Does changing units change r or the regression slope?

Q: Does changing units change the slope of the regression line?

Yes. The slope is b = r(s_y/s_x), and while r contributes no units, the ratio s_y/s_x carries all of them. If you multiply every x by c and every y by k, the new slope is k/c times the old slope.

By Jude Wallis · Published July 28, 2026

Changing units does not change r, and it does not change r-squared. It does change the slope: convert x from inches to centimeters and the slope is divided by 2.54. The intercept moves only if y is rescaled or shifted, or the x scale is shifted. The predictions stay the same.

AP Statistics: Unit 5 (topics 5.2 Correlation, 5.5 Least-Squares Regression). Correlation is Unit 5 topic 5.2 and the slope and intercept of the least-squares line are topic 5.5 in the Fall 2026 AP Statistics course, where you are expected to know that r has no units while the slope is a rate that carries the units of both variables.

The short answer

Convert a variable to different units and the correlation coefficient $r$ does not move. Neither does $r^2$ . The slope does move. The intercept moves only in some conversions, and a pure rescaling of x is not one of them.

The reason is one line long. ** $r$ is built from z-scores, which have no units. The slope is a rate, and every rate carries units.** Everything below follows from that.

Quantity	Does a unit conversion change it?
$r$	No (see the sign note below)
$r^2$	No
Slope	Yes
Intercept	Only if you rescale or shift y, or the x conversion adds an offset
Predicted values	No, they are the same predictions written in the new units

Students get this wrong in one specific way: they assume that because the slope changed, the relationship got stronger or weaker. It did not. You relabeled the axes. The points sit exactly where they sat before.

Why r ignores a change of units

Write the correlation the way the course defines it, as an average product of z-scores:

r = \frac{1}{n-1} \sum \frac{(x_i - \bar{x})}{s_x} \cdot \frac{(y_i - \bar{y})}{s_y}

Now multiply every $x$ value by 2.54 to go from inches to centimeters. Three things happen at once:

Each value $x_i$ is multiplied by 2.54.
The mean $\bar{x}$ is multiplied by 2.54.
The standard deviation $s_x$ is multiplied by 2.54.

So the deviation $x_i - \bar{x}$ is multiplied by 2.54 and then divided by an $s_x$ that is also 2.54 times bigger. The 2.54s cancel and every z-score comes out identical. A z-score says "this value is 1.3 standard deviations above the mean," and that sentence is true in inches, centimeters, or any other length unit.

Adding a constant is even simpler. Shift every $x$ up by 32 and the mean shifts by 32 too, so the deviation $x_i - \bar{x}$ is untouched and $s_x$ is untouched.

Since $r$ is unchanged, $r^2$ is unchanged, so the sentence "about 81% of the variation in y is explained by the linear relationship with x" reads the same in both unit systems. More on the difference between the two numbers in r vs r-squared.

One piece of fine print. This holds for multiplying by a positive number. Multiply one variable by a negative number, for example by recording elevation as depth below sea level instead of height above it, and $r$ keeps its size but flips its sign. Real unit conversions (inches to centimeters, pounds to kilograms, Celsius to Fahrenheit) all use positive factors, so on the exam the sign holds.

Why the slope has to change

The slope of the least-squares line is

b = r \cdot \frac{s_y}{s_x}

Look at what each piece contributes. $r$ has no units. The ratio $s_y / s_x$ carries all of them, because $s_y$ is in y units and $s_x$ is in x units. That makes the slope a rate: predicted y units per one x unit.

So the moment you redefine what "one x unit" means, the number has to change. If 1 inch equals 2.54 centimeters, then one centimeter is a smaller step than one inch, and a smaller step in height buys less predicted weight. Multiply every $x$ by 2.54 and $s_x$ is multiplied by 2.54, so

b_{\text{new}} = r \cdot \frac{s_y}{2.54 \, s_x} = \frac{b_{\text{old}}}{2.54}

The general version: if you multiply every $x$ by $c$ and every $y$ by $k$ , the new slope is $k/c$ times the old slope. Nothing about the strength of the association entered that calculation, which is exactly why $r$ sat still while the slope moved.

The interpretation sentence has to be relabeled too. "Each additional inch predicts 4.5 more pounds" becomes "each additional centimeter predicts 1.77 more pounds." Same claim, different measuring stick. The full mechanics of fitting the line live in the least-squares regression line.

What happens to the intercept

The intercept comes from $a = \bar{y} - b\bar{x}$ , so it inherits changes from both the slope and the means. Here is the complete rule set. Read $c$ , $k$ , $d$ , and $m$ as whatever numbers your conversion uses.

Change you make	$r$	Slope	Intercept
Multiply every x by $c$	unchanged	old slope divided by $c$	unchanged
Multiply every y by $k$	unchanged	old slope times $k$	old intercept times $k$
Add $d$ to every x	unchanged	unchanged	old intercept minus (slope)( $d$ )
Add $m$ to every y	unchanged	unchanged	old intercept plus $m$

Row one surprises people, so read it twice. Rescaling x alone leaves the intercept exactly where it was. The intercept is the predicted y when $x = 0$ , and zero inches is still zero centimeters, so the prediction at that point cannot move.

