How do you calculate the line of best fit by hand?

Find the means x̄ and ȳ. Compute Sxy = Σ(xᵢ−x̄)(yᵢ−ȳ) and Sxx = Σ(xᵢ−x̄)². The slope is b = Sxy/Sxx and the intercept is a = ȳ − b·x̄, so the line is ŷ = a + bx. It always passes through the point (x̄, ȳ). This tool shows every one of those sums in the working panel so you can reproduce the answer on paper.

What is the difference between correlation and linear regression?

Correlation (r) is one number telling you how strongly two variables move together, with no direction of cause. Linear regression gives you the actual predictive line, a slope and intercept, so you can forecast y from x. They are linked: for a simple one-predictor fit, R² equals r², and the regression slope has the same sign as r.

What does the slope of a regression line mean?

The slope b is the average change in y for each one-unit increase in x. If ŷ = 45.8 + 3.2·hours, every extra study hour is associated with about 3.2 more marks. A positive slope trends up, a negative slope trends down, and a slope statistically indistinguishable from zero means x does not linearly predict y.

How do you find the regression equation y = a + bx?

Paste your x values and y values into the two boxes. The calculator computes the slope b = Sxy/Sxx and intercept a = ȳ − b·x̄, then prints the finished equation at the top (for example ŷ = 2.2 + 0.6x). Enter a value in the “Predict y for x” box to get a forecast from that equation.

What is a good R² value for a linear regression?

R² is the share of variance in y explained by x, from 0 to 1. There is no universal cutoff: in physics 0.99 is expected, while in social science or economics 0.3 can be meaningful. Read it alongside the slope p-value, a high R² on very few points can still be non-significant, which this tool flags explicitly.

How many data points do I need?

At least 3 pairs, because the slope t-test needs n − 2 ≥ 1 degree of freedom. With only 3 points the test is very weak (wide confidence interval). More points give a tighter interval and a more trustworthy p-value. The tool rejects fewer than 3 pairs and unequal-length columns with a clear message.

Is the slope significant, and what does the p-value tell me?

The tool tests whether the true slope is zero using t = b/SE(b) with n − 2 degrees of freedom. A two-tailed p-value below 0.05 means the slope is unlikely to be a fluke of your sample, so there is evidence of a real linear relationship. The 95% confidence interval shows the range of slopes consistent with your data.

Is my data uploaded anywhere?

No. Every calculation, the sums, the fit, the p-value, and the plot, runs entirely in your browser using JavaScript. Nothing is sent to a server, there are no accounts, and there is no logging of the numbers you type. You can disconnect from the internet and it still works.

Statistics · Regression

Linear Regression Calculator, Line of Best Fit

Paste two columns of numbers and get the ordinary-least-squares best-fit line ŷ = a + bx, slope, intercept, r, R², standard error, a slope t-test with p-value and 95% confidence interval, a residuals table, and a scatter plot. Free, matched to scipy, and it runs entirely in your browser.

By Induwara Ashinsana· Co-Founder, Ryzera TechnologiesUpdated Jul 10, 2026

Linear regression calculator

X values (predictor)

Numbers separated by commas, spaces, or new lines. Paste two Excel columns here to fill both.

Y values (response)

Must have the same count as X, each X needs a matching Y.

Examples

Predict y for x =

Decimals

Regression equation

ŷ = 2.2000 + 0.6000x

Slope (b)

0.6000

Change in y per +1 in x

Intercept (a)

2.2000

ŷ where x = 0

R² (determination)

0.6000

60.0% of variance explained

Correlation r

0.7746

Moderate fit

Scatter plot

with fitted line ŷ = 2.200 + 0.600x

Solid line is the fit; dashed amber drops are the residuals (yᵢ − ŷᵢ) that least squares minimises.

Slope significance & interval

t = b / SE(b)

2.1213

Degrees of freedom

p-value (two-tailed)

0.1240

At α = 0.05Not significant

Std error of estimate (s)

0.8944

95% CI for slope

[-0.3001, 1.5001]

Cross-check. The deviation-score formula gives slope b = 0.6000; the independent raw-score formula [n·Σxy − Σx·Σy] / [n·Σx² − (Σx)²] gives 0.6000. They reconcile, as they must, and both match scipy.stats.linregress.

Residuals & working

#	xᵢ	yᵢ	ŷᵢ = a + b·xᵢ	residual eᵢ
1	1.0000	2.0000	2.8000	-0.8000
2	2.0000	4.0000	3.4000	0.6000
3	3.0000	5.0000	4.0000	1.0000
4	4.0000	4.0000	4.6000	-0.6000
5	5.0000	5.0000	5.2000	-0.2000
Σ residuals (self-check ≈ 0)				-0.0000

x̄ = 15.0000 / 5 = 3.0000 · ȳ = 20.0000 / 5 = 4.0000

Sxx = 10.0000 · Sxy = 6.0000 · Syy = 6.0000

b = Sxy / Sxx = 6.0000 / 10.0000 = 0.6000

a = ȳ − b·x̄ = 4.0000 − 0.6000·3.0000 = 2.2000

SSE = 2.4000 · SE(b) = 0.2828 · SE(a) = 0.9381

Method: ordinary least squares, b = Σ(xᵢ−x̄)(yᵢ−ȳ)/Σ(xᵢ−x̄)² and a = ȳ − b·x̄; slope test t = b/SE(b) with df = n−2, NIST e-Handbook §4.1.4.1, matched to scipy.stats.linregress. Nothing leaves this page.

