Plotting linearised data and the line of best fit

Experimental readings never fit a law exactly. To test a law and estimate its constants, change the readings into the variables \(X\) and \(Y\) of its linear form, plot them on graph paper and draw the straight line that best fits the points.

Why: if the law is right, the points \((X,\,Y)\) lie close to one straight line; small measuring errors scatter them a little either side. A ruled line through the middle of the points averages out these errors.

Method

1. Write the law in the form \(Y=mX+c\) and state what is plotted on each axis.
2. Make a table of \(X\) and \(Y\). Give logarithms to \(3\) decimal places.
3. Include \(X=0\) on the horizontal axis if the intercept is to be read.
4. On the blank sheet (7 large squares across, 9 up), draw the axes about one large square in from the left and bottom edges and name them with the plotted quantities, for example \(\sqrt{x}\) and \(y\).
5. Choose each scale from the range of the data so the points use most of the sheet: 2 cm to 1, 2, 2.5 or 5 units, or one of these multiplied or divided by a power of 10 (for example 10, 0.5 or 0.025). An axis may start at a convenient value other than 0; write that starting value where the axes cross.
6. Plot each point with a small cross.
7. Draw one ruled straight line with the points spread evenly on both sides. Do not join the dots or force the line through the first and last points.
8. For the gradient, choose two points on your line that are far apart, not data points, and draw the gradient triangle.

Example

Readings of \(x\) and \(y\) are thought to follow \(y=a\sqrt{x}+b\). Plot \(Y=y\) against \(X=\sqrt{x}\).

Using the points \((1.5,\,4.55)\) and \((4.5,\,9.05)\) on the line, gradient \(=\dfrac{9.05-4.55}{4.5-1.5}=\dfrac{4.5}{3.0}=1.5\). The line meets the vertical axis at \(2.3\); the vertical axis starts at \(2\), written where the axes cross. Hence \(a\approx1.5\) and \(b\approx2.3\).