Statistics formulas — 50 solvers

Addition Rule (Mutually Exclusive Events)

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

ProbabilityStatisticsFor events that cannot both happen, the chance that either one occurs is simply the sum of their separate probabilities.

Bayes' Theorem (Two Hypotheses)

P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B \mid A) \, P(A) + P(B \mid A^{c}) \, \left(1 - P(A)\right)}

ProbabilityStatisticsUpdates a prior belief into a posterior after evidence arrives, weighing the true-positive rate against the false-positive rate.

Binomial Distribution Mean

\mu = n p

ProbabilityStatisticsExpected number of successes across n independent trials that each succeed with probability p — the mean of the binomial distribution.

Binomial Distribution Variance

\sigma^{2} = n p (1 - p)

ProbabilityStatisticsSpread of the number of successes across n independent trials, at its largest when the per-trial chance p sits at one half.

Binomial Probability

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

ProbabilityStatisticsChance of exactly k successes in n independent trials that each succeed with the same fixed probability p, as with coin tosses.

Birthday Problem (All Distinct)

P = \frac{N!}{(N - n)! \; N^{\,n}}

ProbabilityStatisticsChance that n independent picks from N equally likely options are all different, the engine behind the birthday paradox.

Chi-Square Contribution of One Cell

\chi^{2}_{\text{cell}} = \frac{(O - E)^{2}}{E}

StatisticsAlgebraHow much a single cell of a contingency or goodness-of-fit table adds to the chi-square statistic, from its observed and expected counts.

Classical Probability

P = \frac{f}{n}

ProbabilityStatisticsProbability of an event as the number of favourable outcomes divided by the total number of equally likely outcomes.

Coefficient of Determination (R²)

R^{2} = r^{2}

StatisticsAlgebraThe share of variation in y explained by a simple linear regression, obtained by squaring the correlation coefficient.

Coefficient of Variation

CV = \frac{s}{\bar{x}}

StatisticsAlgebraRelative variability: the standard deviation expressed as a fraction of the mean, so spreads measured on different scales can be compared.

Cohen's d (Effect Size)

d = \frac{\bar{x}_1 - \bar{x}_2}{s_p}

StatisticsAlgebraStandardised effect size: the gap between two group means measured in pooled standard deviations rather than raw units.

Complement Rule

P(A^{c}) = 1 - P(A)

ProbabilityStatisticsThe chance an event does not happen is one minus the chance it does, because every trial must end in one case or the other.

Conditional Probability

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

ProbabilityStatisticsThe chance of A once B is known to have happened, found by rescaling the overlap to the reduced sample space B.

Confidence Interval Lower Limit

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

StatisticsAlgebraThe lower bound of a confidence interval for a mean, pulling the critical z-value and standard error back from the sample mean.

Confidence Interval Upper Limit

U = \bar{x} + z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraThe upper bound of a confidence interval for a mean, adding the critical z-value times the standard error to the sample mean.

Degrees of Freedom (One-Sample t)

df = n - 1

StatisticsAlgebraThe degrees of freedom used to look up a critical value for a one-sample t-test or confidence interval, one fewer than the sample size.

Expected Trials Until First Success

E[X] = \frac{1}{p}

ProbabilityStatisticsAverage number of independent attempts needed before the first success when each attempt succeeds with probability p.

Expected Value of a Bet

E = p \, W - (1 - p) \, L

ProbabilityStatisticsAverage profit per play of a two-outcome wager that pays W with probability p and costs L the rest of the time, over many plays.

General Addition Rule

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

ProbabilityStatisticsThe chance that either of two events happens, correcting the simple sum by subtracting the overlap that would be counted twice.

General Multiplication Rule

P(A \cap B) = P(A) \, P(B \mid A)

ProbabilityStatisticsChance that both events happen when the second depends on the first, as in drawing two cards without replacement from a deck.

Geometric Distribution (First Success)

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

ProbabilityStatisticsChance that the first success in a run of repeated independent trials arrives exactly on trial number k, after k - 1 failures.

Hypergeometric Probability

P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}

ProbabilityStatisticsChance of drawing exactly k successes in a sample of n taken without replacement from a population of N holding K successes.

Interquartile Range (IQR)

IQR = Q_3 - Q_1

StatisticsAlgebraThe width of the middle half of a data set, from the first quartile to the third, and the spread measure box plots are built on.

Margin of Error for a Mean

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

StatisticsAlgebraHalf-width of a confidence interval for a mean, built from the critical z-value, the standard deviation and the sample size.

Margin of Error for a Proportion

E = z \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraThe plus-or-minus quoted with a poll result, built from the critical z-value, the sample proportion and the number of respondents.

