Simple Linear Regression (Slope) Equation, Derivation & Rearrangements

Core idea

Overview

In simple linear regression, the slope coefficient represents the expected change in the dependent variable for every one-unit increase in the independent variable. This specific formula calculates the slope by adjusting the Pearson correlation coefficient based on the ratio of the variability in the outcome variable to the variability in the predictor.

When to use: This formula is used when you need to establish a predictive relationship between two continuous variables and have already calculated their correlation and standard deviations. It is appropriate when the relationship between variables is linear and you wish to determine the unstandardized effect size in raw units.

Why it matters: In psychological research, understanding the slope allows clinicians and scientists to quantify the impact of variables, such as how much a patient's anxiety score might decrease for every hour of therapy completed. It provides a more practical, unit-based interpretation of data than correlation alone.

Symbols

Variables

b = Slope (b), r = Correlation, SD_y = SD of Y, SD_x = SD of X

b

Slope (b)

V a r iab l e

r

Correlation

V a r iab l e

S D_{y}

SD of Y

V a r iab l e

S D_{x}

SD of X

V a r iab l e

Walkthrough

Derivation

Formula: Simple Linear Regression (Slope)

Formula for the slope of the line of best fit.

Linear relationship.
Homoscedasticity.

1

Calculate slope b:

Based on the correlation r and the relative spread of y and x scores.

b = r \frac{S D _{y}}{S D _{x}}

Result

b = r \frac{S D _{y}}{S D _{x}}

Source: University Psychology — Statistics

Free formulas

Rearrangements

Solve for $b$

Make b the subject

b = r \frac{S D _{y}}{S D _{x}}

The slope 'b' is given by the correlation 'r' multiplied by the ratio of the standard deviation of Y to the standard deviation of X.

Difficulty: 1/5

Solve for $r$

Make r the subject

r = \frac{b \cdot S D _{x}}{S D _{y}}

The correlation 'r' is calculated by multiplying the slope 'b' by the standard deviation of X and then dividing by the standard deviation of Y.

Difficulty: 2/5

Solve for $S D_{y}$

Make SDy the subject

S D_{y} = \frac{b \cdot S D _{x}}{r}

The standard deviation of Y 'SDy' is found by multiplying the slope 'b' by the standard deviation of X 'SDx' and then dividing by the correlation 'r'.

Difficulty: 2/5

Solve for $S D_{x}$

Make SDx the subject

S D_{x} = \frac{r \cdot S D _{y}}{b}

The standard deviation of X 'SDx' is obtained by multiplying the correlation 'r' by the standard deviation of Y 'SDy' and then dividing by the slope 'b'.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a constant horizontal line because the slope (b) is a fixed value derived from the correlation and standard deviations of the variables. Since the formula calculates a single scalar coefficient, the Y-axis value remains unchanged regardless of the independent variable plotted on the X-axis.

Graph type: constant

Why it behaves this way

Intuition

Imagine a scatterplot of data points, where a straight line is drawn to best fit the overall trend. The slope 'b' represents how steeply this line rises or falls, indicating the average vertical change for every one-unit

b

The estimated average change in the dependent variable (Y) for a one-unit increase in the independent variable (X).

It's the 'steepness' of the best-fit line through the data points, showing how much Y is expected to shift for each unit change in X.

r

Pearson product-moment correlation coefficient, quantifying the strength and direction of the linear relationship between X and Y.

A number between -1 and +1 that tells you how closely X and Y move together in a straight line, and whether they move in the same direction (+) or opposite directions (-).

S D_{y}

The standard deviation of the dependent variable Y, representing the typical amount of variation or spread in Y values.

How much individual Y scores typically deviate from the average Y score.

S D_{x}

The standard deviation of the independent variable X, representing the typical amount of variation or spread in X values.

How much individual X scores typically deviate from the average X score.

