Ordinary Least Squares (OLS)
Simple Linear Regression Calculator
Model the linear relationship between an independent variable (Predictor) and a dependent variable (Criterion) to create a predictive equation.
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Analysis of Variance (ANOVA)
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APA 7th edition reporting statement
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Interpretation Report
Step-by-step analysis & plain-language conclusions
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Hypotheses
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Assumptions check
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Model fit & equation
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All interpretations are generated from your data. Review before use in academic submissions.
Reference · Statistical Theory
How Linear Regression Works
Complete theoretical foundation, assumptions, interpretation of coefficients, and primary references for Ordinary Least Squares Regression.
What is Simple Linear Regression?
Simple linear regression models the relationship between a single independent variable (predictor, $X$) and a continuous dependent variable (criterion, $Y$). It uses the Ordinary Least Squares (OLS) method to find the "line of best fit" that minimizes the sum of squared differences (residuals) between the observed $Y$ values and the values predicted by the model ($\hat{Y}$).
Regression Equation:
Ŷ = bX + a
where:
b = SP_xy / SS_x [Slope]
a = ȳ - b(x̄) [Y-intercept]
SP_xy = Σ(x_i - x̄)(y_i - ȳ)
SS_x = Σ(x_i - x̄)²
Interpreting the Coefficients
- Slope (b) — Represents the estimated change in the dependent variable ($Y$) for every one-unit increase in the independent variable ($X$). If the slope is positive, $Y$ increases as $X$ increases. If negative, $Y$ decreases as $X$ increases.
- Intercept (a) — Represents the estimated value of $Y$ when $X$ equals zero. It anchors the line on the Y-axis. Note: If $X=0$ is theoretically impossible or outside the range of your data, the intercept may simply act as an adjustment constant rather than a meaningful prediction.
- R-squared ($R^2$) — Also known as the Coefficient of Determination. It represents the proportion of variance in the dependent variable that can be explained by the predictor. E.g., $R^2 = 0.45$ means 45% of the variation in $Y$ is explained by $X$.
Assumptions (L.I.N.E.)
- Linearity — The relationship between $X$ and the mean of $Y$ is linear. Check via scatterplot.
- Independence — Observations are independent of each other (no autocorrelation).
- Normality of Residuals — The errors (residuals) are normally distributed at any value of $X$.
- Equal Variance (Homoscedasticity) — The variance of the residuals is constant across all levels of $X$.
APA 7th edition reporting format
Template: "A simple linear regression was calculated to predict [Y] based on [X]. A significant regression equation was found (F(1, [df]) = [F-val], p = [p-val]), with an $R^2$ of .[R2]. Participants' predicted [Y] is equal to [intercept] + [slope] (X). [Y] increased [slope] units for each [unit] of [X]."
Primary references
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum Associates.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Montgomery, D. C., Peck, E. A., & Vining, G. G. (2021). Introduction to linear regression analysis (6th ed.). Wiley.