Midrange

M = \frac{x_{\max} + x_{\min}}{2}

StatisticsAlgebraThe midpoint between the largest and smallest observations, a quick centre estimate computed from just the two extremes.

Multiplication Rule (Independent Events)

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

ProbabilityStatisticsWhen one event has no influence on the other, the chance that both occur is the product of their separate probabilities.

Normal Probability Density

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

StatisticsAlgebraThe height of the normal bell curve at a given value, set by the distance from the mean in standard deviations.

Odds and Probability

O = \frac{P}{1 - P}

ProbabilityStatisticsConverts between a probability and odds in favour, the ratio of the chance it happens to the chance it does not.

One-Sample T-Test Statistic

t = \frac{\bar{x} - \mu}{s / \sqrt{n}}

StatisticsAlgebraTests a sample mean against a claimed value when the standard deviation is estimated from the sample itself rather than known.

One-Sample Z-Test Statistic

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

StatisticsAlgebraTests a sample mean against a claimed population mean when the population standard deviation is known, in standard-error units.

Outlier Lower Fence

LF = Q_1 - 1.5 \, IQR

StatisticsAlgebraTukey's lower cutoff for outliers: any observation below one and a half interquartile ranges under the first quartile is flagged.

Outlier Upper Fence

UF = Q_3 + 1.5 \, IQR

StatisticsAlgebraTukey's upper cutoff for outliers: any observation above one and a half interquartile ranges beyond the third quartile is flagged.

Percent Difference

PD = \frac{|x_1 - x_2|}{(x_1 + x_2)/2}

StatisticsAlgebraCompares two measurements of equal standing by dividing their gap by their average, when neither counts as the accepted value.

Percent Error

PE = \frac{|x_{\text{meas}} - x_{\text{acc}}|}{x_{\text{acc}}}

StatisticsAlgebraHow far a measurement strays from the accepted value, expressed as a fraction of that accepted value for lab reports and calibration checks.

Poisson Probability

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

ProbabilityStatisticsChance of exactly k events in a fixed interval when events occur independently at a constant average rate lambda.

Pooled Standard Deviation

s_p = \sqrt{\frac{(n_1 - 1) s_1^{2} + (n_2 - 1) s_2^{2}}{n_1 + n_2 - 2}}

StatisticsAlgebraCombines two sample standard deviations into a single estimate of common spread, weighting each by its degrees of freedom.

Predicted Value from a Regression Line

\hat{y} = a + b x

StatisticsAlgebraReads a prediction off a fitted least-squares line for any chosen value of the predictor, given the intercept and slope.

Probability of At Least One Success

P = 1 - (1 - p)^{n}

ProbabilityStatisticsChance that at least one of n independent attempts succeeds, found as one minus the chance that every single attempt fails.

Range (Max minus Min)

R = x_{\max} - x_{\min}

StatisticsAlgebraThe simplest measure of spread in a data set: the distance from the smallest observation to the largest.

Regression Line Intercept

a = \bar{y} - b \bar{x}

StatisticsAlgebraThe y-intercept of a least-squares line, fixed by the requirement that the line pass through the point of averages.

Regression Slope from Correlation

b = r \frac{s_y}{s_x}

StatisticsAlgebraThe least-squares slope of a regression line, recovered from the correlation coefficient and the two standard deviations.

Sample Size for a Mean

n = \left( \frac{z \sigma}{E} \right)^{2}

StatisticsAlgebraHow many observations a study needs to estimate a mean within a target margin of error at a chosen confidence level.

Sample Size for a Proportion

n = \frac{z^{2} \, p (1 - p)}{E^{2}}

StatisticsAlgebraHow many respondents a survey needs to estimate a percentage within a target margin of error at a chosen confidence level.

Standard Error of a Proportion

SE = \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraHow much a sample percentage typically wanders from the true population proportion, given the proportion and the sample size.

Standard Error of the Mean

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

StatisticsAlgebraHow much a sample mean typically wanders from the true mean, shrinking with the square root of the sample size.

Two-Sample Z-Test Statistic

z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{\sigma_1^{2}}{n_1} + \dfrac{\sigma_2^{2}}{n_2}}}

StatisticsAlgebraCompares two independent sample means when both population standard deviations are known, scaled by the combined standard error.

Variance and Standard Deviation

\sigma^2 = \sigma \cdot \sigma

StatisticsAlgebraConverts between variance and standard deviation: the variance is the square of the standard deviation, and the deviation is its square root.

Weighted Mean of Two Groups

\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

StatisticsAlgebraCombines the averages of two groups into one overall mean, weighting each group by how many observations it contains.

Z-Score (Standard Score)

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

StatisticsAlgebraHow many standard deviations a value sits above or below the mean, turning any measurement into a comparable standard score.