The intercept only shifts when you rescale y (which rescales every predicted value, intercept included) or when the conversion adds an offset. Celsius to Fahrenheit does both a multiply and an add, which is why it moves the slope and the intercept together.

For a conversion that combines steps, apply the rows in the order the conversion applies them, and use the current slope when you handle the shift.

It is the same line, with new axis labels

Here is the check that settles any doubt: convert a value, run it through the converted equation, and compare the prediction to the original. It matches every time. Not approximately. Exactly, up to rounding.

That is because a unit conversion is a relabeling of the axes, not a change to the data. The scatterplot has the identical shape. The points sit in the identical relative positions. The residual plot has the identical pattern, so a curved residual plot stays curved and a linear model that failed still fails. See how to interpret residual plots.

A few related quantities, for completeness:

Residuals are in y units, so they are unchanged if you only convert x, and multiplied by $k$ if you convert y.
The standard deviation of the residuals, $s$ , follows the same rule as the residuals.
$r^2$ is unchanged, so the percent of variation explained is identical.
Which point is influential does not change. You can push a point around in the regression influential point interactive to see what does move $r$ .

The exam trap, and the mistakes that cost points

The multiple-choice version reads something like: "The heights were recorded in inches. If the researcher had recorded them in centimeters instead, which of the following would change?" The distractors are always r, r-squared, the slope, and the intercept in some combination.

Work it in this order and it takes ten seconds:

Cross off $r$ and $r^2$ . Standard unit conversions never touch them.
Did the x and y scales change by the same factor? If not, the slope changed. Converting only x, or only y, always moves it; converting both by the same factor, say two lengths from inches to centimeters, leaves it alone.
Did y get rescaled or shifted at all, or did the x conversion include an offset like the +32 in Fahrenheit? If yes, the intercept changed. If the conversion was a pure multiply on x only, the intercept did not.

The mistakes that show up in student work:

Assuming the slope going down means the association got weaker. Strength lives in $r$ , and $r$ did not move.
Rescaling y but forgetting the intercept. Converting the response variable multiplies the slope and the intercept by the same factor.
Multiplying the slope by 2.54 when going from inches to centimeters instead of dividing. Check the direction with a sentence: a centimeter is a smaller step, so it should predict a smaller change in y.
Leaving the units off the interpretation. "The slope is 1.77" earns nothing; "each additional centimeter of height predicts an increase of about 1.77 pounds" earns the point.
Forgetting to convert your own answer back. If the question asks for a prediction in pounds, do not report kilograms.

Where this sits in AP Statistics

Correlation is topic 5.2 in Unit 5, Regression Analysis, and the slope and intercept of the least-squares line are topic 5.5. Unit 5 is 10 to 20% of the multiple-choice section.

The course states plainly that $r$ has no units. What the exam tests is whether you can carry that fact through a conversion and still say what happened to the slope, which is Practice 4, Interpret Results. Review the unit at Unit 5: Regression Analysis, and check your own conversions against the regression calculator by entering the same data twice, once in each unit.

Inches to centimeters: r holds, the slope drops

Five people are measured for height x (inches) and weight y (pounds): (60, 110), (62, 120), (64, 130), (66, 150), (68, 140). Find r, the slope, and the intercept. Then convert every height to centimeters using 1 inch = 2.54 cm and find all three again.

Find the means with $n = 5$ . $\bar{x} = (60+62+64+66+68)/5 = 320/5 = 64$ inches and $\bar{y} = (110+120+130+150+140)/5 = 650/5 = 130$ pounds.
List the deviations. For x: -4, -2, 0, 2, 4. For y: -20, -10, 0, 20, 10.
Add the products and the squares: $\sum (x_i - \bar{x})(y_i - \bar{y}) = 80 + 20 + 0 + 40 + 40 = 180$ , $\sum (x_i - \bar{x})^2 = 16+4+0+4+16 = 40$ , and $\sum (y_i - \bar{y})^2 = 400+100+0+400+100 = 1000$ .
Correlation in inches: $r = \frac{180}{\sqrt{40 \cdot 1000}} = \frac{180}{\sqrt{40000}} = \frac{180}{200} = 0.9$ .
Slope in inches: $b = \frac{180}{40} = 4.5$ pounds per inch.
Intercept in inches: $a = \bar{y} - b\bar{x} = 130 - 4.5(64) = 130 - 288 = -158$ , so $\hat{y} = -158 + 4.5x$ .
Now convert. Multiply each height by 2.54: 152.4, 157.48, 162.56, 167.64, 172.72 centimeters. The new mean is $64(2.54) = 162.56$ cm and every deviation is 2.54 times its old value: -10.16, -5.08, 0, 5.08, 10.16.
Rebuild the sums. $\sum (x_i' - \bar{x}')(y_i - \bar{y}) = 2.54(180) = 457.2$ and $\sum (x_i' - \bar{x}')^2 = (2.54)^2(40) = 6.4516(40) = 258.064$ . The y sum is untouched at 1000.
Correlation in centimeters: $r = \frac{457.2}{\sqrt{258.064 \cdot 1000}} = \frac{457.2}{\sqrt{258064}} = \frac{457.2}{508} = 0.9$ . Identical to the inches value, and $r^2 = 0.81$ either way.
Slope in centimeters: $b = \frac{457.2}{258.064} = 1.7717$ pounds per centimeter, which is exactly $4.5 / 2.54$ .
Intercept in centimeters: $a = 130 - \frac{4.5}{2.54}(162.56)$ . Since $162.56 / 2.54 = 64$ , this is $130 - 4.5(64) = 130 - 288 = -158$ . The intercept did not move, because the conversion has no offset.
Check one prediction both ways. At 66 inches: $\hat{y} = -158 + 4.5(66) = 139$ pounds. At the same person's 167.64 cm: $\hat{y} = -158 + \frac{4.5}{2.54}(167.64) = -158 + 4.5(66) = -158 + 297 = 139$ pounds.

