A simple linear regression calculator fits the straight line Y = b₀ + b₁X through paired data by least squares, then reports the slope, the intercept, R squared, the t test statistic and the p value.

The slope tells you how much Y changes for a one unit rise in X. R squared tells you what share of the variation in Y the line explains. The p value tells you whether the slope is far enough from zero to be taken seriously. It does not tell you that X causes Y.

Rule of thumb: with 12 study hour and exam score pairs the line is Ŷ = 42.36 + 3.43X, R squared = 0.89, t(10) = 9.12, p < 0.001. A p value below 0.05 and a 95 percent slope interval that excludes zero mean the relationship is statistically significant.

1.1 Plain English definition

Simple linear regression models the linear relationship between one continuous predictor (X) and one continuous outcome (Y). It estimates a slope and an intercept by ordinary least squares, which means it picks the one line that makes the sum of the squared vertical gaps between the points and the line as small as possible.

It is called simple because there is exactly one predictor. Add a second predictor and it becomes multiple linear regression. It is called linear because the model is a straight line in the parameters, not because your scatter must look perfectly straight.

1.2 What it actually measures

Three things at once. First, the direction and size of the relationship, carried by the slope. Second, the tightness of the relationship, carried by R squared and the residual standard error. Third, the evidence that the relationship is not just sampling noise, carried by the t test statistic and the p value.

1.3 A worked one liner

Twelve students recorded weekly study hours and their exam score. The mean of X is 7.50 hours and the mean of Y is 68.08 points. S_xy = 490.50 and S_xx = 143.00, so the slope is 490.50 ÷ 143.00 = 3.4301 and the intercept is 68.0833 − 3.4301 × 7.5 = 42.3578. The regression line is Ŷ = 42.36 + 3.43X.

1.4 Who uses it, and for what

Students use it in first statistics courses. Educational researchers use it to link study time to attainment. Biologists use it for dose and response, and for length and weight relationships. Economists use it for income and spending. Engineers use it for calibration curves. Health researchers use it to link a single exposure to a single continuous outcome before moving to a multivariable model.

1.5 Simple linear regression vs correlation vs multiple regression

In simple linear regression, R squared is exactly the square of the Pearson correlation, so r = 0.9448 gives R squared = 0.8926.

2.1 Analysis options

Run the analysis above and the numbers in this section fill in automatically with your own results.

4.1 What does the slope mean in simple linear regression?

The slope is the engine of the whole model. It answers one question in one number: if X goes up by one unit, how much does Y move on average? Run the analysis above to see this written out with your own values.

Three details decide whether a slope is worth reporting. The size matters in the units of your outcome, so a slope of 0.02 can be huge for a proportion and trivial for a test score. The sign tells you the direction, and a negative slope is not a weaker result, only an opposite one. The precision comes from the standard error, and a slope of 3.4 with a standard error of 3.0 is barely an estimate at all.

Never read the slope past the range of X you actually measured. If your students studied between 2 and 13 hours, the model has nothing honest to say about 40 hours. That mistake is called extrapolation and it is the single most common misuse of a regression line.

4.2 What does the intercept mean, and when is it meaningful?

The intercept is the predicted Y when X equals zero. Run the analysis to see your own intercept and whether X equal to zero is inside your data range.

An intercept is only a real prediction when X equal to zero is plausible and observed. Predicting the exam score of a student who studies zero hours is defensible if some students really did study zero hours. Predicting the weight of a person of height zero is nonsense. When zero is far outside the data, say plainly in your write up that the intercept is a mathematical anchor rather than a meaningful value. Centring X on its mean fixes this: the intercept then becomes the predicted Y at the average X, which is always interpretable.

4.3 What is a good R squared value in simple linear regression?

R squared is the share of the variation in Y that the line explains. Run the analysis to see yours.

R squared is a proportion between 0 and 1. It is computed as SSR divided by SST, which is the explained sum of squares over the total sum of squares. Multiply by 100 and you have a percentage. The leftover percentage is what the line misses, which includes every variable you did not measure plus pure measurement error.

Four things R squared does not mean. It is not the probability that the model is correct. It is not a measure of whether the model is unbiased, since a badly curved relationship can still produce a high R squared. It does not tell you that X causes Y. And it cannot be compared across datasets with different ranges of X, because widening the range of X inflates R squared without changing the underlying relationship at all.

The residual standard error is often the more useful companion number, because it carries the units of Y. Saying "our predictions are typically off by 4.5 exam points" lands better with a reader than "R squared is 0.89".

