Lines of Best Fit

Drawing a line of best fit by eye requires following these rules:

1. Calculate the mean point $(\bar{x}, \bar{y})$: find the mean of all x values and the mean of all y values. Plot this point.
2. Draw a straight line through the mean point that follows the general trend of the data.
3. Balance the points: Roughly half the points should be above the line and half below. The line should not have all points on one side.
4. Extend the line to cover the full range of x values in the dataset.
5. Ignore outliers when drawing, draw the line to fit the main cluster, not to include extreme points.

A line of best fit is drawn through the middle of a scatterplot so that roughly half the points fall above and half below. It must pass through the mean point (x̄, ȳ) and should minimise overall vertical distance from points.

Pause, copy the two line-of-best-fit rules: equal numbers of points above and below the line (or a balance of distances), and the line should pass through the mean point (x̄, ȳ) into your book.

Drawing a line of best fit by eye requires balancing points above and below the line and passing it through (or near) the point of means. Once the line is drawn, its equation y = a + bx is found by reading the y-intercept a directly from the vertical axis, then calculating the gradient b = rise/run using two clearly identified points on the line (not data points).

Once you have drawn the line, you can find its equation $y = mx + b$:

1. Find the y-intercept ($b$): Read where the line crosses the y-axis.
2. Find the gradient ($m$): Choose two clearly readable points on the line (not necessarily data points). Calculate $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.
3. Write the equation using your values of $m$ and $b$: $y = mx + b$.

Example: A line of best fit passes through (0, 30) and (5, 55). Reading the graph: y-intercept = 30, so $b = 30$. Gradient: $m = \dfrac{55-30}{5-0} = \dfrac{25}{5} = 5$. Equation: $y = 5x + 30$.

Read the equation of a line of best fit from its graph by identifying the y-intercept (where line crosses y-axis) and gradient (rise ÷ run between two clearly-read points). Express as y = a + bx.

Pause, copy the two-step equation-reading procedure: (1) read the y-intercept a from where the line crosses the y-axis; (2) calculate gradient b = rise ÷ run using two points on the line (not data points) into your book.

With the equation y = a + bx in hand, a from the y-intercept and b = rise/run, predictions are straightforward: substitute the x-value into the equation and calculate y. The key caveat is range: substituting an x-value inside the data range (interpolation) gives a reliable prediction; substituting outside the range (extrapolation) may be unreliable because the trend may not continue.

Once you have the equation $y = mx + b$, you can predict y for any x value by substituting.

Example: The equation $y = 5x + 30$ relates study hours (x) to score (y). Predict the score for a student who studies 6 hours:

$y = 5(6) + 30 = 30 + 30 = 60$

The predicted score is 60%.

Reading directly from the graph: You can also read a prediction from the graph by drawing a vertical line from x = 6 up to the line of best fit, then reading across to the y-axis. This gives the same result as substituting into the equation.

To make predictions using the equation of a line of best fit, substitute the known x-value into y = a + bx to find the predicted y. Reliability depends on whether the x-value falls within or outside the data range.

Pause, copy the prediction method (substitute x into y = a + bx), the interpolation vs extrapolation distinction, and the rule: predictions within the data range are more reliable than those outside it into your book.

Data: daily temperature °C (x): 18, 22, 25, 28, 32 and ice-cream sales (y): 40, 60, 75, 90, 120.

1. Calculate the mean point $(\bar{x}, \bar{y})$.
2. A line of best fit passes through (18, 35) and (32, 125). Find the equation of this line.
3. Use the equation to predict ice-cream sales on a 27°C day.
4. Explain what the gradient means in this context.