Hard GCSE Scatter graphs and correlation Questions

Read the plotted points off the scatter graph

Each cross is one of the $9$ days. Reading its $x$-value off the horizontal axis and its $y$-value off the vertical axis gives the pairs above.

Write down the range of the x-values in the data

The smallest $x$-value plotted is $2$ and the largest is $9$. This range is what decides later whether an estimate is interpolation or extrapolation, so it is always worth writing down.

Write down the range of the y-values in the data

The $y$-values run from $5$ to $24$ tens of ice creams.

Decide which way the points slope

As the temperature increases, the number sold increases too: the points climb from bottom left to top right. That is positive correlation.

Write down the two points the line of best fit passes through

The question gives two points on the line: $(0, 5)$ and $(9, 23)$. Everything about the line — its gradient, its equation, every estimate read from it — comes from these two points.

Divide the rise by the run to get the gradient

The gradient is rise divided by run, so $m = 2$. A positive gradient is exactly what positive correlation looks like.

Write down where the line crosses the y-axis

The line passes through $(0, 5)$, so the intercept is $c = 5$.

Write the equation of the line of best fit

With gradient $2$ and intercept $5$ the line of best fit is $y = 2x + 5$. Using the equation is more accurate than squinting at the graph.

Measure how far each point is from the line of best fit

Every other point is within $1$ tens of ice creams of the line, but $(8, 5)$ is $16$ tens of ice creams away from it. That is what makes it an outlier: not that it is the biggest or the smallest, but that it does not follow the trend.

Say what the outlier does to the line of best fit

An outlier drags the line of best fit towards itself and makes the correlation look weaker than it really is, which is why it is worth identifying before drawing any line.

Take the outlier out and look again

With $(8, 5)$ removed, the other $8$ points lie close to $y = 2x + 5$ — that is the line quoted in the question.

Describe the correlation of the remaining points

Without the outlier the remaining points show strong positive correlation, so the outlier was hiding a clear trend.

Decide what to do about the outlier

An outlier may be a recording mistake, or it may be a genuine unusual case. You should say that it does not fit the trend — you cannot simply cross it out because it is inconvenient.

Note the mistake to avoid

Two traps on scatter graphs: a strong correlation never proves that one quantity causes the other, and a line of best fit is only trustworthy inside the range of the data.

Write down the outlier

The point $(8, 5)$ is the one that does not follow the trend, so it is the outlier.

Read the plotted points off the scatter graph

Each cross is one of the $9$ students. Reading its $x$-value off the horizontal axis and its $y$-value off the vertical axis gives the pairs above.

Write down the range of the x-values in the data

The smallest $x$-value plotted is $2$ and the largest is $9$. This range is what decides later whether an estimate is interpolation or extrapolation, so it is always worth writing down.

Write down the range of the y-values in the data

The $y$-values run from $10$ to $26$ marks.

Decide which way the points slope

As the revision time increases, the test score increases too: the points climb from bottom left to top right. That is positive correlation.

Write down the two points the line of best fit passes through

The question gives two points on the line: $(0, 7)$ and $(9, 25)$. Everything about the line — its gradient, its equation, every estimate read from it — comes from these two points.

Divide the rise by the run to get the gradient

The gradient is rise divided by run, so $m = 2$. A positive gradient is exactly what positive correlation looks like.

Write down where the line crosses the y-axis

The line passes through $(0, 7)$, so the intercept is $c = 7$.

Write the equation of the line of best fit

With gradient $2$ and intercept $7$ the line of best fit is $y = 2x + 7$. Using the equation is more accurate than squinting at the graph.

Measure how far each point is from the line of best fit

Every other point is within $1$ marks of the line, but $(4, 26)$ is $11$ marks away from it. That is what makes it an outlier: not that it is the biggest or the smallest, but that it does not follow the trend.

Say what the outlier does to the line of best fit

An outlier drags the line of best fit towards itself and makes the correlation look weaker than it really is, which is why it is worth identifying before drawing any line.

Take the outlier out and look again

With $(4, 26)$ removed, the other $8$ points lie close to $y = 2x + 7$ — that is the line quoted in the question.

Describe the correlation of the remaining points

Without the outlier the remaining points show strong positive correlation, so the outlier was hiding a clear trend.

Decide what to do about the outlier

An outlier may be a recording mistake, or it may be a genuine unusual case. You should say that it does not fit the trend — you cannot simply cross it out because it is inconvenient.

Note the mistake to avoid

Two traps on scatter graphs: a strong correlation never proves that one quantity causes the other, and a line of best fit is only trustworthy inside the range of the data.

Write down the outlier

The point $(4, 26)$ is the one that does not follow the trend, so it is the outlier.

Read the plotted points off the scatter graph

Each cross is one of the $9$ students. Reading its $x$-value off the horizontal axis and its $y$-value off the vertical axis gives the pairs above.

Write down the range of the x-values in the data

The smallest $x$-value plotted is $1$ and the largest is $9$. This range is what decides later whether an estimate is interpolation or extrapolation, so it is always worth writing down.

Write down the range of the y-values in the data

The $y$-values run from $15$ to $26$ marks.

Decide which way the points slope

As the TV time increases, the exam mark decreases: the points fall from top left to bottom right. That is negative correlation.

Decide how strong the trend is

A trend is there, but the points are scattered quite widely either side of the line rather than hugging it, so the correlation is weak.

Put the direction and the strength together

Direction and strength together give the description: weak negative correlation.

Work out the mean point

The mean point is $(\bar{x}, \bar{y}) = (5, 20)$: the mean of all the $x$-values paired with the mean of all the $y$-values. Every correct line of best fit goes through it.