4.4 How do you calculate the simple linear regression test statistic t?

The t test statistic is the slope divided by its standard error, tested against the t distribution with n minus 2 degrees of freedom. Run the analysis to see your value.

The null hypothesis is that the true slope is exactly zero, meaning X carries no linear information about Y. A large absolute t means your estimated slope is many standard errors away from zero, which is hard to explain by sampling noise alone. The p value converts that t into a probability: the chance of seeing a slope at least this far from zero if the true slope really were zero.

A p value below your alpha means reject the null. It does not mean the effect is large, important, or causal. It also does not mean the probability that the null is true, which is the single most repeated misreading of a p value in the literature.

In simple linear regression the slope t test, the ANOVA F test and the test of the Pearson correlation are all the same test. You will find t squared equals F, exactly, every time. If they disagree in your output, something has gone wrong with the degrees of freedom.

4.5 How do you read a regression confidence interval?

The confidence interval gives the range of slope values compatible with your data. Run the analysis to see yours.

Read the interval before the p value. If the interval is [2.59, 4.27], every value inside is a defensible estimate of the true slope, and zero is nowhere near it. If the interval is [0.02, 6.40], the p value may still be under 0.05 but the honest conclusion is that you know the direction and almost nothing about the size. A narrow interval far from zero is the strong result. A wide interval that just misses zero is a call for more data, not a failed study.

4.6 How your result compares with published values

Source bands are the conventional ranges reported in the methodological literature listed in Section 22. Compare yourself with your own field, never with 1.00.

4.7 How many data points do you need for simple linear regression?

Sample size does not change the expected slope, but it changes everything about your confidence in it. The standard error of the slope shrinks roughly with the square root of n, so quadrupling your sample halves your interval width. With n = 10 a moderate relationship often fails to reach significance. With n = 200 an almost meaningless relationship will pass easily, which is why a large study should always report the effect size and the interval rather than leaning on the p value.

The spread of X matters just as much as n. S_xx sits in the denominator of the slope standard error, so measuring across a wide range of X buys precision as effectively as adding cases. If you can choose your X values, spread them out.

4.8 What is the difference between correlation and simple linear regression?

4.9 The decision guide

4.10 Common mistakes this calculator prevents

Every template below fills in with your own numbers once you click Run Analysis. Click Copy on the one that matches where you are writing. Before the run they show underscores.

5.1 What a complete regression report contains

A reader who never sees your data should be able to reconstruct your conclusion from the sentence alone. That needs six ingredients: the model test (F with both degrees of freedom), the model fit (R squared), the coefficient (b with its standard error), the coefficient test (t with its degrees of freedom), the exact p value, and the confidence interval. Leave any one of them out and a reviewer will ask for it.

5.2 How do you report simple linear regression in APA format?

5.3 Thesis or dissertation

5.4 Plain language for a non technical reader

5.5 Structured abstract or conference poster

5.6 Replication or pre-registration

5.7 Table and figure conventions

5.8 Final reporting checklist

Seven formulas do all the work in simple linear regression. They are shown in the order a reader needs them: the line first, then the pieces it is built from, then the tests and intervals around it.

Worked example. Load sample dataset 1, leave alpha at 0.05, and click Run. You should see Ŷ = 42.3578 + 3.4301X, R squared = 0.8926, t(10) = 9.1158 and p < 0.001. If you see anything different, one of the two value lists has an extra or a missing number.

Excel handles simple linear regression better than most people expect. Four built in functions give you the whole line in seconds, and the Analysis ToolPak gives you the full table with t statistics and p values. What trips people up is that the direct functions take the ranges in the order Y first, then X, which is the opposite of how you read the equation.

8.1 The functions you need

8.2 Step by step

8.3 The awkward cases in Excel

8.4 Why your Excel answer might differ from R or Python

With clean data, Excel, R, Python and this calculator all agree to at least four decimal places for simple linear regression. Unlike the t-test, there is no competing default here, because there is only one ordinary least squares fit.

8.5 Charting the result in Excel

Select both data columns, then Insert, Scatter, and pick the plain markers option with no connecting lines. Right click any point, choose Add Trendline, pick Linear, and tick both Display Equation on chart and Display R-squared value on chart. Two formatting changes make it publication ready: delete the automatic chart title and the gridlines, and label both axes with their units. For the residual plot, chart column D against column C using the same scatter type, then add a horizontal line at zero.

8.6 Excel errors you will probably hit

In R the whole model is one line, lm(y ~ x), but the one liner is rarely what you should report. The script below fits the model, prints every number a journal will ask for, checks all five assumptions, and saves one publication ready figure at 300 dpi.

