Find the equation of a line fitted to a scatterplot

Which points should be used to calculate the slope of the fitted line?

These synthetic clay samples show mass against length. What overall pattern should a useful straight-line fit follow?

Lower the line towards the middle of the cloud. Then change its slope while keeping its height at length 4 cm near 30 g. Compare both ends and restore a reasonable fit. Several nearby lines can be reasonable by eye

What should guide the placement?

Equation of a fitted line

A fitted line is written with the response variable on the left and the explanatory variable on the right. Use two points that lie on the line to find its slope, then use one of those points to find the intercept

1. 1Keep the response on the left Write the response variable as y and the explanatory variable as x \(\displaystyle \text{RV}=y=a+bx,\quad\text{EV}=x\)
2. 2Use line points Choose two points that are on the fitted line, not just two points in the cloud \(\displaystyle b=\frac{15-7}{6-2}=2\)
3. 3Find the intercept Substitute one line point and the slope to find the intercept \(\displaystyle a=y-bx=7-2\times2=3\)
4. 4Check the equation Substitute an anchor into the equation to check that it gives the anchor's response value \(\displaystyle y=3+2x,\quad 15=3+2\times6\)

Calculate the slope from the two points on the fitted line

Use the point (2, 7) and the slope to find the intercept

A fitted line passes through (2, 7) and (6, 15). Scatterplot observations are (2, 8) and (6, 14). Which calculation gives the slope of the fitted line?

Fit a tendency, not every observation

Use a straight line only when the cloud is approximately linear. Place it through the main pattern, considering points on both sides and their vertical distances

1. 1Check suitability A clear curve may need a different model. A weak pattern gives uncertain predictions
2. 2Investigate unusual cases Check measurement and context. Do not automatically delete a separated observation; explain any decision to model the main group
3. 3Estimate honestly A by-eye fit is approximate. Different reasonable lines can give slightly different predictions

Make and explain your own fit

Use the seven synthetic pairs shown earlier: (1,13), (2,23), (3,24), (4,30), (5,36), (6,37), (7,47), in cm and g. Draw the scatterplot and a reasonable straight-line fit on paper or with graphing software

1. 1Use your own fitted line Choose two well-separated points on your own line, not necessarily observations. Calculate and write its complete equation. Interpret the slope with units and discuss whether its intercept has a supported practical meaning. Check both chosen points in the equation
2. 2Predict and compare Estimate mass at lengths 4 cm and 9 cm. Classify each using the observed length range, and compare a second reasonable fit
3. 3Report limits Explain the pattern, possible unusual cases, uncertainty and why association alone does not establish cause. This is synthetic teaching data, not a representative scientific study
4. 4Review your work Your drawing and written reasoning need self or teacher review; the app does not automatically assess them

Review an example fit, not a unique answer

One reasonable fit is M=10+5L, where L is length in centimetres and M is predicted mass in grams. Two points on this line are (2,20) and (6,40), even though those are not observed pairs. The slope is (40−20)/(6−2)=5 g/cm and the intercept is 10 g. At length 2 cm, 10 plus 5 times 2 gives 20 g; at length 6 cm, 10 plus 5 times 6 gives 40 g. The intercept is the model value at zero length, outside the observed 1–7 cm range; it is not a verified mass of a zero-length sample

1. 1Predict and qualify At L=4, the model predicts 30 g, an interpolation. At L=9, it predicts 55 g, an extrapolation. Report approximate predictions conditional on the trend continuing, not guaranteed observations
2. 2Compare another reasonable fit For example, M=8+5.5L predicts 30 g at 4 cm and 57.5 g at 9 cm. Its vertical departures at the seven observations are −0.5,4,−0.5,0,0.5,−4,0.5 g. Both lines follow the increasing approximately linear cloud. Different reasonable fits can agree at one input yet differ elsewhere; no unique exact by-eye answer is claimed
3. 3Review your complete response Check every ordered pair, labelled axes/units, a sensible window, the placement of your own line through the whole cloud, two on-line points and a consistent equation. Explain unusual values, prediction uncertainty and why this synthetic association does not establish a cause. Do not replace your line by a model solely because the coefficients differ

The displayed fitted line passes through (2,20) and (6,40), with length in cm and mass in g. What mass does it predict at length 4 cm?

Using the displayed fitted line through (2,20) and (6,40), predict the mass at length 9 cm