Draw a line of best fit through the points

Draw a straight line that follows the trend, passes through the mean point, and leaves about as many points above it as below it. For this data that line is $y = -x + 25$.

Divide the rise by the run to get the gradient

The gradient is rise divided by run, so $m = -1$. A negative gradient is exactly what negative correlation looks like.

Write down where the line crosses the y-axis

The line passes through $(0, 25)$, so the intercept is $c = 25$.

Write the equation of the line of best fit

With gradient $-1$ and intercept $25$ the line of best fit is $y = -x + 25$. Using the equation is more accurate than squinting at the graph.

Say what the spread about the line means

The wider the points are spread either side of the line of best fit, the weaker the correlation. It is the spread, not the steepness, that decides strong versus weak.

Check whether any point is an outlier

Before describing the correlation, look for a point that sits well away from all the others. Here every point lies in the same band, so there is no outlier to comment on.

Note the mistake to avoid

Two traps on scatter graphs: a strong correlation never proves that one quantity causes the other, and a line of best fit is only trustworthy inside the range of the data.

State the correlation

The scatter graph shows weak negative correlation.

Read the plotted points off the scatter graph

Each cross is one of the $8$ children. Reading its $x$-value off the horizontal axis and its $y$-value off the vertical axis gives the pairs above.

Write down the range of the x-values in the data

The smallest $x$-value plotted is $2$ and the largest is $9$. This range is what decides later whether an estimate is interpolation or extrapolation, so it is always worth writing down.

Write down the range of the y-values in the data

The $y$-values run from $7$ to $24$ marks.

Decide which way the points slope

As the shoe size increases, the reading score increases too: the points climb from bottom left to top right. That is positive correlation.

Decide how strong the trend is

The points lie in a narrow band about a straight line, so the correlation is strong. Strong does not mean steep: it means the points hug the line.

Put the direction and the strength together

Direction and strength together give the description: strong positive correlation.

Work out the mean point

The mean point is $(\bar{x}, \bar{y}) = (5.5, 16)$: the mean of all the $x$-values paired with the mean of all the $y$-values. Every correct line of best fit goes through it.

Draw a line of best fit through the points

Draw a straight line that follows the trend, passes through the mean point, and leaves about as many points above it as below it. For this data that line is $y = 2x + 5$.

Divide the rise by the run to get the gradient

The gradient is rise divided by run, so $m = 2$. A positive gradient is exactly what positive correlation looks like.

Write down where the line crosses the y-axis

The line passes through $(0, 5)$, so the intercept is $c = 5$.

Write the equation of the line of best fit

With gradient $2$ and intercept $5$ the line of best fit is $y = 2x + 5$. Using the equation is more accurate than squinting at the graph.

Recall what correlation can and cannot show

Correlation means two quantities move together. It does not say that one makes the other happen. Only a fair experiment can show cause.

Look for a third factor that could explain both

The age of the child would push up both the shoe size and the reading score. A third factor like that explains the correlation with no need for either quantity to cause the other.

Note the mistake to avoid

Two traps on scatter graphs: a strong correlation never proves that one quantity causes the other, and a line of best fit is only trustworthy inside the range of the data.

State what can and cannot be concluded

Ella is wrong. The graph shows a strong positive correlation, but correlation is not causation: the age of the child is a third factor that would raise both the shoe size and the reading score, which explains the pattern without either one causing the other.

Read the plotted points off the scatter graph

Each cross is one of the $8$ fires. Reading its $x$-value off the horizontal axis and its $y$-value off the vertical axis gives the pairs above.

Write down the range of the x-values in the data

The smallest $x$-value plotted is $2$ and the largest is $9$. This range is what decides later whether an estimate is interpolation or extrapolation, so it is always worth writing down.

Write down the range of the y-values in the data

The $y$-values run from $7$ to $27$ thousand pounds.

Decide which way the points slope

As the number of engines increases, the cost of the damage increases too: the points climb from bottom left to top right. That is positive correlation.

Decide how strong the trend is

The points lie in a narrow band about a straight line, so the correlation is strong. Strong does not mean steep: it means the points hug the line.

Put the direction and the strength together

Direction and strength together give the description: strong positive correlation.

Work out the mean point

The mean point is $(\bar{x}, \bar{y}) = (5.5, 18.5)$: the mean of all the $x$-values paired with the mean of all the $y$-values. Every correct line of best fit goes through it.

Draw a line of best fit through the points

Draw a straight line that follows the trend, passes through the mean point, and leaves about as many points above it as below it. For this data that line is $y = 3x + 2$.

Divide the rise by the run to get the gradient

The gradient is rise divided by run, so $m = 3$. A positive gradient is exactly what positive correlation looks like.

Write down where the line crosses the y-axis

The line passes through $(0, 2)$, so the intercept is $c = 2$.

Write the equation of the line of best fit

With gradient $3$ and intercept $2$ the line of best fit is $y = 3x + 2$. Using the equation is more accurate than squinting at the graph.

Recall what correlation can and cannot show

Correlation means two quantities move together. It does not say that one makes the other happen. Only a fair experiment can show cause.

Look for a third factor that could explain both

The size of the fire would push up both the number of fire engines sent and the cost of the damage. A third factor like that explains the correlation with no need for either quantity to cause the other.

Note the mistake to avoid

Two traps on scatter graphs: a strong correlation never proves that one quantity causes the other, and a line of best fit is only trustworthy inside the range of the data.

State what can and cannot be concluded

Tom is wrong. The graph shows a strong positive correlation, but correlation is not causation: the size of the fire is a third factor that would raise both the number of fire engines sent and the cost of the damage, which explains the pattern without either one causing the other.