No installation is needed. Base R alone does everything here, which is exactly what reviewers expect to see in a methods section.

9.1 The complete script

simple_linear_regression.R

# Simple linear regression calculator, base R only
# Fits Y = b0 + b1*X by ordinary least squares and checks every assumption

set.seed(42)  # reproducible jitter in the figure

# ---- 1. YOUR DATA. Change only these two lines ------------------------
x <- c(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13)   # predictor: study hours
y <- c(54, 52, 53, 52, 63, 71, 70, 75, 79, 76, 90, 82)  # outcome: exam score

# From a CSV instead:
# dat <- read.csv("mydata.csv")
# dat <- dat[complete.cases(dat$hours, dat$score), ]   # drop rows with NA
# x <- dat$hours ; y <- dat$score

xlab <- "Study hours per week"
ylab <- "Exam score"
alpha <- 0.05

# ---- 2. CLEAN AND CHECK ----------------------------------------------
ok <- complete.cases(x, y)
x  <- x[ok]; y <- y[ok]
n  <- length(x)
if (n < 3)      stop("Need at least 3 complete pairs.")
if (var(x) == 0) stop("The predictor does not vary, so no slope exists.")

# ---- 3. FIT THE MODEL -------------------------------------------------
fit  <- lm(y ~ x)
s    <- summary(fit)
ci   <- confint(fit, level = 1 - alpha)
av   <- anova(fit)

b0   <- coef(fit)[1]
b1   <- coef(fit)[2]
se1  <- s$coefficients[2, 2]
tval <- s$coefficients[2, 3]
pval <- s$coefficients[2, 4]
r2   <- s$r.squared
adj  <- s$adj.r.squared
rse  <- s$sigma
fst  <- s$fstatistic[1]
df2  <- n - 2
res  <- resid(fit)
fitv <- fitted(fit)

# ---- 4. ASSUMPTION CHECKS --------------------------------------------
sw   <- shapiro.test(res)                       # normality of residuals
dw   <- sum(diff(res)^2) / sum(res^2)        # Durbin-Watson, want near 2
bp   <- summary(lm(res^2 ~ fitv))$r.squared * n  # Breusch-Pagan LM statistic
bp_p <- pchisq(bp, df = 1, lower.tail = FALSE)
cook <- cooks.distance(fit)
infl <- which(cook > 4 / n)

# ---- 5. PRINTED REPORT -----------------------------------------------
cat("\n================ SIMPLE LINEAR REGRESSION ================\n")
cat(sprintf("n                    : %d\n", n))
cat(sprintf("Equation             : Y = %.4f + %.4f * X\n", b0, b1))
cat(sprintf("Slope b1 (SE)        : %.4f (%.4f)\n", b1, se1))
cat(sprintf("t(%d)                 : %.4f\n", df2, tval))
cat(sprintf("p value              : %.6f\n", pval))
cat(sprintf("%.0f%% CI for slope    : [%.4f, %.4f]\n", 100*(1-alpha), ci[2,1], ci[2,2]))
cat(sprintf("R squared / adjusted : %.4f / %.4f\n", r2, adj))
cat(sprintf("Residual std error   : %.4f on %d df\n", rse, df2))
cat(sprintf("F(1, %d)              : %.4f\n", df2, fst))
cat("---------------- assumption checks ----------------------\n")
cat(sprintf("Shapiro-Wilk p       : %.4f  (want above 0.05)\n", sw$p.value))
cat(sprintf("Durbin-Watson        : %.4f  (want 1.5 to 2.5)\n", dw))
cat(sprintf("Breusch-Pagan p      : %.4f  (want above 0.05)\n", bp_p))
cat(sprintf("Influential points   : %s\n",
    if (length(infl)) paste(infl, collapse = ", ") else "none"))

# ---- 6. PLAIN ENGLISH VERDICT ----------------------------------------
verdict <- if (pval < alpha) {
  sprintf("p is %.5f, below %.2f, so the slope differs significantly from zero. Each extra unit of X predicts a change of %.4f in Y, and the model explains %.1f%% of the variation.",
          pval, alpha, b1, 100 * r2)
} else {
  sprintf("p is %.5f, not below %.2f, so there is not enough evidence that the slope differs from zero.", pval, alpha)
}
cat("\nVERDICT: ", verdict, "\n\n", sep = "")

