Search Results

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.

Permutations

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

Number of ordered arrangements of r items from n.

Combinations

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

Number of unordered selections of r items from n.

Factorial

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

Product of all positive integers up to n.

Permutations with Repetition

n^r

Ordered arrangements of r items from n when repetition is allowed.

Permutations of a Multiset

\frac{n!}{n_1! n_2! \cdots n_k!}

Arrangements of n items where some are identical.

Classical Probability

P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

Probability of an event with equally likely outcomes.

Complement Rule

P(A') = 1 - P(A)

Probability that an event does not occur.

Addition Rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability that A or B occurs.

Multiplication Rule (Independent)

P(A \cap B) = P(A) \cdot P(B)

Probability of both independent events occurring.

Conditional Probability

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

Probability of A given that B has occurred.

Bayes Theorem

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

Reverses conditional probabilities using prior knowledge.

Expected Value

E[X] = \sum_{i} x_i\, p_i

Long-run average value of a random variable.

Variance of a Random Variable

Var(X) = E[X^2] - (E[X])^2

Measures the spread of a random variable around its mean.

Binomial Probability

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

Probability of k successes in n independent trials.

Poisson Probability

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

Probability of k events in a fixed interval given average rate λ.

Geometric Probability

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

Probability that the first success occurs on trial k.

Normal Distribution PDF

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

Bell-curve probability density function.

Odds

Odds = \frac{P(A)}{1 - P(A)}

Ratio of the probability of an event to its non-occurrence.

Pigeonhole Principle

\lceil n/k \rceil

If n items go into k containers, some container holds at least this many.

Inclusion-Exclusion (3 sets)

|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Counts the union of overlapping sets without double counting.

Catalan Number

C_n = \frac{1}{n+1}\binom{2n}{n}

Counts many recursive structures such as balanced parentheses.