MW SysArcKnowledge Platform
Menu

Mathematics · Statistics

Linear Regression Calculator

Fit a least-squares line to three paired observations.

Runs locally
Your numbers

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains
Slope2
Intercept0
Prediction at mean x4

Calculation steps

  1. x̄=2; ȳ=4.
  2. Slope=2; intercept=0; line y=2x+0.

Understand Linear regression

One idea, three depths

Choose how deeply to explain Linear regression

Linear regression: Fit a least-squares line to three paired observations.

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Linear regression to answer this question: fit a least-squares line to three paired observations? Enter x₁, y₁, x₂, and 3 other inputs; the calculator shows Slope. For example: (1,2),(2,4),(3,6) fits y=2x. The answer tells you Slope.

Age 15Explain it to a 15-year-oldConnect it to the formula

Least squares chooses the line minimizing summed squared vertical residuals. The rule is slope=Σdx·dy/Σdx²; intercept=ȳ−slope·x̄. Its input values are x₁, y₁, x₂, y₂, x₃, y₃, and the main result is Slope. For example: (1,2),(2,4),(3,6) fits y=2x.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated linear regression relation over the valid real-number domain stated below. The implemented relation is slope=Σdx·dy/Σdx²; intercept=ȳ−slope·x̄, evaluated from x₁, y₁, x₂, y₂, x₃, y₃ to produce Slope. Least squares chooses the line minimizing summed squared vertical residuals. Extrapolating beyond the observed x range may be unreliable.

Inputs and valid domain

  • x₁ must be a finite real number.
  • y₁ must be a finite real number.
  • x₂ must be a finite real number.
  • y₂ must be a finite real number.
  • x₃ must be a finite real number.
  • y₃ must be a finite real number.

Important boundary: Extrapolating beyond the observed x range may be unreliable.

The formula

slope=Σdx·dy/Σdx²; intercept=ȳ−slope·x̄

How the calculator works through it

It substitutes x₁, y₁, x₂, y₂, x₃, y₃ into the formula and exposes every numerical step above. The main output is Slope, accompanied by Intercept, Prediction at mean x.

Read the result correctly

The Slope is the direct answer to “fit a least-squares line to three paired observations.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

(1,2),(2,4),(3,6) fits y=2x.

Where this model stops being reliable

Extrapolating beyond the observed x range may be unreliable.

Learn it by changing one value

Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.

Dictionary terms behind this calculator

Browse the mathematics dictionary
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.

These foundations help you understand why Linear regression works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Linear regression uses slope=Σdx·dy/Σdx²; intercept=ȳ−slope·x̄. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.

    Review this foundation about 4 min

Strong support

  • Averages and representative values

    Representative values help you judge what the Linear regression inputs summarise and what the result can legitimately describe.

    Review this foundation about 5 min

Optional enrichment

  • Spread and measurement variation

    Variation is not always part of the Linear regression formula, but it helps you judge how stable a reported result may be.

    Review this foundation about 6 min
Learn the missing foundationsI already know these — show the code

Mathematics → algorithm → program

Implement this calculation in code

These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.

Algorithm

  1. Read x₁, y₁, x₂, y₂, x₃, y₃.
  2. Evaluate the principal relationship: slope=Σdx·dy/Σdx²; intercept=ȳ−slope·x̄.
  3. Return Slope and check the domain conditions described above.
Python
            from math import *

def linear_regression_three_pairs(v1, v2, v3, v4, v5, v6) -> float:
    return (((((v1 - (((v1 + v3) + v5) / 3.0)) * (v2 - (((v2 + v4) + v6) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v4 - (((v2 + v4) + v6) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v6 - (((v2 + v4) + v6) / 3.0)))) / ((((v1 - (((v1 + v3) + v5) / 3.0)) * (v1 - (((v1 + v3) + v5) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v3 - (((v1 + v3) + v5) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v5 - (((v1 + v3) + v5) / 3.0)))))

assert abs(linear_regression_three_pairs(1, 2, 2, 4, 3, 6) - 2) < 1e-6 * max(1.0, abs(2))
Current calculator valuesUpdates when you change an input above.
C
            #include <assert.h>
#include <math.h>