# ---- 7. ONE 300 dpi FIGURE, FOUR PANELS ------------------------------
png("simple_linear_regression.png", width = 2400, height = 2000, res = 300)
par(mfrow = c(2, 2), mar = c(4.2, 4.2, 2.6, 1))

plot(x, y, pch = 19, col = "#2563eb", xlab = xlab, ylab = ylab,
     main = "1. Scatter with least squares line")
abline(fit, col = "#16a34a", lwd = 3)
legend("topleft", bty = "n", cex = 0.9,
       legend = sprintf("Y = %.2f + %.2f X\nR2 = %.3f", b0, b1, r2))

plot(fitv, res, pch = 19, col = "#9333ea", xlab = "Fitted values",
     ylab = "Residuals", main = "2. Residuals vs fitted")
abline(h = 0, lty = 2, col = "#c2410c", lwd = 2)
lines(lowess(fitv, res), col = "#0d9488", lwd = 2)

qqnorm(res, pch = 19, col = "#0d9488", main = "3. Normal Q-Q of residuals")
qqline(res, col = "#c2410c", lwd = 2)

hist(res, breaks = "FD", col = "#bbf7d0", border = "#15803d",
     xlab = "Residual", main = "4. Histogram of residuals")
abline(v = 0, lty = 2, col = "#c2410c", lwd = 2)

dev.off()
cat("Figure saved as simple_linear_regression.png (300 dpi)\n")
cat("Generated by STATS UNLOCK - statsunlock.com\n")

9.2 What each part does

9.3 What the figure shows

Panel 1 is the result: the points and the fitted line with the equation and R squared printed in the corner. Panel 2 is the honesty check, and the teal smoother should stay flat along zero. If it arches, the relationship is curved and a straight line is the wrong model. Panel 3 puts the residuals against a normal reference line, and a slight wobble at the ends is normal in small samples while a strong S shape is not. Panel 4 should look roughly symmetric and centred on zero.

9.4 Common R problems and fixes

9.5 Useful one-liners

coef(lm(y ~ x))                                  # just the intercept and slope
confint(lm(y ~ x), level = 0.99)                   # 99 percent interval
predict(lm(y ~ x), data.frame(x = 7.5), interval = "prediction")
predict(lm(y ~ x), data.frame(x = 7.5), interval = "confidence")
cor.test(x, y)                                     # same p value as the slope test
summary(lm(y ~ I(x - mean(x))))                    # centred, so the intercept is meaningful
write.csv(data.frame(x, y, fit = fitted(fit), res = resid(fit)), "regression_output.csv")
plot(fit)                                          # the four standard base R diagnostics

Python gives you two honest routes. scipy.stats.linregress returns the slope, intercept, r, p and standard error in a single call, which is perfect for a quick check. For anything you plan to publish, use statsmodels, which prints the full coefficient table, the ANOVA and the confidence intervals in the layout a journal expects.

Install once with pip install numpy scipy matplotlib pandas. Nothing else is required.

10.1 The complete script

regression.py

# Simple linear regression calculator in Python
# Fits Y = b0 + b1*X by ordinary least squares and checks every assumption

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

np.random.seed(42)  # reproducible figure

# ---- 1. YOUR DATA. Change only these two lines ------------------------
x = np.array([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], dtype=float)
y = np.array([54, 52, 53, 52, 63, 71, 70, 75, 79, 76, 90, 82], dtype=float)

# From a CSV instead:
# import pandas as pd
# dat = pd.read_csv("mydata.csv").dropna(subset=["hours", "score"])
# x = dat["hours"].to_numpy(float); y = dat["score"].to_numpy(float)

xlab, ylab = "Study hours per week", "Exam score"
alpha = 0.05

# ---- 2. CLEAN AND CHECK ----------------------------------------------
mask = ~(np.isnan(x) | np.isnan(y))
x, y = x[mask], y[mask]
n = x.size
if n < 3:
    raise ValueError("Need at least 3 complete pairs.")
if np.var(x) == 0:
    raise ValueError("The predictor does not vary, so no slope exists.")

