How Measurement Units Change the Slope

Why the Same Relationship Can Have Different Slopes

A regression slope describes the predicted change in a response for a one-unit increase in an explanatory variable. But a “one-unit increase” depends on how the variables are measured. One dollar is 100 cents, and one meter is 100 centimeters. The physical quantities have not changed, but the numerical size of one unit has. As a result, a slope’s numerical value changes when measurement units change.

This change is systematic, not a flaw in the regression. The fitted predictions remain the same physical quantities when translated into the new units. To interpret or compare slopes correctly, keep track of both the units and which variable has been converted.

Definition: Suppose $x_{\text{new}}=c x_{\text{old}}$ and $y_{\text{new}}=d y_{\text{old}}$, where $c$ and $d$ are positive unit-conversion factors. If the original least-squares line has slope $b_{\text{old}}$, then the slope after both variables are expressed in the new units is $$ b_{\text{new}}=\frac{d}{c}b_{\text{old}}. $$ The new slope is measured in units of $y_{\text{new}}$ per unit of $x_{\text{new}}$.

Converting the Explanatory Variable

Suppose the response variable stays in the same units, so $d=1$, while the explanatory variable is converted according to $x_{\text{new}}=c x_{\text{old}}$. Then $b_{\text{new}}=b_{\text{old}}/c$. For example, changing an explanatory variable from meters to centimeters multiplies its numerical values by 100, so the slope is divided by 100. A centimeter is a smaller unit than a meter, and a one-centimeter increase is only one hundredth of a meter.

Worked Example: Dollars to Cents for the Explanatory Variable

In a hypothetical data set, let $x$ be a customer’s monthly purchase amount in dollars and $y$ be a predicted monthly account charge in dollars. Suppose the fitted line is $\hat{y}=12+0.08x$. Re-express the purchase amount in cents, leaving the response in dollars, and find the new slope.

Let $x_{\text{cents}}=100x_{\text{dollars}}$. Thus $c=100$, while $d=1$, because the response remains in dollars. The new slope is $$ b_{\text{new}}=\frac{1}{100}(0.08)=0.0008 \quad\text{dollars per cent}. $$

The converted line is $\hat{y}=12+0.0008x_{\text{cents}}$. For a purchase amount of 50 dollars, the original line predicts $12+0.08(50)=16$ dollars. The same amount is 5,000 cents, and the converted line predicts $12+0.0008(5000)=16$ dollars. The slope’s number is smaller because one cent is a smaller explanatory-variable unit than one dollar.

Converting the Response Variable

Now suppose the explanatory variable stays in the same units and the response is converted by $y_{\text{new}}=d y_{\text{old}}$. Then $c=1$, and $b_{\text{new}}=d b_{\text{old}}$. If a response measured in dollars is converted to cents, each response value is multiplied by 100, so the slope is also multiplied by 100. The slope’s units change along with the response.

Worked Example: Dollars to Cents for the Response Variable

In a hypothetical analysis, let $x$ be delivery distance in kilometers and $y$ be a delivery fee in dollars. Suppose the fitted line is $\hat{y}=4.25+1.60x$. Express the response in cents and recalculate the slope.

The response conversion is $y_{\text{cents}}=100y_{\text{dollars}}$, so $d=100$. Distance remains in kilometers, so $c=1$. Therefore, $$ b_{\text{new}}=\frac{100}{1}(1.60)=160 \quad\text{cents per kilometer}. $$

The intercept is also multiplied by 100, giving the converted line $\hat{y}_{\text{cents}}=425+160x$. At 3 kilometers, the original line predicts $4.25+1.60(3)=9.05$ dollars. The converted line predicts $425+160(3)=905$ cents, which is the same predicted fee expressed in cents.

Converting Both Variables

If both variables change units, apply both conversion factors: multiply the old slope by the response conversion factor $d$, then divide by the explanatory conversion factor $c$. The order does not matter, since $d b_{\text{old}}/c$ gives the same result either way. This rule keeps track of two different effects: enlarging the numerical response values enlarges the slope, while enlarging the numerical explanatory values reduces it.

Worked Example: Kilometers and Dollars to Meters and Cents

In a hypothetical delivery analysis, let $x$ be distance in kilometers and $y$ be a fee in dollars. Suppose the fitted line is $\hat{y}=3.50+0.12x$. Convert distance to meters and the fee to cents.