double linear_regression_three_pairs(double v1, double v2, double v3, double v4, double v5, double v6) {
    return (((((v1 - (((v1 + v3) + v5) / 3.0)) * (v2 - (((v2 + v4) + v6) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v4 - (((v2 + v4) + v6) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v6 - (((v2 + v4) + v6) / 3.0)))) / ((((v1 - (((v1 + v3) + v5) / 3.0)) * (v1 - (((v1 + v3) + v5) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v3 - (((v1 + v3) + v5) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v5 - (((v1 + v3) + v5) / 3.0)))));
}

int main(void) {
    const double expected = 2;
    const double actual = linear_regression_three_pairs(1, 2, 2, 4, 3, 6);
    assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
Current calculator valuesUpdates when you change an input above.
C++
            #include <cassert>
#include <cmath>
#include <numbers>

double linear_regression_three_pairs(double v1, double v2, double v3, double v4, double v5, double v6) {
    return (((((v1 - (((v1 + v3) + v5) / 3.0)) * (v2 - (((v2 + v4) + v6) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v4 - (((v2 + v4) + v6) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v6 - (((v2 + v4) + v6) / 3.0)))) / ((((v1 - (((v1 + v3) + v5) / 3.0)) * (v1 - (((v1 + v3) + v5) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v3 - (((v1 + v3) + v5) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v5 - (((v1 + v3) + v5) / 3.0)))));
}

int main() {
    constexpr double expected = 2;
    const double actual = linear_regression_three_pairs(1, 2, 2, 4, 3, 6);
    assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Current calculator valuesUpdates when you change an input above.
Linux x86-64 assembly

x86-64 NASM · System V ABI · Linux · SSE2 with libm where required

            ; double linear_regression_three_pairs(double v1, double v2, double v3, double v4, double v5, double v6)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global linear_regression_three_pairs
section .text