# ---- 3. FIT THE MODEL -------------------------------------------------
res = stats.linregress(x, y)
b1, b0 = res.slope, res.intercept
fitted = b0 + b1 * x
resid = y - fitted

df2 = n - 2
sxx = np.sum((x - x.mean()) ** 2)
sse = np.sum(resid ** 2)
sst = np.sum((y - y.mean()) ** 2)
ssr = sst - sse
mse = sse / df2
rse = np.sqrt(mse)                       # residual standard error
se1 = rse / np.sqrt(sxx)                 # standard error of the slope
se0 = rse * np.sqrt(1 / n + x.mean() ** 2 / sxx)
tval = b1 / se1
pval = 2 * stats.t.sf(abs(tval), df2)
tcrit = stats.t.ppf(1 - alpha / 2, df2)
ci = (b1 - tcrit * se1, b1 + tcrit * se1)
r2 = ssr / sst
adj = 1 - (1 - r2) * (n - 1) / df2
fstat = ssr / mse
pf = stats.f.sf(fstat, 1, df2)

# ---- 4. ASSUMPTION CHECKS --------------------------------------------
sw_p = stats.shapiro(resid).pvalue if 3 <= n <= 5000 else float("nan")
dw = np.sum(np.diff(resid) ** 2) / sse
bp_r2 = stats.linregress(fitted, resid ** 2).rvalue ** 2
bp_p = stats.chi2.sf(n * bp_r2, 1)
lev = 1 / n + (x - x.mean()) ** 2 / sxx
cook = (resid ** 2 / (2 * mse)) * (lev / (1 - lev) ** 2)
infl = np.where(cook > 4 / n)[0] + 1

# ---- 5. PRINTED REPORT -----------------------------------------------
print("\n================ SIMPLE LINEAR REGRESSION ================")
print(f"n                    : {n}")
print(f"Equation             : Y = {b0:.4f} + {b1:.4f} * X")
print(f"Slope b1 (SE)        : {b1:.4f} ({se1:.4f})")
print(f"Intercept b0 (SE)    : {b0:.4f} ({se0:.4f})")
print(f"t({df2})                 : {tval:.4f}")
print(f"p value              : {pval:.6f}")
print(f"{100*(1-alpha):.0f}% CI for slope    : [{ci[0]:.4f}, {ci[1]:.4f}]")
print(f"R squared / adjusted : {r2:.4f} / {adj:.4f}")
print(f"Residual std error   : {rse:.4f} on {df2} df")
print(f"F(1, {df2})              : {fstat:.4f}  (p = {pf:.6f})")
print("---------------- assumption checks ----------------------")
print(f"Shapiro-Wilk p       : {sw_p:.4f}  (want above 0.05)")
print(f"Durbin-Watson        : {dw:.4f}  (want 1.5 to 2.5)")
print(f"Breusch-Pagan p      : {bp_p:.4f}  (want above 0.05)")
print(f"Influential rows     : {list(infl) if infl.size else 'none'}")

# ---- 6. PLAIN ENGLISH VERDICT ----------------------------------------
if pval < alpha:
    verdict = (f"p is {pval:.5f}, below {alpha:.2f}, so the slope differs significantly "
               f"from zero. Each extra unit of X predicts a change of {b1:.4f} in Y, "
               f"and the model explains {100*r2:.1f}% of the variation.")
else:
    verdict = (f"p is {pval:.5f}, not below {alpha:.2f}, so there is not enough "
               f"evidence that the slope differs from zero.")
print("\nVERDICT:", verdict, "\n")

# ---- 7. ONE 300 dpi FIGURE, FOUR PANELS ------------------------------
fig, ax = plt.subplots(2, 2, figsize=(11, 9))

ax[0, 0].scatter(x, y, color="#2563eb", s=45, zorder=3)
xs = np.linspace(x.min(), x.max(), 100)
ax[0, 0].plot(xs, b0 + b1 * xs, color="#16a34a", lw=3)
ax[0, 0].set(xlabel=xlab, ylabel=ylab, title="1. Scatter with least squares line")
ax[0, 0].text(0.04, 0.92, f"Y = {b0:.2f} + {b1:.2f}X\nR2 = {r2:.3f}",
              transform=ax[0, 0].transAxes, va="top", fontsize=10)

ax[0, 1].scatter(fitted, resid, color="#9333ea", s=45)
ax[0, 1].axhline(0, ls="--", color="#c2410c", lw=2)
ax[0, 1].set(xlabel="Fitted values", ylabel="Residuals", title="2. Residuals vs fitted")

stats.probplot(resid, dist="norm", plot=ax[1, 0])
ax[1, 0].set_title("3. Normal Q-Q of residuals")

ax[1, 1].hist(resid, bins="auto", color="#bbf7d0", edgecolor="#15803d")
ax[1, 1].axvline(0, ls="--", color="#c2410c", lw=2)
ax[1, 1].set(xlabel="Residual", ylabel="Frequency", title="4. Histogram of residuals")

fig.tight_layout()
fig.savefig("simple_linear_regression.png", dpi=300, bbox_inches="tight")
print("Figure saved as simple_linear_regression.png (300 dpi)")
print("Generated by STATS UNLOCK - statsunlock.com")

10.2 What each part does

10.3 What the figure shows

The four panels answer four questions in order. Is there a relationship, and how steep is it? Is a straight line the right shape? Are the residuals close enough to normal for the p value to mean what it claims? And is the spread of errors symmetric, or is one long tail doing all the work? Read them in that order, and only trust the p value once panels 2, 3 and 4 look reasonable.

