Formula solver index — formula.expert

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.