Signs and relationships

r: The sign of the slope 'b' is directly determined by the sign of the Pearson correlation coefficient 'r'. If 'r' is positive, 'b' will be positive, indicating that as X increases, Y tends to increase.

Free study cues

Insight

Canonical usage

The slope coefficient 'b' will have units corresponding to the ratio of the units of the dependent variable (Y) to the independent variable (X), while the Pearson correlation coefficient 'r' is dimensionless.

Common confusion

Students often forget that the slope 'b' has units, unlike the correlation coefficient 'r', and misinterpret its magnitude without considering the scales of X and Y.

Dimension note

The Pearson correlation coefficient ('r') is inherently dimensionless, as it is a ratio of covariation to the product of standard deviations.

Unit systems

$b$ [unit of Y]/[unit of X] · The unit of the slope 'b' is determined by the units of the dependent variable (Y) divided by the units of the independent variable (X). For example, if Y is 'score' and X is 'hours', 'b' would be in 'score/hour'.

$r$ dimensionless · The Pearson correlation coefficient is a dimensionless measure of linear association, ranging from -1 to +1.

$S D_{y}$ [unit of Y] · The standard deviation of Y has the same units as the variable Y itself.

$S D_{x}$ [unit of X] · The standard deviation of X has the same units as the variable X itself.

One free problem

Practice Problem

A psychologist finds that the correlation between weekly exercise hours (X) and subjective well-being scores (Y) is 0.50. If the standard deviation of well-being scores is 12 and the standard deviation of exercise hours is 2, calculate the regression slope.

Correlation0.5

SD of Y12

SD of X2

Solve for: $b$

Hint: Divide the standard deviation of Y by the standard deviation of X, then multiply by the correlation.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Predicting exam performance from hours of sleep.

Study smarter

Tips

The sign of the slope (b) will always be the same as the sign of the correlation coefficient (r).
If the standard deviations of both variables are equal, the slope is exactly equal to the correlation coefficient.
Always ensure the predictor (SDx) is the denominator in the fraction to avoid inverse scaling errors.

Avoid these traps

Common Mistakes

Swapping SDx and SDy.
Extrapolating outside the data range.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Formula for the slope of the line of best fit.

This formula is used when you need to establish a predictive relationship between two continuous variables and have already calculated their correlation and standard deviations. It is appropriate when the relationship between variables is linear and you wish to determine the unstandardized effect size in raw units.

In psychological research, understanding the slope allows clinicians and scientists to quantify the impact of variables, such as how much a patient's anxiety score might decrease for every hour of therapy completed. It provides a more practical, unit-based interpretation of data than correlation alone.

Swapping SDx and SDy. Extrapolating outside the data range.

Predicting exam performance from hours of sleep.

The sign of the slope (b) will always be the same as the sign of the correlation coefficient (r). If the standard deviations of both variables are equal, the slope is exactly equal to the correlation coefficient. Always ensure the predictor (SDx) is the denominator in the fraction to avoid inverse scaling errors.

References

Sources

Gravetter, F. J., Wallnau, L. B., Forzano, L. B. (2018). Statistics for the Behavioral Sciences (10th ed.). Cengage Learning.
Wikipedia: Simple linear regression
Field, A. (2018). Discovering statistics using IBM SPSS Statistics (5th ed.). SAGE Publications.
Gravetter, F. J., Wallnau, L. B., Forzano, L. B., & Witnauer, J. E. (2021). Essentials of statistics for the behavioral sciences (10th ed.).
Wikipedia: Pearson correlation coefficient
Wikipedia: Standard deviation
Howell, D. C. (2013). Statistical Methods for Psychology (8th ed.). Wadsworth Cengage Learning.
University Psychology — Statistics

Simple Linear Regression (Slope)

Overview

Variables

Derivation

Calculate slope b:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Pearson's r (Calculation)

Coefficient of Determination (R²)

ANOVA F-Ratio

Frequently Asked Questions

Sources