Correlation and regressionIB MYP Maths Extended: Revision notes

A scatter graph plots two variables for the same items, with one point for each item. Correlation describes the relationship between the variables.

• Positive correlation: as one variable increases, the other tends to increase (points rise from left to right).
• Negative correlation: as one variable increases, the other tends to decrease (points fall from left to right).
• No correlation: no clear pattern. Correlation describes a pattern; it does not prove that one variable causes the other.

The correlation coefficient $r$ is found with technology. It is always between $-1$ and $1$.

• The sign gives the direction: $r>0$ is positive, $r<0$ is negative.
• The size $|r|$ gives the strength: close to $1$ is strong, around $0.5$ is moderate, close to $0$ is weak or none.
• $r=1$ or $r=-1$ means all points lie exactly on a straight line. For example, $r=-0.82$ is a strong negative correlation and $r=0.1$ is a very weak correlation. The strength depends on how close the points are to a line, not on how steep the line is.

The line of best fit is a straight line that follows the trend of the points. Using technology (a graphic display calculator or spreadsheet) enter the data in two lists and ask for linear regression. This gives the equation $y=mx+c$ and $r$. Example: temperature $T$ and drinks sold $n$ give $n=3.02T-43.2$ with $r=0.990$. Interpret the numbers in context. The gradient $m$ is the change in $y$ for each increase of $1$ in $x$: here about 3 more drinks for every extra degree. The intercept $c$ is the value of $y$ when $x=0$, which may have no meaning in practice (here $-43.2$ drinks).

You can use the equation to predict a value.

• Interpolation: predicting a value inside the range of the data you collected. This is usually reliable, especially when the correlation is strong.
• Extrapolation: predicting a value outside the range of the data. This is usually unreliable, because the pattern may not continue. Example: data for $T$ from $24$ to $36$. At $T=30$ the equation gives $47.5$ (reliable). At $T=45$ it gives $92.8$ (unreliable, outside the data). Check for impossible values too: a car price of $-4.6$ thousand AED shows that a model has failed.

When you comment on a prediction, mention these things:

1. Strength of correlation: a strong correlation (close to $\pm1$) makes predictions more reliable.
2. Interpolation or extrapolation: inside the data is reliable, outside is not.
3. Sensible answer: is the predicted value possible? Prices and counts cannot be negative.
4. Amount of data: the more points, the better the line. Write a short conclusion, for example: 'The prediction for 30 °C is reliable because it is within the data and $r=0.990$.'