10.4 Common Python problems and fixes

10.5 Useful one-liners

from scipy import stats
stats.linregress(x, y)                          # slope, intercept, r, p, stderr
np.polyfit(x, y, 1)                             # [slope, intercept], no inference
stats.pearsonr(x, y)                            # same p value as the slope test
stats.spearmanr(x, y)                           # rank based fallback when linearity fails

import statsmodels.api as sm
m = sm.OLS(y, sm.add_constant(x)).fit(); print(m.summary())
print(m.conf_int(alpha=0.05))                    # confidence intervals
print(m.get_prediction([1, 7.5]).summary_frame())  # prediction at X = 7.5

Running it. Save the script and run python regression.py from a terminal. In Jupyter, drop the savefig line and add plt.show() instead. Both routes give identical numbers.

11.1 Critical t values for the slope test, df = n − 2

What to conclude: if your absolute t exceeds the tabled value at your alpha, reject the null of zero slope. With n = 12 the 0.05 threshold is 2.228, and the worked example t of 9.116 is far beyond it. Note how slowly the critical value falls after n = 30, which is why the jump from n = 10 to n = 30 buys you far more than the jump from n = 100 to n = 500.

11.2 Critical F values for the model test, df = 1 and n − 2

What to conclude: the last column proves the identity that catches people out. In simple linear regression the F test and the slope t test are the same test, and F is exactly t squared. If your software reports an F whose square root is not your t, one of the two used the wrong degrees of freedom.

11.3 Interpreting R squared by field

What to conclude: R squared above 0.98 in behavioural data usually means the two variables are two measurements of the same thing, not a discovery. Judge against your own field, and always report the residual standard error beside it. Bands follow the conventions summarised by Cohen (1988) and the field surveys listed in Section 22.

11.4 Critical correlation r for significance at α = 0.05, two tailed

What to conclude: the same relationship is significant at n = 100 and not significant at n = 10. That is why a non significant result in a small study is weak evidence of absence, and why a significant result in a huge study needs an effect size before it means anything.

11.5 Sample size needed for 80 percent power at α = 0.05

What to conclude: plan your sample from the smallest effect you would care about, not from the effect you hope to see. If a correlation of 0.20 would still matter in your field, you need roughly 193 cases, and a study of 30 will almost certainly miss it.

11.6 Assumption check thresholds used by this calculator

What to conclude: a caution badge does not invalidate your result, it tells you to mention the issue in the limitations. A fail badge means the p value as printed cannot be defended, and you should either transform the data, switch to a robust or rank based method, or explain very clearly why you proceeded anyway.

11.7 Durbin-Watson quick guide

What to conclude: the Durbin-Watson statistic only makes sense when your rows have a meaningful order, such as time or spatial sequence. For a shuffled cross sectional sample it carries no information, and you should judge independence from the study design instead.

Eight real shaped datasets, each fitted with the same least squares method this calculator uses. They deliberately span the full range of outcomes, from a textbook fit to a model that should not be reported at all. Every figure is drawn from the actual numbers in the table beside it.

Design statement. Simple linear regression assumes one continuous predictor and one continuous outcome measured on the same independent units. Each row is one unit. The predictor may be observed, as in a survey, or set by the researcher, as in a dose response experiment. Only the second design supports a causal claim.

13.1 How to collect data for a simple linear regression calculator

13.2 How to structure your data for regression analysis

The calculator expects exactly two numeric lists of equal length. In a spreadsheet that means two columns and one header row.

The ID column is for your own audit trail and is ignored by the calculator. Do not type units into the value cells, and never merge cells.

13.3 Minimum sample size for a simple linear regression study

Three pairs is the mathematical minimum, because you need at least one residual degree of freedom. It is not a usable minimum. For a moderate relationship, around r = 0.4, you need roughly 46 cases for 80 percent power at alpha 0.05. For a small relationship, around r = 0.2, you need roughly 193. Below n = 20 the confidence interval for the slope is usually so wide that the study cannot settle anything.

