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Statistics Formulas
Browse 4 reviewed statistics formulas with concrete inputs, outputs, calculations, applicability checks, and LaTeX.
Statistics
Sample Mean
\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_iCalculates the arithmetic average of n observed values.
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Sample Variance
s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2Measures sample spread using squared deviations and Bessel’s correction.
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Standard Error of the Mean
SE_{\bar{x}}=\frac{s}{\sqrt{n}}Estimates the standard deviation of the sample mean’s sampling distribution.
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Z-Score Formula
z=\frac{x-\mu}{\sigma}Expresses how many population standard deviations a value lies from the mean.
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Statistics
Chi-Square Statistic
\chi^2=\sum\frac{(O-E)^2}{E}Compares observed and expected categorical counts.
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Coefficient of Determination
R^2=1-\frac{SS_{res}}{SS_{tot}}Measures the fraction of response variation explained by a regression model relative to a constant-mean baseline.
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Coefficient of Variation
CV=\frac{s}{\bar{x}}\times100\%Expresses sample standard deviation relative to the sample mean.
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Covariance
\operatorname{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]Measures how two variables vary together.
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Covariance Definition
\operatorname{Cov}(X,Y)=\mathbb E[(X-\mu_X)(Y-\mu_Y)]Measures joint linear variation of two random variables.
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Covariance Matrix
\Sigma=\mathbb E[(X-\mu)(X-\mu)^T]Collects variances and pairwise covariances of a random vector.
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Cramér–Rao Lower Bound
\operatorname{Var}(\hat\theta)\ge\frac{1}{I(\theta)}Bounds the variance of an unbiased estimator using Fisher information.
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F Statistic for Two Variances
F=\frac{s_1^2}{s_2^2}Compares two sample variances under an F-distribution model.
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Fisher Information
I(\theta)=\mathbb E\left[\left(\frac{\partial}{\partial\theta}\log p(X;\theta)\right)^2\right]Measures how much an observable random variable tells us about an unknown parameter.
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Geometric Mean
G=\left(\prod_{i=1}^{n}x_i\right)^{1/n}Averages positive values multiplicatively.
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Harmonic Mean
H=\frac{n}{\sum_{i=1}^{n}1/x_i}Averages positive rates or ratios.
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Least-Squares Normal Equation
\hat\beta=(X^TX)^{-1}X^TyComputes ordinary least-squares coefficients when XᵀX is invertible.
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Logistic Regression Probability
p(x)=\frac{1}{1+e^{-(\beta_0+\beta^Tx)}}Maps a linear predictor to a probability between zero and one.
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Logistic Sigmoid
\sigma(z)=\frac{1}{1+e^{-z}}Maps a real-valued score to a number between zero and one.
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Mean Absolute Error
MAE=\frac{1}{n}\sum_{i=1}^{n}|y_i-\hat y_i|Averages absolute prediction errors.
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Mean Confidence Interval
\bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt n}Estimates a population mean when population standard deviation is known.
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Mean Confidence Interval
\bar{x}\pm t_{\alpha/2,n-1}\frac{s}{\sqrt n}Gives a t-based confidence interval for a population mean.
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Mean Squared Error
MSE=\frac{1}{n}\sum_{i=1}^{n}(y_i-\hat y_i)^2Averages squared prediction errors.
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One-Sample t Statistic
t=\frac{\bar x-\mu_0}{s/\sqrt n}Standardizes a sample mean when population standard deviation is unknown.
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PCA Eigenvalue Problem
\Sigma v_k=\lambda_k v_kFinds principal directions as covariance-matrix eigenvectors.
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Pearson Correlation
r=\frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}}Measures linear association between two variables.
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Population Variance
\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2Measures average squared deviation from the population mean.
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Root Mean Square
x_{\mathrm{rms}}=\sqrt{\frac1n\sum_{i=1}^{n}x_i^2}Measures the quadratic mean of a set of values.
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Root Mean Squared Error
RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i-\hat y_i)^2}Returns the square root of mean squared error in the response variable’s units.
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Shannon Entropy
H(X)=-\sum_i p_i\log_2 p_iMeasures uncertainty in a discrete probability distribution in bits.
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Simple Regression Intercept
b_0=\bar y-b_1\bar xComputes the intercept of a fitted simple regression line.
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Simple Regression Slope
b_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}Computes the least-squares slope in simple linear regression.
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Softmax Function
p_i=\frac{e^{z_i}}{\sum_j e^{z_j}}Converts a vector of real scores into positive values that sum to one.
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Softmax Normalization Formula
\operatorname{softmax}(z)_i=\frac{e^{z_i}}{\sum_j e^{z_j}}Converts a vector of scores into positive values summing to one.
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Weighted Mean
\bar x_w=\frac{\sum_i w_ix_i}{\sum_i w_i}Calculates an average where observations have different weights.
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