Effect of Changing Units on the Regression Line

Same Fitted Relationship, Different Units

In “Interpreting Slope and Intercept Together,” the slope was described as the change in predicted response for a one-unit increase in the predictor, and the intercept as the predicted response at zero. But a “one-unit increase” depends on how the predictor is measured. One hour is 60 minutes; one kilometer is 1,000 meters. When units change, the numerical values of the slope—and sometimes the intercept—change too.

A changed coefficient does not necessarily mean the data show a different relationship. If the same observations are expressed in new units, their positions are rescaled or shifted in a consistent way. The regression equation must use the new units, but its predictions still correspond to the same fitted pattern. This tutorial shows how to convert the equation directly and check that the converted predictions agree.

Key idea: Changing units changes the numerical form of a regression equation. For ordinary unit conversions, it does not change which cases are paired or the underlying fitted relationship between them. The slope’s units and numerical value change; the intercept changes when the response is shifted or when the predictor’s zero point is shifted.

How Unit Conversions Affect the Equation

Start with a regression line \(\hat{y}=a+bx\). Suppose a new predictor value is defined by \(x^*=c+dx\), and a new response value is defined by \(y^*=e+fy\). Here, \(c\) and \(e\) are possible shifts, while \(d\) and \(f\) are scale factors. For the usual unit conversions considered here, \(d\) and \(f\) are positive.

To write the converted line, first solve the predictor conversion for \(x\): \(x=(x^*-c)/d\). Then convert the original predicted response into the new response units. Substituting into the original line gives the equation in the new units:

\hat{y}^* =e+f\left(a+b\frac{x^*-c}{d}\right) =\left(e+fa-\frac{fb c}{d}\right)+\frac{fb}{d}x^*

So the new slope is \(fb/d\), and the new intercept is \(e+fa-fbc/d\). This formula is useful even when a unit conversion includes an offset, such as converting Celsius to Fahrenheit. In more familiar cases, one or both shifts are zero, and the calculation simplifies.

Formula: If \(x^*=c+dx\) and \(y^*=e+fy\), then the converted regression line is \(\hat{y}^*=(e+fa-fbc/d)+(fb/d)x^*\). The slope is multiplied by the response scale factor and divided by the predictor scale factor. The intercept also reflects any shifts in the units.

The formula follows the slope’s units: slope units are response units per predictor unit. If the response is converted to a unit 1,000 times smaller, the slope’s numerical value is multiplied by 1,000. If the predictor is converted to a unit 60 times smaller, the slope’s numerical value is divided by 60. Applying both changes multiplies the slope by the response scale factor and divides it by the predictor scale factor.

The intercept is the predicted response at predictor value zero, as discussed in “Interpreting the Y-Intercept in Context.” If the predictor conversion keeps zero at zero, changing only predictor scale leaves the intercept alone. If the predictor’s zero point shifts, the intercept changes because the converted line’s value at \(x^*=0\) corresponds to a different original \(x\). A shift in the response’s zero point changes the intercept as well.

Convert One Variable at a Time

When only the predictor’s units change, keep the response as it is and substitute the new predictor expression into the original equation. For example, converting hours to minutes means \(x^*=60x\), so \(x=x^*/60\). The slope is divided by 60 because each minute is a smaller predictor unit than an hour. If zero hours and zero minutes refer to the same point, the intercept stays the same.

When only the response’s units change, multiply the entire predicted response by the conversion factor, including both the intercept and the slope term. If a response measured in kilometers is converted to meters, multiply both coefficients by 1,000. The predictor values remain unchanged.

If both variables change units, apply both conversions. One reliable check is to pick a predictor value, calculate the original predicted response, convert both values, and then calculate the prediction from the new equation. The two predicted responses should match after conversion, apart from rounding.

Worked Examples

Worked Example: Converting the Predictor From Hours to Minutes

A fictional model predicts a route’s distance in miles from travel time in hours: \(\widehat{\text{distance}}=2.4+6.2(\text{time})\). Convert the equation so that the predictor is travel time in minutes.

Let \(x\) represent time in hours and \(x^*\) represent time in minutes. Since \(x^*=60x\), we have \(x=x^*/60\). Substitute that expression into the original equation:

\widehat{\text{distance}} =2.4+6.2\left(\frac{x^*}{60}\right) =2.4+0.1033x^*

The converted slope is \(6.2/60=0.1033\) miles per minute, rounded to four decimal places. The intercept remains 2.4 miles because zero hours and zero minutes both represent the same predictor value. In context, the converted line predicts an increase of about 0.1033 miles for each additional minute of travel time. This is the same slope expressed using a smaller predictor unit than an hour.

Check a travel time of 1.5 hours, or 90 minutes. The original equation predicts:

2.4+6.2(1.5)=11.7\text{ miles}

The converted equation gives \(2.4+0.1033(90)=11.697\) miles, which rounds to 11.7 miles. Using the unrounded converted slope, \(2.4+(6.2/60)(90)=11.7\) miles exactly. The slight difference using 0.1033 is due to rounding; both equations give the same prediction before rounding.

Worked Example: Converting the Response From Kilometers to Meters

A fictional line predicts a cycling route’s distance in kilometers from a rider’s time in hours: \(\widehat{\text{distance}}=1.8+0.42(\text{time})\). Write the line with distance in meters instead.