13.4 Pre-registration line

Copy this into your pre-registration, then fill in the blanks before you collect any data: "We will fit a simple linear regression of [outcome] on [predictor] using ordinary least squares, testing the slope against zero with a two tailed t test at alpha = 0.05 on n minus 2 degrees of freedom. Our target sample size is [n], chosen for 80 percent power to detect a slope corresponding to r = [r]. If the residual plot shows non constant variance or the Shapiro-Wilk test on residuals gives p below 0.01, we will report the log transformed model as the primary analysis and the untransformed model in a supplement."

13.5 Common mistakes that make the data unusable

This free simple linear regression calculator is designed for anyone who needs one continuous predictor, one continuous outcome and a defensible published style output. Use the checklists below before you run anything.

14.1 Use it when

• You have exactly one predictor and one outcome, both measured on a continuous scale.
• Each row is a different independent unit.
• The scatter plot looks broadly like a straight band, even a noisy one.
• You want a prediction equation, not just a strength of association number.
• You need slope, intercept, R squared, t, p and confidence intervals in one place.
• You need APA or thesis ready wording you can paste straight into a manuscript.

14.2 Do not use it when

• Your outcome is binary, such as pass or fail. Use logistic regression.
• Your outcome is a count, such as number of visits. Use Poisson or negative binomial regression.
• Your outcome is an ordered rating from 1 to 5. Use ordinal regression.
• You have two or more predictors you need to control for. Use multiple linear regression.
• The same units were measured repeatedly over time. Use a mixed model or a time series model.
• The scatter is clearly curved. Use polynomial regression or transform a variable.
• Your predictor is a group label rather than a number. Use a t-test or ANOVA.

14.3 Four real world examples

14.4 Study design guidance

If you can set X yourself, do so, and spread the levels evenly across the widest range that is still realistic. Replicate at each level so you can separate pure error from lack of fit. If you can only observe X, sample deliberately across its range rather than taking whatever is convenient, since a convenience sample clustered in the middle gives you a small S_xx and a wide confidence interval for no extra cost.

14.5 Decision tree: is simple linear regression the right test?

16.1 What are the assumptions of simple linear regression?

1. Linearity. Requires the mean of Y to be a straight line function of X. Check with the scatter plot and the residuals versus fitted plot. If it fails, the slope is a biased summary of a curved relationship, and predictions at the two ends of the X range are systematically wrong in opposite directions.
2. Independence of errors. Requires each observation to carry its own information. Check with the study design first, and with the Durbin-Watson statistic when the rows have a meaningful order. If it fails, standard errors are usually too small, so p values are too small and confidence intervals too narrow.
3. Constant variance, also called homoscedasticity. Requires the spread of residuals to be the same across the fitted range. Check with the residuals versus fitted plot and the Breusch-Pagan test. If it fails, the slope stays unbiased but its standard error is wrong, usually too small at the high variance end, which inflates significance.
4. Approximate normality of residuals. Requires residuals, not raw Y values, to be roughly normal. Check with the Q-Q plot, the histogram and the Shapiro-Wilk test. If it fails at small n, the p value and the confidence interval are unreliable. At large n the central limit theorem largely rescues the inference.
5. No extreme influential points. Requires no single case to control the slope. Check with Cook's distance against the 4 divided by n rule and with leverage. If it fails, the slope reflects one observation rather than the sample, and the direction of the bias is toward that one point.
6. The predictor is measured without substantial error. Requires X to be reliable. Check with the reliability of your instrument. If it fails, the slope is biased toward zero, an effect called regression dilution, so a real relationship looks weaker than it is.

16.2 Limitations

1. It cannot establish causation. Use a randomised experiment, or a design with a credible identification strategy such as instrumental variables or a regression discontinuity.
2. It handles only one predictor. A confounder can create or hide the whole relationship. Use multiple linear regression when other variables matter.
3. It is not robust to outliers. Squaring the residuals gives extreme points enormous weight. Use robust regression, such as an M estimator, or Theil-Sen regression.
4. It cannot model a curve. Use polynomial regression, a spline, or a generalised additive model.
5. It assumes an additive constant effect. If the effect of X depends on a third variable, you need an interaction term, which requires multiple regression.
6. Predictions are only valid inside the observed range of X. There is no statistical method that fixes extrapolation. Collect data across the range you care about.
7. It treats the outcome as unbounded. With a proportion or a percentage the model can predict values below 0 or above 100. Use beta regression or a logit transformation.
8. R squared can be inflated by a wide X range. Two studies of the same relationship can report very different R squared values purely because of sampling. Compare slopes across studies, not R squared.

