Math formula sheet

Math & Data All Math & Data tools

• Riemann Sum and Integral Explorer

  Watch rectangles converge on an integral, and compare how fast each rule gets there.

  Riemann, Habilitationsschrift (1854)

  $$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x$$

• Derivative and Tangent Line Explorer

  Sweep a tangent along a curve and watch the derivative draw itself underneath, one gradient at a time.

  Differential calculus, Newton and Leibniz

  $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

• Derivative Calculator

  Exact derivatives with every rule named, from the product rule to the chain rule.

  Leibniz (1684), rules for differentiating products and quotients

  $$\frac{d}{dx}\left[u v\right] = u' v + u v'$$

• Standard Deviation Calculator

  Mean, SD, SEM and a t-based confidence interval from pasted measurements.

  Pearson (1894), standard deviation

  $$s = \sqrt{\frac{\sum_i (x_i - \bar{x})^2}{n-1}}$$

• Error Propagation Calculator

  Uncertainty through sums, differences, products, quotients, powers and multipliers, properly rounded.

  First-order propagation of uncertainty, ISO/IEC Guide 98-3 (GUM)

  $$\sigma_f = \sqrt{\sum_i \left(\frac{\partial f}{\partial x_i}\right)^{2}\sigma_{x_i}^{2}}$$

• Percent Error Calculator

  Compare a measurement to the accepted value and see percent, absolute and signed error.

  Definition of relative error

  $$\%\,\text{error} = \frac{|x_{\text{meas}} - x_{\text{true}}|}{|x_{\text{true}}|} \times 100$$

• Unit Converter

  Convert between every unit this site uses, with the whole group shown at once.

  SI Brochure, BIPM

  $$x_{\text{to}} = \frac{x_{\text{from}} f_{\text{from}} + o_{\text{from}} - o_{\text{to}}}{f_{\text{to}}}$$

• 3D Vector and Cross Product Explorer

  See where the cross product points, and why its length is an area.

  Vector analysis, Gibbs and Wilson (1901)

  $$\vec{a} \times \vec{b} = \begin{pmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{pmatrix}, \quad |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$$

• Fourier Series Visualizer

  Harmonics added one at a time and drawn by turning circles, overshoot and all.

  Fourier, Théorie analytique de la chaleur (1822)

  $$f(x) = \frac{4A}{\pi}\sum_{n\,\text{odd}} \frac{1}{n}\sin nx$$

• Significant Figures Calculator

  Count sig figs digit by digit, round to any number of them, and carry them through arithmetic.

  NIST Guide to the SI, SP 811 (2008), sections 7.9 and B.7.1

  $$x_n = \mathrm{round}\left(\frac{x}{10^{k}}\right) \times 10^{k}, \quad k = \mathrm{floor}(\log_{10}|x|) - n + 1$$

• Unit Circle Explorer

  Drag a point round the circle and read sin, cos and tan as lengths, with exact values at the special angles.

  Unit circle definitions, NCERT Class XI Mathematics, chapter 3

  $$(x, y) = (\cos\theta, \sin\theta), \quad \tan\theta = \frac{y}{x}, \quad x^{2} + y^{2} = 1$$

• Linear Regression Calculator

  Least-squares line of best fit from pasted data, with uncertainties on the slope and intercept.

  Method of least squares, Legendre (1805) and Gauss (1809)

  $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^{2} - (\sum x)^{2}}, \quad c = \frac{\sum y - m\sum x}{n}$$

• Percent Difference Calculator

  The gap between two values as a percentage of their average, with the percent change each way.

  Relative percent difference, US EPA SW-846 Chapter One (2014)

  $$\%\,\text{difference} = \frac{|a - b|}{(|a| + |b|)/2} \times 100$$

• Chi-Square Calculator

  Goodness of fit and contingency tables, with expected counts, p-value, critical values and working.

  Pearson (1900), chi-squared test

  $$\chi^{2} = \sum \frac{(O - E)^{2}}{E}$$

• T-Test Calculator

  One-sample, two-sample and paired t-tests, with p-values, a confidence interval and working.

  Student (1908), with Welch (1947) for unequal variances

  $$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, \quad t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_1^{2}/n_1 + s_2^{2}/n_2}}$$

• Spearman’s Rank Correlation Calculator

  Spearman’s rs from paired data, with tied ranks, the ranks table and a p-value.

  Spearman (1904), rank correlation

  $$r_s = 1 - \frac{6\sum d^{2}}{n(n^{2} - 1)}$$

• Scientific Notation Converter

  Scientific, E and engineering notation both ways, with working, plus arithmetic in scientific notation.

  NCERT Class XI Chemistry, section 1.4.1, Scientific Notation

  $$x = a \times 10^{n}, \quad n = \mathrm{floor}(\log_{10}|x|), \quad a = \frac{x}{10^{n}}$$

• Frequency Density and Histogram Calculator

  Frequency density for each class, a histogram where area is frequency, and the grouped mean, median and modal class.

  Freedman, Pisani and Purves, Statistics, 4th ed. (2007), ch. 3

  $$\text{frequency density} = \frac{\text{frequency}}{\text{class width}}$$

• Error Bars and Worst-Fit Line Calculator

  Steepest and shallowest lines through every error bar, with the uncertainties in gradient and intercept they give.

  IB Diploma Physics guide (2014), topic 1.2: uncertainty of gradient and intercepts

  $$\Delta m = \frac{m_{\text{max}} - m_{\text{min}}}{2}, \quad \text{percentage uncertainty} = \frac{\Delta m}{m} \times 100$$

• Sensitivity and Specificity Visualizer

  Move a test’s cut-off and watch sensitivity, specificity and PPV change, on two bell curves and 1,000 people.

  Yerushalmy, Public Health Reports (1947)

  $$\text{sensitivity} = \frac{\text{TP}}{\text{TP} + \text{FN}}, \quad \text{specificity} = \frac{\text{TN}}{\text{TN} + \text{FP}}$$

• Linear Transformation and Eigenvector Visualizer

  Drag two basis vectors to warp the grid and read off the determinant and the eigenvectors.

  Definition of the determinant and the characteristic equation

  $$\det A = ad - bc, \quad \lambda^{2} - (a + d)\lambda + (ad - bc) = 0$$

• Monty Hall Problem Simulator

  Play the game or auto-run thousands of rounds and watch switching settle at 2/3.

  Selvin (1975), The American Statistician

  $$P(\text{switch wins}) = \frac{N - 1}{N}, \quad P(\text{stay wins}) = \frac{1}{N}$$

• Normal Distribution Calculator

  Shade any area under a bell curve, read the z-scores and steps, or fit a curve to your data.

  The standard normal distribution; quantiles by Wichura (1988), Algorithm AS 241

  $$z = \frac{x - \mu}{\sigma}, \quad P(a < X < b) = \Phi(z_b) - \Phi(z_a)$$

• Margin of Error Calculator

  Margin of error and confidence interval for a proportion or a mean, with z* or t*.

  Neyman (1937), confidence intervals; Student (1908), the t distribution

  $$E = z^{*}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}, \quad E = z^{*}\frac{\sigma}{\sqrt{n}}, \quad E = t^{*}\frac{s}{\sqrt{n}}$$

• Sample Size Calculator

  Sample size for a survey proportion or a mean, with the finite population correction.

  Cochran, Sampling Techniques, 3rd edition (1977), chapter 4

  $$n = \frac{z^{*2}\,p(1 - p)}{E^{2}}, \quad n = \left(\frac{z^{*}\sigma}{E}\right)^{2}$$