FS6, HS7 - Scatter graphs and prediction

Bivariate data and correlation

Bivariate data records two variables for each individual or item. A scatter graph plots each pair as one point. Positive correlation means larger values of one variable tend to accompany larger values of the other; negative correlation means they tend to accompany smaller values. No apparent correlation means there is no clear increasing or decreasing trend. The closer the points lie to a clear trend, the stronger that correlation.

Worked example 1: plotting a positive relationship

Question. Plot these six learners' study times and test scores, and describe the association.

Plan. Use study time on the horizontal axis and score on the vertical axis; plot one point per learner.

For the fourth learner, move across to four hours and up to $58\%$, marking $(4,58)$. Plot the remaining pairs in the same way. In the left panel below, the cloud rises from left to right, with some scatter rather than a perfect line.

Answer. There is a clear positive correlation: learners with longer study times tend to have higher scores in these data.

Check. There are six plotted points for six paired records. The points $(1,40)$ and $(6,73)$ match the first and last rows, and the fourth point has not been reversed to $(58,4)$.

Worked example 2: plotting a negative relationship

Question. Plot these used phones' ages and selling prices, then describe the correlation.

Plan. Put age on the horizontal axis and price on the vertical axis, keeping every age with its own price.

For example, the four-year-old phone gives point $(4,134)$. The points move generally downwards as age increases, as in the centre panel.

Answer. Negative correlation: older phones tend to sell for less in this sample.

Check. The youngest and oldest records plot at $(1,181)$, $(6,101)$; the intermediate points also follow the downward pattern. A negative correlation describes the direction of association, not a negative selling price.

Watch out

Use the variables in your description: “price tends to decrease as age increases” communicates more than the word “negative” alone. A trend does not require every successive pair to move in exactly the same direction.

Best fit and interpolation

An estimated line of best fit represents the overall linear trend; it need not pass through actual data points or the origin. Interpolation predicts within the range supported by the observed data. Read from an input to the line, then across to the other axis. A prediction estimates an outcome, rather than reproducing every observation.

Worked example 3: a forward prediction

Question. For the study data, a representative line is $S=7t+32$, where $t$ is study time in hours and $S$ is predicted score in percent. Estimate a score for $3.5$ hours, to the nearest whole percent.

Plan. Move vertically from $t=3.5$ to the chosen line and horizontally to the score axis; use its equation to check the reading.

The left fit diagram gives a height of about $56.5$. From the rule,

The time lies inside the observed one-to-six-hour range, so this is interpolation.

Answer. A predicted score of about $57\%$.

Check. The chosen line has heights $53$ at three hours and $60$ at four. Halfway between those heights is $56.5$, matching the calculation. These are line heights; the actual observations need not equal them.

Worked example 4: a reverse prediction

Question. A representative line for the phone data is $P=196-16a$, with age $a$ in years and predicted price $P$ in pounds. Estimate the age corresponding to $£156$.

Plan. Move horizontally from $P=156$ to the line, then vertically down to age, or rearrange the line equation.

This age is within the observed one-to-six-year range.

Answer. About $2.5$ years, an interpolated estimate.

Check. The line predicts $£164$ at two years and $£148$ at three; $£156$ is halfway between them, so the age estimate is halfway as well. The data do not guarantee that every phone selling for this price has that age.