Since $x_{\text{meters}}=1000x_{\text{kilometers}}$, $c=1000$. Since $y_{\text{cents}}=100y_{\text{dollars}}$, $d=100$. The new slope is $$ b_{\text{new}}=\frac{100}{1000}(0.12)=0.012 \quad\text{cents per meter}. $$

The converted line is $\hat{y}_{\text{cents}}=350+0.012x_{\text{meters}}$. For a 2-kilometer delivery, the original line predicts $3.50+0.12(2)=3.74$ dollars. In the new units, 2 kilometers is 2,000 meters, and the converted line predicts $350+0.012(2000)=374$ cents. Since 374 cents is 3.74 dollars, the two fitted predictions agree.

Why the Conversion Rule Works

The rule follows directly from rewriting the original fitted line. Suppose the original line is $\hat{y}_{\text{old}}=a+b_{\text{old}}x_{\text{old}}$. Because $x_{\text{new}}=c x_{\text{old}}$, we have $x_{\text{old}}=x_{\text{new}}/c$. Also, $y_{\text{new}}=d y_{\text{old}}$, so the converted prediction is

\begin{aligned} \hat{y}_{\text{new}} &=d\hat{y}_{\text{old}}\\ &=d\left(a+b_{\text{old}}\frac{x_{\text{new}}}{c}\right)\\ &=da+\frac{d}{c}b_{\text{old}}x_{\text{new}}. \end{aligned}

The coefficient of $x_{\text{new}}$ is the new slope, $(d/c)b_{\text{old}}$. The intercept becomes $da$. The algebra shows that each original fitted prediction is simply expressed in the new units; no physical prediction has been changed.

The same conclusion holds for the least-squares fit to the transformed data, not only for rewriting a line. Least squares chooses the line that minimizes the sum of squared residuals. Under these conversions, each transformed residual is $d$ times its original residual, so the transformed sum of squared residuals is $d^2$ times the original one. Because $c$ and $d$ are nonzero, each candidate line in one set of units corresponds to exactly one candidate line in the other. Multiplying every candidate’s squared-error total by the same positive constant $d^2$ does not change which line minimizes it. Thus the converted slope is the least-squares slope for the converted data.

Unit Conversions That Include an Offset

Not every conversion is a multiplication alone. For example, Fahrenheit temperature is related to Celsius temperature by $x_F=32+(9/5)x_C$. The added 32 shifts the numerical values, while the factor $9/5$ changes their scale. More generally, if $x_{\text{new}}=r+c x_{\text{old}}$ and $y_{\text{new}}=q+d y_{\text{old}}$, then the slope is still $(d/c)b_{\text{old}}$: additive shifts do not change the slope. They do change the intercept, which is the value needed to express the transformed line at $x_{\text{new}}=0$.

For instance, if the response is unchanged and the explanatory variable is converted from Celsius to Fahrenheit, the explanatory scale factor is $c=9/5$. The slope per degree Fahrenheit is therefore $b_C/(9/5)=(5/9)b_C$, where $b_C$ is the slope per degree Celsius. The added 32 does not enter the slope calculation.

Common Mistakes and a Practical Check

A frequent error is to multiply the slope by 100 whenever one variable is converted by a factor of 100. That is only correct when the response is multiplied by 100 and the explanatory variable is unchanged. When the explanatory variable is multiplied by 100, the slope is divided by 100. If both variables are converted, use $d/c$, not just one conversion factor.

Another mistake is to compare slope numbers without their units. A slope of 0.08 dollars per dollar and a slope of 0.0008 dollars per cent can describe the same fitted relationship. Their values differ because the explanatory units differ. Always name the response unit first and the explanatory unit second when stating a slope.

A useful check is to choose a specific physical value of $x$, express it in both unit systems, and compare the resulting fitted responses. After translating the response predictions to the same units, they should agree. If they do not, check whether the response conversion factor belongs in the numerator and the explanatory conversion factor in the denominator.

Key takeaway: If $x_{\text{new}}=c x_{\text{old}}$ and $y_{\text{new}}=d y_{\text{old}}$, then $b_{\text{new}}=(d/c)b_{\text{old}}$. Converting the response multiplies the slope; converting the explanatory variable divides it; converting both does both.

Check Your Understanding

For each question, keep track of which variable is the response and state the units of the resulting slope.

A slope is 2.4 dollars per hour. The response is converted from dollars to cents, while hours are unchanged. What is the new slope?
A slope is 0.6 kilograms per meter. The explanatory variable is converted from meters to centimeters, while kilograms are unchanged. What is the new slope?
A slope is 5 meters per second. The response is converted from meters to centimeters, and the explanatory variable from seconds to milliseconds. What is the new slope?
Why does converting only the response from dollars to cents multiply the slope by 100, while converting only the explanatory variable from dollars to cents divides it by 100?
A conversion adds a constant to every explanatory-variable value but does not change its scale. Does this change the slope? Explain briefly.