linear_regression_three_pairs:
    push rbp
    mov rbp, rsp
    sub rsp, 624
    movsd [rbp-8], xmm0
    movsd [rbp-16], xmm1
    movsd [rbp-24], xmm2
    movsd [rbp-32], xmm3
    movsd [rbp-40], xmm4
    movsd [rbp-48], xmm5
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-112], xmm0
    movsd xmm0, [rbp-112]
    addsd xmm0, [rbp-40]
    movsd [rbp-104], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-120], xmm0
    movsd xmm0, [rbp-104]
    divsd xmm0, [rbp-120]
    movsd [rbp-96], xmm0
    movsd xmm0, [rbp-8]
    subsd xmm0, [rbp-96]
    movsd [rbp-88], xmm0
    movsd xmm0, [rbp-16]
    addsd xmm0, [rbp-32]
    movsd [rbp-152], xmm0
    movsd xmm0, [rbp-152]
    addsd xmm0, [rbp-48]
    movsd [rbp-144], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-160], xmm0
    movsd xmm0, [rbp-144]
    divsd xmm0, [rbp-160]
    movsd [rbp-136], xmm0
    movsd xmm0, [rbp-16]
    subsd xmm0, [rbp-136]
    movsd [rbp-128], xmm0
    movsd xmm0, [rbp-88]
    mulsd xmm0, [rbp-128]
    movsd [rbp-80], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-200], xmm0
    movsd xmm0, [rbp-200]
    addsd xmm0, [rbp-40]
    movsd [rbp-192], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-208], xmm0
    movsd xmm0, [rbp-192]
    divsd xmm0, [rbp-208]
    movsd [rbp-184], xmm0
    movsd xmm0, [rbp-24]
    subsd xmm0, [rbp-184]
    movsd [rbp-176], xmm0
    movsd xmm0, [rbp-16]
    addsd xmm0, [rbp-32]
    movsd [rbp-240], xmm0
    movsd xmm0, [rbp-240]
    addsd xmm0, [rbp-48]
    movsd [rbp-232], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-248], xmm0
    movsd xmm0, [rbp-232]
    divsd xmm0, [rbp-248]
    movsd [rbp-224], xmm0
    movsd xmm0, [rbp-32]
    subsd xmm0, [rbp-224]
    movsd [rbp-216], xmm0
    movsd xmm0, [rbp-176]
    mulsd xmm0, [rbp-216]
    movsd [rbp-168], xmm0
    movsd xmm0, [rbp-80]
    addsd xmm0, [rbp-168]
    movsd [rbp-72], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-288], xmm0
    movsd xmm0, [rbp-288]
    addsd xmm0, [rbp-40]
    movsd [rbp-280], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-296], xmm0
    movsd xmm0, [rbp-280]
    divsd xmm0, [rbp-296]
    movsd [rbp-272], xmm0
    movsd xmm0, [rbp-40]
    subsd xmm0, [rbp-272]
    movsd [rbp-264], xmm0
    movsd xmm0, [rbp-16]
    addsd xmm0, [rbp-32]
    movsd [rbp-328], xmm0
    movsd xmm0, [rbp-328]
    addsd xmm0, [rbp-48]
    movsd [rbp-320], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-336], xmm0
    movsd xmm0, [rbp-320]
    divsd xmm0, [rbp-336]
    movsd [rbp-312], xmm0
    movsd xmm0, [rbp-48]
    subsd xmm0, [rbp-312]
    movsd [rbp-304], xmm0
    movsd xmm0, [rbp-264]
    mulsd xmm0, [rbp-304]
    movsd [rbp-256], xmm0
    movsd xmm0, [rbp-72]
    addsd xmm0, [rbp-256]
    movsd [rbp-64], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-392], xmm0
    movsd xmm0, [rbp-392]
    addsd xmm0, [rbp-40]
    movsd [rbp-384], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-400], xmm0
    movsd xmm0, [rbp-384]
    divsd xmm0, [rbp-400]
    movsd [rbp-376], xmm0
    movsd xmm0, [rbp-8]
    subsd xmm0, [rbp-376]
    movsd [rbp-368], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-432], xmm0
    movsd xmm0, [rbp-432]
    addsd xmm0, [rbp-40]
    movsd [rbp-424], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-440], xmm0
    movsd xmm0, [rbp-424]
    divsd xmm0, [rbp-440]
    movsd [rbp-416], xmm0
    movsd xmm0, [rbp-8]
    subsd xmm0, [rbp-416]
    movsd [rbp-408], xmm0
    movsd xmm0, [rbp-368]
    mulsd xmm0, [rbp-408]
    movsd [rbp-360], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-480], xmm0
    movsd xmm0, [rbp-480]
    addsd xmm0, [rbp-40]
    movsd [rbp-472], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-488], xmm0
    movsd xmm0, [rbp-472]
    divsd xmm0, [rbp-488]
    movsd [rbp-464], xmm0
    movsd xmm0, [rbp-24]
    subsd xmm0, [rbp-464]
    movsd [rbp-456], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-520], xmm0
    movsd xmm0, [rbp-520]
    addsd xmm0, [rbp-40]
    movsd [rbp-512], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-528], xmm0
    movsd xmm0, [rbp-512]
    divsd xmm0, [rbp-528]
    movsd [rbp-504], xmm0
    movsd xmm0, [rbp-24]
    subsd xmm0, [rbp-504]
    movsd [rbp-496], xmm0
    movsd xmm0, [rbp-456]
    mulsd xmm0, [rbp-496]
    movsd [rbp-448], xmm0
    movsd xmm0, [rbp-360]
    addsd xmm0, [rbp-448]
    movsd [rbp-352], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-568], xmm0
    movsd xmm0, [rbp-568]
    addsd xmm0, [rbp-40]
    movsd [rbp-560], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-576], xmm0
    movsd xmm0, [rbp-560]
    divsd xmm0, [rbp-576]
    movsd [rbp-552], xmm0
    movsd xmm0, [rbp-40]
    subsd xmm0, [rbp-552]
    movsd [rbp-544], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-24]
    movsd [rbp-608], xmm0
    movsd xmm0, [rbp-608]
    addsd xmm0, [rbp-40]
    movsd [rbp-600], xmm0
    mov rax, 0x4008000000000000
    movq xmm0, rax
    movsd [rbp-616], xmm0
    movsd xmm0, [rbp-600]
    divsd xmm0, [rbp-616]
    movsd [rbp-592], xmm0
    movsd xmm0, [rbp-40]
    subsd xmm0, [rbp-592]
    movsd [rbp-584], xmm0
    movsd xmm0, [rbp-544]
    mulsd xmm0, [rbp-584]
    movsd [rbp-536], xmm0
    movsd xmm0, [rbp-352]
    addsd xmm0, [rbp-536]
    movsd [rbp-344], xmm0
    movsd xmm0, [rbp-64]
    divsd xmm0, [rbp-344]
    movsd [rbp-56], xmm0
    movsd xmm0, [rbp-56]
    leave
    ret
Current calculator valuesUpdates when you change an input above.
MATLAB
            function result = linear_regression_three_pairs(v1, v2, v3, v4, v5, v6)
    result = (((((v1 - (((v1 + v3) + v5) / 3.0)) * (v2 - (((v2 + v4) + v6) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v4 - (((v2 + v4) + v6) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v6 - (((v2 + v4) + v6) / 3.0)))) / ((((v1 - (((v1 + v3) + v5) / 3.0)) * (v1 - (((v1 + v3) + v5) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v3 - (((v1 + v3) + v5) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v5 - (((v1 + v3) + v5) / 3.0)))));
end
Current calculator valuesUpdates when you change an input above.
Wolfram Language
            ClearAll[mwCalculate];
mwCalculate[v1_, v2_, v3_, v4_, v5_, v6_] := (((((v1 - (((v1 + v3) + v5) / 3.0)) * (v2 - (((v2 + v4) + v6) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v4 - (((v2 + v4) + v6) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v6 - (((v2 + v4) + v6) / 3.0)))) / ((((v1 - (((v1 + v3) + v5) / 3.0)) * (v1 - (((v1 + v3) + v5) / 3.0))) + ((v3 - (((v1 + v3) + v5) / 3.0)) * (v3 - (((v1 + v3) + v5) / 3.0)))) + ((v5 - (((v1 + v3) + v5) / 3.0)) * (v5 - (((v1 + v3) + v5) / 3.0)))));
Current calculator valuesUpdates when you change an input above.

