When the points in a scatter plot cluster around a line, one line can sum up the trend. You can write its equation in slope-intercept form and use it to predict.
Por qué funciona
A line of fit trades many points for one simple rule. Its slope gives the typical change in $y$ for each 1 more in $x$, and its intercept gives a starting value. Because it is drawn to follow all the points, with some above and some below, it describes the trend better than a line through any two particular points.
Idea clave
Draw a line that follows the trend, with about as many points above it as below. Pick two points on the line (they don’t have to be data points), find its equation, and use it to predict. The slope is the average change per unit; the intercept is the predicted value at $x = 0$.
Worked examples
Ejemplo resuelto: Write the equation
A line of fit for the study data passes through $(2, 56)$ and $(7, 81)$. Write its equation.
- Slope: $\frac{81 - 56}{7 - 2} = \frac{25}{5} = 5$.
- Intercept: $b = 56 - 5 \cdot 2 = 46$.
Respuesta: $y = 5x + 46$
Ejemplo resuelto: Predict
Use $y = 5x + 46$ to predict the score of a student who studies 5 hours.
- $y = 5 \cdot 5 + 46 = 71$.
- It is a prediction from the trend; a real student could score a bit higher or lower.
Respuesta: about 71
Ejemplo resuelto: Interpret
What do the 5 and the 46 in $y = 5x + 46$ mean?
- Slope 5: each extra hour of study goes with about 5 more points, on average.
- Intercept 46: the predicted score with 0 hours of study.
Respuesta: about 5 points per hour; a starting value of 46
Error común
Drawing the line through the first and last points. Those two points might be unusual, and the line can end up above or below most of the others. Look at the whole cloud, and balance the points above and below the line.