Let \(y\) be distance in kilometers and \(y^*\) be distance in meters. Then \(y^*=1000y\). Multiply the entire predicted response by 1,000:

\widehat{\text{distance}}^* =1000\left(1.8+0.42x\right) =1800+420x

The converted line is \(\widehat{\text{distance}}=1800+420(\text{time})\), where distance is in meters and time is in hours. Its slope is \(420\) meters per hour, and its intercept is \(1{,}800\) meters. Both coefficients are 1,000 times their kilometer versions because the response is measured in meters.

At 5 hours, the original equation predicts \(1.8+0.42(5)=3.9\) kilometers. That is \(3{,}900\) meters. The converted equation predicts \(1800+420(5)=3{,}900\) meters. Thus the numerical response changes from 3.9 to 3,900, but the predicted distance is the same quantity in different units.

Worked Example: Converting Both Variables, Including an Offset

A fictional model predicts temperature in degrees Celsius from elapsed time in minutes: \(\widehat{\text{temperature}}=24-0.5(\text{time})\). Convert it to predict temperature in degrees Fahrenheit from elapsed time in seconds.

Let \(x\) be time in minutes and \(x^*\) be time in seconds, so \(x^*=60x\) and \(x=x^*/60\). Let \(y\) be temperature in Celsius and \(y^*\) be temperature in Fahrenheit, so \(y^*=32+1.8y\). Substitute the time conversion into the original line, then convert its predicted temperature:

\hat{y}^* =32+1.8\left(24-0.5\frac{x^*}{60}\right) =75.2-0.015x^*

The new intercept is \(32+1.8(24)=75.2\) degrees Fahrenheit. This is the converted prediction at zero seconds, corresponding to zero minutes. The new slope is \(1.8(-0.5)/60=-0.015\) degrees Fahrenheit per second. It combines the response scale factor, 1.8, with division by 60 for the smaller predictor unit.

Check a time of 8 minutes, which is 480 seconds. The original line predicts \(24-0.5(8)=20\) degrees Celsius. Converting gives \(32+1.8(20)=68\) degrees Fahrenheit. The new equation also predicts \(75.2-0.015(480)=68\) degrees Fahrenheit. Both calculations agree.

The negative slope still describes a decreasing fitted temperature as time increases. The coefficient’s numerical value has changed because the units changed, not because the pattern among the observations changed.

What Changes—and What Stays the Same

For standard conversions with positive scale factors, the slope’s numerical value and units change to match the new measurements. The intercept may change or remain the same depending on whether the conversions shift the response or the predictor’s zero point. The equation must be rewritten so it predicts in the new units.

For the same cases, however, each transformed prediction represents the original prediction converted into the new response units. The residuals are also expressed in the new response units. As covered in “Effect of Changing Units on \(r\),” positive unit conversions leave the correlation unchanged; the coefficient of determination \(R^2\) is unchanged as well. These summaries describe the same linear pattern, not the particular size of a measurement unit.

Key takeaway: Convert the slope according to the response and predictor scale factors, and recalculate the intercept if either conversion shifts a zero point. Check the equation by predicting for a value in the original units and converting that prediction; the new equation should agree.

Common Mistakes and AP Exam Tips

Changing the slope but not its units. A slope of 0.1033 in the first example is miles per minute, not miles per hour. State the new response and predictor units.
Multiplying when you should divide. Converting hours to minutes makes the predictor numbers 60 times larger, so the slope number becomes 60 times smaller. Check the units: miles per minute must be smaller than miles per hour.
Converting only one coefficient when changing the response unit. If the response is multiplied by 1,000, multiply both the intercept and slope by 1,000.
Assuming the intercept never changes. It stays the same when only the predictor scale changes and its zero point remains zero. A shifted predictor origin or a shifted response scale can change it.
Confusing a changed numerical value with a changed relationship. A slope written in meters per hour will be numerically different from the same slope written in kilometers per hour. Compare the units and predictions before concluding the association changed.
Rounding too early. Keep the converted slope unrounded during calculations when possible. If a rounded coefficient creates a tiny mismatch in a prediction check, identify rounding as the reason.
Claiming a causal effect from the converted slope. A unit conversion does not change what the data can establish. As discussed in “Why Correlation Does Not Imply Causation,” an association alone does not show that changing the predictor causes the response to change.

A full-credit explanation gives the converted equation, identifies the units of its slope and intercept, and connects the new coefficients to the conversion. When appropriate, verify the result by checking that a prediction from the converted line matches the converted prediction from the original line.

Check Your Understanding

For each question, track the units as well as the coefficient values.

A line predicts distance in miles from time in hours: \(\hat{y}=3+5x\). Write the equation when time is measured in minutes instead.
A line predicts a chemical solution’s volume in liters from mixing time in minutes: \(\hat{y}=0.6+0.08x\). Convert the response to milliliters. What happens to the slope and intercept?
A line predicts temperature in Celsius from time in hours. If time is converted to minutes but the temperature remains in Celsius, which coefficient changes and why?
Why can the intercept change when converting the predictor from one scale to another that shifts the zero point?
Describe a prediction check you could use to confirm that a converted equation gives the same fitted relationship in new units.