In inches: $r = 0.9$ , $\hat{y} = -158 + 4.5x$ . In centimeters: $r = 0.9$ still, $\hat{y} = -158 + 1.7717x$ . The correlation and $r^2 = 0.81$ are unchanged, the slope is divided by 2.54, and the intercept holds at -158 pounds because the conversion multiplies without shifting. Both equations predict 139 pounds for the person measured at 66 inches, or 167.64 cm.

Celsius to Fahrenheit: the slope and the intercept both move

A shop fits the model $\hat{y} = 12 + 3.5x$ , where x is the daily high temperature in degrees Celsius and y is ice cream sales in dollars, with r = 0.85. Rewrite the model with temperature in degrees Fahrenheit and state the new r.

Write the conversion so you can substitute for the old variable. Fahrenheit is $F = 1.8C + 32$ , so Celsius is $C = \frac{F - 32}{1.8}$ .
Substitute into the fitted model: $\hat{y} = 12 + 3.5 \cdot \frac{F - 32}{1.8}$ .
Distribute the 3.5 over the fraction: the coefficient on F is $\frac{3.5}{1.8} = 1.9444$ , and the constant it subtracts is $\frac{3.5(32)}{1.8} = \frac{112}{1.8} = 62.2222$ .
Collect the constants: $12 - 62.2222 = -50.2222$ , so $\hat{y} = -50.2222 + 1.9444F$ .
State r. The conversion multiplied x by the positive number 1.8 and then added 32, and neither operation changes a z-score, so r stays at 0.85 and $r^2$ stays at $0.85^2 = 0.7225$ .
Check a prediction. A high of 20 degrees Celsius is $1.8(20) + 32 = 68$ degrees Fahrenheit. Celsius model: $12 + 3.5(20) = 82$ dollars. Fahrenheit model, keeping the slope unrounded as $3.5/1.8$ : $-50.2222 + \frac{3.5}{1.8}(68) = -50.2222 + 132.2222 = 82$ dollars. The rounded slope 1.9444 gives 81.997, which is 82.00 to the cent.
Sanity check the slope in words. One Celsius degree is 1.8 Fahrenheit degrees, so a Fahrenheit degree should predict less: $3.5 / 1.8 = 1.94$ dollars, which is what you got.

$\hat{y} = -50.22 + 1.94F$ with F in degrees Fahrenheit. The slope fell from 3.50 dollars per degree Celsius to 1.94 dollars per degree Fahrenheit, and the intercept moved from 12 to -50.22 dollars, because this conversion multiplies and shifts. The correlation is still 0.85 and $r^2$ is still 0.7225, and both equations predict 82 dollars on a 20 degree Celsius (68 degree Fahrenheit) day.

Frequently asked questions

Does converting inches to centimeters change the correlation coefficient?

No. Any conversion that multiplies by a positive number and adds a constant leaves $r$ exactly the same, because $r$ is an average of products of z-scores and z-scores have no units. $r^2$ is unchanged for the same reason.

Does changing units change the slope of the regression line?

Yes. The slope is $b = r(s_y/s_x)$ , and while $r$ contributes no units, the ratio $s_y/s_x$ carries all of them. If you multiply every x by $c$ and every y by $k$ , the new slope is $k/c$ times the old slope.

Does the intercept always change when the units change?

No. Rescaling x alone leaves it alone, since zero inches is still zero centimeters. The intercept changes when you rescale y (it gets multiplied by the same factor) or when the x conversion includes an offset, as Celsius to Fahrenheit does with its +32.

Can a unit conversion flip the sign of r?

Only if you multiply one variable by a negative number, such as recording elevation as depth below sea level rather than height above it. Standard unit conversions all use positive factors, so the sign holds. $r^2$ is unchanged in either case.