Continue in mathematical software

The downloaded file includes your current inputs and first calculated result. It is created locally.

Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.

Build the whole topic

Continue learning and calculating

Browse all mathematics toolsExtended statistics learning stageMathematics dictionarySearch every MW SysArc tool

Keep going

math

Correlation Explained-Variance Percentage

Calculate explained variance percentage from percentage scale and correlation coefficient.

math

Correlation Explained-Variance Percentage: solve correlation coefficient

Rearrange the correlation explained-variance percentage relationship and solve for correlation coefficient.

math

Correlation Explained-Variance Percentage: solve percentage scale

Rearrange the correlation explained-variance percentage relationship and solve for percentage scale.

math

Definite linear integral

Integrate mx+b over a finite interval.

Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations

Standards, reading and academic references

Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.

Introductory Statistics 2e

Read the free OpenStax statistics textbook
Cite this book
APA 7
Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
MLA 9
Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
Chicago author-date
Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.

OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.

Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS

These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.

Organizational author: MW SysArc. Published:. Canonical page:https://math.mwsysarc.com/statistics/linear-regression-three-pairs.

APA 7

MW SysArc. (2026, July 21). Linear Regression Calculator. MW SysArc Tools. https://math.mwsysarc.com/statistics/linear-regression-three-pairs

MLA 9

MW SysArc. “Linear Regression Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/statistics/linear-regression-three-pairs. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Linear Regression Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/statistics/linear-regression-three-pairs.

Harvard

MW SysArc (2026) ‘Linear Regression Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/statistics/linear-regression-three-pairs (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_linear_regression_three_pairs_2026,
  author = {{MW SysArc}},
  title = {Linear Regression Calculator},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/statistics/linear-regression-three-pairs},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Linear Regression Calculator
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/statistics/linear-regression-three-pairs
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Linear regression do?

Fit a least-squares line to three paired observations.

How does the Linear regression work?

The calculator applies slope=Σdx·dy/Σdx²; intercept=ȳ−slope·x̄. Least squares chooses the line minimizing summed squared vertical residuals.

What can I learn from the Linear regression?

It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.

Does MW SysArc receive or store what I enter?

No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.

How should I use the result?

Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.

Last reviewed . Calculations tested .

MW SysArc Certified