Statistics & Probability

Averages, spread, distributions and probability

Statistics & Probability formulas help you make sense of data, uncertainty, and chance, from simple averages and spread to distributions, counting methods, and conditional probability. Whether you’re a student preparing for exams or using data in real work, these formulas are essential for interpreting results, predicting outcomes, and making smarter decisions with confidence.

Arithmetic Mean

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

The average of a data set.

Median

M = \text{middle value of ordered data}

The middle value separating the higher and lower halves.

Variance

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2

Average squared deviation from the mean.

Standard Deviation

\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2}

Square root of the variance; typical spread.

Z-Score

z = \frac{x - \mu}{\sigma}

Number of standard deviations a value is from the mean.

Bayes' Theorem

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Updates a probability given new evidence.

Combinations

\binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of ways to choose r items from n (order irrelevant).

Permutations

P(n,r) = \frac{n!}{(n-r)!}

Number of ordered arrangements of r items from n.

Normal Distribution

f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Probability density of the bell curve.

Binomial Distribution

P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}

Probability of k successes in n trials.

Poisson Distribution

P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval.

Expected Value

E[X] = \sum_i x_i\,p_i

The long-run average value of a random variable.

Conditional Probability

P(A|B) = \frac{P(A \cap B)}{P(B)}

Probability of A given B has occurred.

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 correlation between two variables.

Standard Error of the Mean

SE = \frac{\sigma}{\sqrt{n}}

Standard deviation of the sample mean.

Mode

\text{Mode} = \text{most frequent value}

The value that appears most often in a data set.

Range

R = x_{max} - x_{min}

The difference between the largest and smallest values.

Factorial

n! = n\times(n-1)\times\cdots\times 2\times 1

Product of all positive integers up to n.

Geometric Mean

GM = \sqrt[n]{x_1 x_2 \cdots x_n}

The nth root of the product of n values.

Interquartile Range

IQR = Q_3 - Q_1

The spread of the middle 50% of the data.

Coefficient of Variation

CV = \frac{\sigma}{\bar{x}}\times 100\%

Relative variability as a percentage of the mean.

Range

R = x_{max} - x_{min}

Difference between the largest and smallest values.

Interquartile Range

IQR = Q_3 - Q_1

Spread of the middle 50% of the data.

Covariance

\text{cov}(X,Y) = \frac{1}{n}\sum (x_i-\bar{x})(y_i-\bar{y})

Measures how two variables vary together.

Coefficient of Variation

CV = \frac{\sigma}{\mu}\times 100

Relative variability as a percentage of the mean.

Percentile Rank

PR = \frac{\text{below}}{n}\times 100

Percentage of values below a given score.

Margin of Error

E = z\frac{\sigma}{\sqrt{n}}

Half-width of a confidence interval for the mean.

Confidence Interval (Mean)

\bar{x} \pm z\frac{\sigma}{\sqrt{n}}

Range likely to contain the population mean.

Z-Test Statistic

z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}

Tests whether a sample mean differs from a hypothesized value.

Chi-Square Statistic

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

Tests goodness of fit between observed and expected counts.

Mode

\text{most frequent value}

The value that appears most often in a data set.

Weighted Mean

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

Average where values carry different weights.

Odds Form of Bayes

O(A|B) = O(A)\times \frac{P(B|A)}{P(B|\neg A)}

Updates odds using a likelihood ratio.

Sample Variance

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

Unbiased estimate of population variance.