17.1 What was found

Run the analysis above and a plain summary of your own result appears here, with the equation, the effect size and the p value written out in sentences.

17.2 What it means in practice

This block translates the slope into a practical statement about your own units once you click Run Analysis.

17.3 Cautions

The assumption checks are summarised here after the run, together with what each failure would do to the p value.

17.4 Next steps

Suggested next analyses appear here after the run, chosen from what your own diagnostics show.

17.5 The short version

A simple linear regression calculator turns two columns of numbers into a sentence you can defend. The slope gives the size and direction, the confidence interval gives the precision, R squared and the residual standard error give the fit, and the four diagnostic plots decide whether any of it can be trusted. Report all of them together, keep your predictions inside the range you measured, and be careful never to promote a significant slope into a causal claim your design cannot support. Used that way, the simple linear regression calculator is one of the most reliable tools in applied statistics.

All six links point to live StatsUnlock tools. Each one is the tool you move to when this page tells you simple linear regression is the wrong choice.

Twenty peer reviewed journal articles behind the methods, thresholds and interpretation bands used by this simple linear regression calculator. Every entry has a resolvable DOI.

1. Aiken, L. S., West, S. G., & Millsap, R. E. (2008). Doctoral training in statistics, measurement, and methodology in psychology. American Psychologist, 63(1), 32-50. https://doi.org/10.1037/0003-066X.63.1.32
2. Altman, D. G., & Bland, J. M. (1995). Statistics notes: The normal distribution. BMJ, 310(6975), 298. https://doi.org/10.1136/bmj.310.6975.298
3. Anscombe, F. J. (1973). Graphs in statistical analysis. The American Statistician, 27(1), 17-21. https://doi.org/10.1080/00031305.1973.10478966
4. Belsley, D. A. (1984). Demeaning conditioning diagnostics through centering. The American Statistician, 38(2), 73-77. https://doi.org/10.1080/00031305.1984.10483169
5. Breusch, T. S., & Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica, 47(5), 1287-1294. https://doi.org/10.2307/1911963
6. Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155-159. https://doi.org/10.1037/0033-2909.112.1.155
7. Cook, R. D. (1977). Detection of influential observation in linear regression. Technometrics, 19(1), 15-18. https://doi.org/10.1080/00401706.1977.10489493
8. Cumming, G. (2014). The new statistics: Why and how. Psychological Science, 25(1), 7-29. https://doi.org/10.1177/0956797613504966
9. Durbin, J., & Watson, G. S. (1951). Testing for serial correlation in least squares regression. II. Biometrika, 38(1-2), 159-178. https://doi.org/10.1093/biomet/38.1-2.159
10. Ernst, A. F., & Albers, C. J. (2017). Regression assumptions in clinical psychology research practice. PeerJ, 5, e3323. https://doi.org/10.7717/peerj.3323
11. Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337-350. https://doi.org/10.1007/s10654-016-0149-3
12. Hoekstra, R., Morey, R. D., Rouder, J. N., & Wagenmakers, E.-J. (2014). Robust misinterpretation of confidence intervals. Psychonomic Bulletin & Review, 21(5), 1157-1164. https://doi.org/10.3758/s13423-013-0572-3
13. Kutner, M. H., & Neter, J. (1983). The analysis of covariance structure in regression. Journal of the American Statistical Association, 78(383), 615-620. https://doi.org/10.1080/01621459.1983.10478020
14. Osborne, J. W., & Waters, E. (2002). Four assumptions of multiple regression that researchers should always test. Practical Assessment, Research, and Evaluation, 8(2), 1-5. https://doi.org/10.7275/r222-hv23
15. Royston, P. (1992). Approximating the Shapiro-Wilk W-test for non-normality. Statistics and Computing, 2(3), 117-119. https://doi.org/10.1007/BF01891203
16. Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3-4), 591-611. https://doi.org/10.1093/biomet/52.3-4.591
17. Schielzeth, H. (2010). Simple means to improve the interpretability of regression coefficients. Methods in Ecology and Evolution, 1(2), 103-113. https://doi.org/10.1111/j.2041-210X.2010.00012.x
18. Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129-133. https://doi.org/10.1080/00031305.2016.1154108
19. White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817-838. https://doi.org/10.2307/1912934
20. Zuur, A. F., Ieno, E. N., & Elphick, C. S. (2010). A protocol for data exploration to avoid common statistical problems. Methods in Ecology and Evolution, 1(1), 3-14. https://doi.org/10.1111/j.2041-210X.2009.00001.x