R-Squared Calculator
Variance explained: R^2 = 1 - SSE/SST with r companion.
Variance explained: R^2 = 1 - SSE/SST with r companion.
- Grade level
- 9–12
- Curriculum band
- Difficulty
- Intermediate
- For this topic
- Time to master
- 2 weeks
- With daily practice
- Tools on page
- 10 on page
- Calculator + practice
Calculator
Pairs — R-Squared Calculator
Result Summary
Results
Headline result
0.727273
Formula output
Least-squares normal equations with rank handling for Spearman/Kendall; display rounds to six decimals.
Statistics
Live readouts from your current pairs.
Headline
0.727273
fitted
Visualizations
Headline vs bounds
Strength (|r|-style)
Formula
Variance explained: R^2 = 1 - SSE/SST with r companion.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| r | Correlation | -1 to 1 | Linear association |
| b, a | Slope, intercept | y/x, y | Line coefficients |
| R^2 | Explained share | 0 to 1 | Fit quality |
Explanation
Least squares minimizes squared misses; correlation standardizes covariance into [-1, 1].
Derivation
Setting SSE partial derivatives to zero yields normal equations; solving gives slope, intercept and plane coefficients.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Parse both lists and check equal lengths.
Center each list at its mean.
Solve the normal equations for coefficients.
Cross-check with the companion identity.
Round for display to six decimals.
State the finished answer.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Run the page defaults once.
Inputs
Calculation
Center, cross-sum, solve
Result
See live result
Interpretation
Confirms the engine path end to end.
Understanding Your Result
Your Result: Headline result: 0.727273
Typical range: |r| > 0.7 strong; R^2 near 1 tight; SEE in y units.
What This Means
The headline number is the fitted coefficient or association for your paired data.
Why It Matters
Prediction, attribution and calibration all run through regression and correlation.
Professional Interpretation
Plot first: curves and outliers break linear summaries silently.
Recommendations
- •Scatter before fitting
- •Compare Pearson vs Spearman
- •Report SEE with R^2
Learn — The Complete Guide
R-Squared Calculator turns paired observations into one calibrated number.
The core concept
Covariation normalized by variation: slope in units, correlation unit-free.
- Scatter first
- Slope has units; r does not
- Residuals judge fits
Derivations — why the formulas work
Why b = Sxy/Sxx
- 1Write SSE(a, b)
- 2Set partial derivatives to zero
- 3Solve: b = Sxy/Sxx, a = my - b mx
Where you meet this in real life
Learn this first — prerequisites in order
- 1Scatter reading
- 2Means
- 3Line equations
Careers that use this daily
- •Analyst — forecasts
- •Scientist — calibration
- •Marketer — drivers
- •Engineer — models
Why it helps
- •Live scatter feedback
- •Moment plus rank views
- •Interval companions
Best practices
- •Plot first
- •Check residuals
- •Beware extrapolation
Limitations — what this tool does not do
- •Linear lens only (mostly)
- •Correlation is not causation
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Algebra | 8–9 | 8.SP | Scatter plots · Lines |
| Statistics | 10–12 | S-ID | Correlation · Least squares |
Timeline — how this idea grew up
1886
Regression named
Galton on heredity.
1896
Pearson r
Correlation formalized.
A little history
- • Gauss-Legendre least squares (1805-09).
- • Spearman (1904) ranked it robust.
Exam-day checklist
- • Plot first
- • Give slope units
- • Check residuals
Myths vs facts
Practical Tips
- •Paste equal-length lists
- •Scatter first
- •Compare robust variants
- •Mind extrapolation
- •Copy with share bar
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
R^2 = 0.85 means?
Limits?
How do I use this calculator?
Does it show steps?
Can I print or share results?
Is my data stored?
Related Calculators
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Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Correlation
- Standardized covariation in [-1, 1].
- Slope
- y change per x unit.
- R-squared
- Explained variance share.
- Residual
- Observed minus fitted.
- SEE
- Typical miss in y units.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
Galton invents regression
Tall parents, less-tall children — regression to the mean (1886).
r is unit-free
Slope carries units; correlation strips them.
Ranks resist
Spearman shrugs at outliers that swing Pearson.
Compare Side by Side
Sign, mirror and strength — sanity views of one fit.
| View | Measure | Value | Note |
|---|---|---|---|
| Headline | 0.727273 | answer | this page |
| Mirror | -0.727273 | negated | sign check |
| Abs | 0.727273 | strength | magnitude |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
R-Squared Calculator: 0.727273
Current inputs — live
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Scatter plots
- ▸Mean and variance
- ▸Straight lines
Mastery progress — 0%
- I scatter first
- I report SEE with R^2
- I never claim causation
- 1
Week 1 · ~15 min/day
Associate then fit
Goals
- Read r honestly
- Fit lines by least squares
- Check residuals
Practice
One dataset daily on this page.
Exam tips
- •Show scatter
- •State units of slope
Practice Problems
0/3 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Guess r for points nearly on a rising line.
If cov = 8 and var(x) = 4, slope?
SSE = 10, SST = 50. R^2?
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| b = Sxy/Sxx | Slope | Rise per run |
| r in [-1, 1] | Correlation | Sign + strength |
| R^2 = 1 - SSE/SST | Fit quality | Share explained |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.