All Tools

Experimental Data & Uncertainty Analyzer

A general-purpose data analysis tool for science labs. Enter your experimental data, run linear regression, propagate uncertainties, and calculate percent error with significant figures.

Scatter Plot

0.020.040.060.080.10.120.140.161234Extension (m)Force (N)

Data Entry

#Extension (m)Force (N)δ(Force (N))
1
2
3
4
5
6
7
8

Paste CSV or tab-delimited data directly into the table (columns: x, y, and optionally δy).

Regression Analysis

Best-Fit Line

y^=25.1786 x+0.0214\hat{y} = 25.1786\,x + 0.0214
Slope (m)
25.1786
SEm=0.4251\text{SE}_m = 0.4251
m = 25.1786 ± 0.4251
Intercept (b)
0.0214
SEb=0.0429\text{SE}_b = 0.0429
b = 0.0214 ± 0.0429
R²
0.998293
Excellent fit
SSₘₑₛ
0.0182
Sum of squared residuals
SSₜₒₜ
10.6687
Total sum of squares
Data Summary
n = 8
x̄ = 0.0900
ȳ = 2.2875
Degrees of freedom = 6
Formulas Used
m=nΣxy−ΣxΣynΣx2−(Σx)2m = \frac{n\Sigma xy - \Sigma x \Sigma y}{n\Sigma x^2 - (\Sigma x)^2}
b=Σy−mΣxnb = \frac{\Sigma y - m\Sigma x}{n}
R2=1−SSresSStotR^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}
SEm=SSres(n−2)Σ(xi−xˉ)2\text{SE}_m = \sqrt{\frac{SS_{\text{res}}}{(n-2)\Sigma(x_i - \bar{x})^2}}

Residuals Table

#xy (observed)ŷ (predicted)Residual (y − ŷ)
10.020.50.5250-0.0250
20.041.11.0286+0.0714
30.061.51.5321-0.0321
40.0822.0357-0.0357
50.12.62.5393+0.0607
60.1233.0429-0.0429
70.143.53.5464-0.0464
80.164.14.0500+0.0500

Reference Guide

Linear Regression

Least-squares regression fits the best straight line through your data by minimizing the sum of squared residuals.

m=nΣxy−ΣxΣynΣx2−(Σx)2m = \frac{n\Sigma xy - \Sigma x \Sigma y}{n\Sigma x^2 - (\Sigma x)^2}
b=Σy−mΣxnb = \frac{\Sigma y - m\Sigma x}{n}

R-squared and Residuals

R-squared measures how well the model explains the variation in your data. Residuals show the difference between observed and predicted values.

R2=1−SSresSStot=1−Σ(yi−y^i)2Σ(yi−yˉ)2R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}} = 1 - \frac{\Sigma(y_i - \hat{y}_i)^2}{\Sigma(y_i - \bar{y})^2}

An R-squared near 1 indicates a strong linear relationship.

Uncertainty Propagation

When you combine measurements that each have their own uncertainty, the total uncertainty depends on the operation.

δ(A±B)=δA2+δB2\delta(A \pm B) = \sqrt{\delta A^2 + \delta B^2}
δ(A×B)∣A×B∣=(δAA)2+(δBB)2\frac{\delta(A \times B)}{|A \times B|} = \sqrt{\left(\frac{\delta A}{A}\right)^2 + \left(\frac{\delta B}{B}\right)^2}
δ(An)∣An∣=∣n∣⋅δA∣A∣\frac{\delta(A^n)}{|A^n|} = |n| \cdot \frac{\delta A}{|A|}

Percent Error

Percent error compares your experimental result to a known accepted value. It tells you how close your measurement was.

% error=∣experimental−accepted∣∣accepted∣×100%\text{\% error} = \frac{|\text{experimental} - \text{accepted}|}{|\text{accepted}|} \times 100\%

Significant figures in your result should match the least precise measurement used in the calculation.