📊 Statistics & Probability
📉

Lines of Best Fit

Draw an informal line of best fit through a scatter plot to model and predict from the data.

A line of best fit (trend line) is a straight line drawn through a scatter plot that best represents the linear trend in the data. It should have roughly equal numbers of points above and below it. Use it to make predictions — but be cautious about extrapolating far beyond your data.

Drawing an informal line of best fit:

1Look for the general trend in the scatter plot
2Draw a straight line that passes through the middle of the data cloud
3Aim for roughly equal numbers of points above and below the line
4The line does not need to pass through any specific data point
5Use two points on your line to find its equation: calculate slope and y-intercept
y^=mx+b(predicted y for a given x)\hat{y} = mx + b \quad (\text{predicted } y \text{ for a given } x)
🌡️Predicting temperature
A scatter plot shows altitude (x, in thousands of feet) vs. temperature (y, in °F).

Line of best fit: y = −3.5x + 70

At 4,000 ft: y = −3.5(4) + 70 = −14 + 70 = 56°F

This is interpolation (within the data range).
y=−3.5x+70⇒y(4)=−14+70=56∘Fy = -3.5x + 70 \Rightarrow y(4) = -14 + 70 = 56^\circ F

Interpolation (reliable)

Predicting within the range of x-values in your dataExample: data ranges x = 1 to 10, predicting at x = 5

Extrapolation (risky)

Predicting outside the data rangeExample: data ranges x = 1 to 10, predicting at x = 50Results may be unreliable

Always state whether your prediction is an interpolation or an extrapolation.

💡

Remember This!

The slope of the line of best fit has meaning in context. If y = −3.5x + 70 models altitude vs. temperature, the slope −3.5 means temperature drops approximately 3.5°F for every 1,000-foot rise in altitude.

✏️ Try It!

A line of best fit for a data set is y = 2.5x + 10. What does the model predict for x = 6?

Practice with CalcVerse
📉Regression & Best Fit Line
Take Quiz 📝 — 25 Questions