How it works

The calculator fits a straight line to your paired data using ordinary least squares (OLS), the method that chooses the slope and intercept minimising the sum of squared vertical distances (residuals) between the points and the line. The formulas below are the canonical simple-regression definitions from the NIST/SEMATECH e-Handbook of Statistical Methods §4.1.4.1.

For n pairs (xᵢ, yᵢ), first take the means x̄ and ȳ, then the sums of squares and cross-products of the deviations:

Sxx = Σ(xᵢ − x̄)²
Syy = Σ(yᵢ − ȳ)²
Sxy = Σ(xᵢ − x̄)(yᵢ − ȳ)

The slope is b = Sxy / Sxx and the intercept is a = ȳ − b·x̄, so the fitted line is ŷ = a + bx and always passes through (x̄, ȳ). The strength of the fit comes from the correlation r = Sxy / √(Sxx·Syy) and R² = r², which is the proportion of the variance in y explained by the line (exact for one-predictor OLS).

To judge whether the slope is real rather than sampling noise, the tool computes the error sum of squares SSE = Syy − b·Sxy, the standard error of the estimate s = √(SSE / (n − 2)), and the standard error of the slope SE(b) = s / √Sxx. The slope test statistic is t = b / SE(b) on df = n − 2 degrees of freedom; its two-tailed p-value uses the regularized incomplete beta identity p = I_x(df/2, ½) with x = df/(df + t²), the same identity behind scipy.stats.linregress. The 95% confidence interval for the slope is b ± t*₀.₉₇₅,df · SE(b), where the critical value comes from the same Student-t distribution.

Every result is cross-checked two ways: the deviation-score slope above is reconciled against the independent raw-score formula b = [n·Σxy − Σx·Σy] / [n·Σx² − (Σx)²], and the residuals are shown summing to zero, a property every correct least-squares fit must satisfy. All of this runs client-side; nothing is uploaded.

Worked examples

Classic five points

ŷ = 2.2 + 0.6x

X = [1, 2, 3, 4, 5], Y = [2, 4, 5, 4, 5]; n = 5
x̄ = 3, ȳ = 4 · Sxy = 6, Sxx = 10, Syy = 6
b = Sxy/Sxx = 6/10 = 0.6
a = ȳ − b·x̄ = 4 − 0.6·3 = 2.2
r = 6/√60 = 0.7746 → R² = 0.6000 (60% explained)
SSE = 6 − 0.6·6 = 2.4 → s = √(2.4/3) = 0.8944
SE(b) = 0.8944/√10 = 0.2828 → t = 0.6/0.2828 = 2.121, df = 3
p = 0.1240 → not significant at α = 0.05
Predict x = 6: ŷ = 2.2 + 0.6·6 = 5.8

Study hours vs exam marks (significant fit)

ŷ = 45.8 + 3.2x

X = [2, 4, 6, 8, 10] hours, Y = [52, 58, 67, 70, 78] marks; n = 5
x̄ = 6, ȳ = 65 · Sxy = 128, Sxx = 40, Syy = 416
b = 128/40 = 3.2 → each extra hour ≈ +3.2 marks
a = 65 − 3.2·6 = 45.8
r = 128/√16640 = 0.9923 → R² = 0.9846 (98.5% explained)
SSE = 416 − 3.2·128 = 6.4 → s = √(6.4/3) = 1.4606
SE(b) = 1.4606/√40 = 0.2309 → t = 3.2/0.2309 = 13.86, df = 3
p ≈ 0.0008 → highly significant; 95% CI = [2.465, 3.935]
Predict x = 7 hours: ŷ = 45.8 + 3.2·7 = 68.2 marks

Negative slope (downward trend)

ŷ = 1.6 − 1.7x

X = [−2, −1, 0, 1, 2], Y = [5, 3, 2, 0, −2]; n = 5
x̄ = 0, ȳ = 1.6 · Sxy = −17, Sxx = 10, Syy = 29.2
b = −17/10 = −1.7 (y falls 1.7 per +1 in x)
a = 1.6 − (−1.7)·0 = 1.6
r = −17/√292 = −0.9948 → R² = 0.9897
Residuals (0, −0.3, 0.4, 0.1, −0.2) sum to 0 ✓
t = −1.7/0.1 = −17.0, df = 3 → p ≈ 0.0004, significant

Frequently asked questions

Sources & references

These are stable mathematical definitions, not rates that change. The formulas and worked examples on this page were last cross-checked against the NIST e-Handbook and scipy.stats.linregress on 2026-07-10.

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