Linear Regression Calculator

This free linear regression calculator finds the line of best fit for your x and y data, with the slope, intercept, correlation coefficient r, r², p-values, a scatter plot, a residuals table and the full step-by-step working using your own numbers. It follows the TI-84 LinReg(ax+b) format, compares quadratic, exponential, power and logarithmic fits, and shows the exact TI-84 keys for the same data. Results update as you type.

How to Use the Linear Regression Calculator

  1. Enter your data. In X, Y pairs, type one pair per line, such as 3, 5. You can paste two columns straight from Excel or Google Sheets, and a header row like Hours Score is skipped. Prefer two lists? Switch to Separate X and Y lists, just like L1 and L2 on a TI-84.
  2. Read the equation. The calculator shows y = ax + b with the slope a, the intercept b, r and r² in the highlighted boxes.
  3. Predict a value. Type an x in Predict y at x = to get ŷ, plus a 95% confidence interval for the mean and a 95% prediction interval for a single new value. You get a warning if x is outside your data range.
  4. Try other models. Pick quadratic, exponential, power or logarithmic in Model, or press Plot in the comparison table, to draw that curve next to the straight line.
  5. Check, copy or share. Open the residuals table and residual plot to check the fit. Copy results copies a text summary, and Copy share link copies a link that reopens the calculator with your data.

Linear Regression Formulas

Least-squares regression picks the line that makes the sum of the squared vertical distances from the points to the line (the residuals) as small as possible. With n pairs:

Slope: a = Sxy / Sxx = [nΣxy − ΣxΣy] / [nΣx² − (Σx)²]

Intercept: b = ȳ − a·x̄

Correlation coefficient: r = Sxy / √(Sxx · Syy), and the coefficient of determination is r².

Here Sxx = Σx² − (Σx)²/n, Syy = Σy² − (Σy)²/n and Sxy = Σxy − (Σx)(Σy)/n. The standard error of the estimate is s = √[SSE / (n − 2)], the slope’s standard error is s / √Sxx, and the slope t-test uses t = a / SE(a) with n − 2 degrees of freedom.

Worked Example

Take the example data (1, 2), (2, 4), (3, 5), (4, 4), (5, 6), (6, 8), (7, 7). Here n = 7, Σx = 28, Σy = 36, Σx² = 140, Σxy = 168 and Σy² = 210.

  • Sxx = 140 − 28²/7 = 28, and Sxy = 168 − 28 × 36 / 7 = 24.
  • Slope a = 24 / 28 = 0.8571, and the intercept b = 36/7 − 0.8571 × 4 = 1.7143.
  • Syy = 210 − 36²/7 = 24.857, so r = 24 / √(28 × 24.857) = 0.9097 and r² = 0.8276.

The regression line is y = 0.8571x + 1.7143. About 83% of the variation in y is explained by x, and the slope’s p-value is 0.0045, so the relationship is statistically significant at the 5% level. At x = 8, the predicted value is ŷ = 8.5714.

How to Do Linear Regression on a TI-84 Plus CE

  1. Turn on diagnostics once. Press 2nd 0 for CATALOG, scroll to DiagnosticOn and press ENTER twice. Without this step the TI-84 hides r and r². On newer TI-84 Plus CE operating systems, you can set STAT DIAGNOSTICS to ON in MODE instead.
  2. Enter the data. Press STAT, choose 1:Edit, type the x values in L1 and the y values in L2.
  3. Run LinReg. Press STAT, arrow right to CALC and choose 4:LinReg(ax+b). Set Xlist to L1, Ylist to L2, and Store RegEQ to Y1 (ALPHA TRACE, 1:Y1). Highlight Calculate and press ENTER.
  4. Read the screen. The TI-84 shows a (the slope), b (the intercept), r² and r.
  5. Graph it. Turn on Plot1 under 2nd Y= (STAT PLOT), then press ZOOM 9 for ZoomStat. The line in Y1 is drawn through the scatter plot.

The calculator above repeats these steps with your own numbers and shows what the TI-84 screen should display, so you can check your work. Other models are in the same CALC menu: 5:QuadReg, 9:LnReg, 0:ExpReg and A:PwrReg. 8:LinReg(a+bx) gives the same line with the letters swapped. You can practise on our free TI-84 Plus CE calculator online, and use the standard deviation calculator to describe each variable on its own.

How to Read the Results

  • Slope (a): how much y changes, on average, for each 1-unit increase in x.
  • Intercept (b): the predicted y when x = 0. This value only has a real meaning if x = 0 is inside or near your data.
  • r: the strength and direction of the linear relationship, from −1 to 1. Values above about 0.7 or below −0.7 are usually called strong.
  • r²: the share of the variation in y that the line explains. An r² of 0.83 means 83%.
  • p-value: tests whether the true slope is zero. A p-value below 0.05 is usual evidence of a real linear relationship.
  • Residuals: the actual y minus the predicted y. They should scatter randomly around 0. A curve in the residual plot means a straight line is the wrong model. Points more than 2 standard errors away are highlighted as possible outliers.

Linear vs Quadratic, Exponential, Power and Log Regression

Not every relationship is a straight line. Growth that doubles at a steady rate suits exponential regression (y = a·bˣ). Curves that rise and then fall suit quadratic regression (y = ax² + bx + c). Scaling laws suit power regression (y = a·xᵇ), and fast growth that levels off suits logarithmic regression (y = a + b·ln x). The comparison table fits all five TI-84 models at once. Like the TI-84, it fits ExpReg, PwrReg and LnReg on log-transformed data. It also gives an “R² on y” for every model, so you can compare them fairly. Pick the model whose shape makes sense for your data, not just the highest R².

Correlation is not causation: a strong r shows that two variables move together, not that one causes the other.

Frequently Asked Questions

It fits the least-squares line y = ax + b through paired x and y data. It gives the slope, intercept, correlation coefficient r and r², so you can describe the relationship and predict y for new x values.

Enter x in L1 and y in L2 with STAT, 1:Edit. Then press STAT, go to CALC, choose 4:LinReg(ax+b), set L1 and L2 and press Calculate. The screen shows a (slope) and b (intercept).

Diagnostics are off. Press 2nd, 0 for CATALOG, scroll to DiagnosticOn and press ENTER twice, then run LinReg again. On newer TI-84 Plus CE models you can also turn on STAT DIAGNOSTICS in MODE.

Nothing in the result. Both give the same line. In ax+b, a is the slope. In a+bx, b is the slope and a is the intercept. This calculator uses the ax+b form shown by option 4.

It depends on the field. In physics r² above 0.95 is common. In social science, 0.3 can be meaningful. Always check the residual plot and the p-value as well as r².

Yes. Copy two columns and paste them into the X, Y pairs box. A header row is skipped automatically. You can also paste each column into the Separate X and Y lists boxes.

Ti84CalcPro is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.