimate values of the dependent variable for values which lie within the range of the values of the graph. # 3

The heights and masses of a group of students are recorded. The shortest student has height 156 cm and the tallest student has height 172 cm. Explain whether making a prediction for the mass of a student of height 180 cm using this data is interpolation.

The range of heights for which the data is recorded is from 156 to 172. A height of 180 cm lies outside this range, therefore making a prediction for it would not be interpolation. # 4

The number of books read in a year by a set of students is recorded alongside their percentage grade in English Literature and the two variables are found to be positively correlated. The number of books read by the students ranged from 3 to 55. Explain whether an estimation for the English Literature grade of a student who read 38 books or one who read 0 books would be more accurate.

Estimating the grade of a student who read 38 books would be done by interpolation, whereas for the one who read 0 books would be done by extrapolation. Since interpolation is done for values within the recorded range, it will be more accurate, therefore the predicted English Literature grade for the student who read 38 books will be more accurate. # 5

Match the given scatter plots to the following pairs of variables: (1) number of hours spent studying in a week and the percentage grade received by the student, (2) number of weeks since a student studies a topic and their memory of that topic, (3) the number of guitars sold in a year and the number of coats sold in a year.

(1) is the plot C, (2) is the plot B and (3) is the plot A. End of page