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Math & statistics

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Math & statistics

Absolute and relative error

Absolute and relative error with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Arithmetic mean

Arithmetic mean: Sum of values ÷ count

Math & statistics

Arithmetic sequence and sum

Find a term and the sum of the first n terms of a sequence with a constant difference.

Math & statistics

Bayesian binary-classifier posterior

Bayesian binary-classifier posterior: Apply Bayes’ theorem: posterior after a positive result = prior × true-positive rate / [prior × true-positive rate + (1−prior) × false-positive rate].

Math & statistics

Binomial probability

Binomial probability with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Combinations

Combinations: n! ÷ [r!(n−r)!]; order does not matter

Math & statistics

Complex number arithmetic

Complex number arithmetic: Use componentwise addition/subtraction, (a+bi)(c+di)=(ac−bd)+(ad+bc)i, and multiply by the divisor’s conjugate for division.

Math & statistics

Covariance of paired data

Covariance of paired data with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Event probability

Event probability: Favorable outcomes ÷ total outcomes

Math & statistics

Exact fraction arithmetic

Exact fraction arithmetic: Parse signed integer fractions with BigInt, apply the selected operation, normalize the denominator sign and reduce by the greatest common divisor.

Math & statistics

Factorial

Factorial: n! = 1 × 2 × … × n; 0! = 1

Math & statistics

Frequency-table mean and variance

Frequency-table mean and variance: Treat each nonnegative integer frequency as a count of repeated observations.

Math & statistics

GCD and LCM of integers

Find the greatest common divisor and least common multiple of two or more integers, including exact large integers.

Math & statistics

Geometric sequence and sum

Find a term and finite sum of a geometric sequence, including ratio zero, one or negative ratios.

Math & statistics

Histogram bin and frequency table

Histogram bin and frequency table: Split the observed range into equal-width bins, using left-inclusive/right-exclusive intervals except that the last bin includes the maximum.

Math & statistics

Interpolate values from an XY table

Interpolate values from an XY table: Sort distinct x knots, find the two surrounding knots for each query, and linearly interpolate y between them.

Math & statistics

Logarithm

Logarithm: log(value) ÷ log(base)

Math & statistics

Median

Median: Sort numerically; use the central value or mean of the central two

Math & statistics

Mode

Mode: Count exact numeric frequencies; return every value tied for most frequent

Math & statistics

Moving average time-series worksheet

Moving average time-series worksheet: For each input row with enough observations, average that row and its previous window−1 values.

Math & statistics

Percentile with linear interpolation

Percentile with linear interpolation with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Permutations

Permutations: n! ÷ (n−r)!; order matters

Math & statistics

Poisson event probability

Poisson event probability with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Polynomial definite integral by Simpson’s rule

Polynomial definite integral by Simpson’s rule: Composite Simpson’s rule: h/3 × [f(a)+f(b)+4Σ odd interior samples+2Σ even interior samples], evaluating the entered polynomial with Horner’s method.

Math & statistics

Powers and roots

Powers and roots: Raise the base to the supplied real exponent

Math & statistics

Prime factorization

Decompose a positive whole number into prime factors, exponents and the number of positive divisors.

Math & statistics

Quadratic roots

Quadratic roots: Solve ax² + bx + c = 0 using the discriminant; real roots only

Math & statistics

Range

Range: Maximum minus minimum

Math & statistics

Root mean square (RMS)

Root mean square (RMS) with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Simple linear regression

Simple linear regression with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Solve a linear equation system

Solve a linear equation system: Apply Gaussian elimination with partial pivoting to an n×(n+1) augmented matrix.

Math & statistics

Spearman rank correlation with ties

Spearman rank correlation with ties: Rank each series with average ranks for ties, then calculate Pearson correlation between the two rank series.

Math & statistics

Standard deviation

Standard deviation: Sum squared deviations from mean ÷ N (population) or N−1 (sample); take square root

Math & statistics

Survey sample size for a proportion

Plan a survey sample size for a proportion from a confidence level, absolute margin and optional population correction.

Math & statistics

Variance

Variance: Sum squared deviations from mean ÷ N (population) or N−1 (sample)

Math & statistics

Weighted average

Weighted average: Σ(value × weight) ÷ Σweights

Math & statistics

Weighted median

Weighted median with an explained formula, a worked example and checks for invalid or unsupported inputs.

Math & statistics

Z-score standardization

Express an observation as the number of population standard deviations above or below the mean.

Math & statistics

Real nth root

Real nth root: Solve y^n = x over the real numbers.

Math & statistics

Exponential growth doubling time

Exponential growth doubling time: Doubling time = ln(2) / ln(1 + growth/100), with a strictly positive growth rate.

Math & statistics

Circular sector area

Circular sector area: Sector area = π × radius² × central angle / 360 for angles from zero through 360 degrees.

Math & statistics

Circular arc length

Circular arc length: Arc length = radius × angle in radians = radius × angle in degrees × π/180.

Math & statistics

Ellipse area

Ellipse area: Ellipse area = π × semi-major axis × semi-minor axis.

Math & statistics

Ellipsoid volume

Ellipsoid volume: Ellipsoid volume = (4/3) × π × a × b × c, where a, b and c are the three semi-axes.

Math & statistics

Circular frustum volume

Circular frustum volume: A circular cone frustum has volume πh(R² + Rr + r²)/3.

Math & statistics

Torus volume

Torus volume: For a ring torus, volume = 2π² × major radius × tube radius².

Math & statistics

Distance from a point to a line

Distance from a point to a line: For line ax + by + c = 0, perpendicular distance is |ax₀ + by₀ + c| / √(a² + b²).

Math & statistics

Angle between two 2D vectors

Angle between two 2D vectors: Angle = arccos[(A·B)/(|A||B|)], returned in degrees from 0 to 180.

Math & statistics

Harmonic mean

Harmonic mean: Harmonic mean = count / Σ(1/value).

Math & statistics

Geometric mean

Geometric mean: Compute exp(mean(ln(value))) for positive observations.

Math & statistics

Coefficient of variation

Coefficient of variation: Sample coefficient of variation = sample standard deviation / arithmetic mean × 100.

Math & statistics

Mean absolute deviation about the mean

Mean absolute deviation about the mean: Find the arithmetic mean, then average the absolute distances of observations from that mean.

Math & statistics

Cohen’s d for independent samples

Cohen’s d for independent samples: Cohen’s d = difference of means divided by pooled sample SD; pooled variance weights each sample variance by n−1.

Math & statistics

Probability of at least one success

Probability of at least one success: For independent equal-probability trials, P(at least one success) = 1−(1−p)^n.

Math & statistics

Modular exponentiation

Modular exponentiation: Compute base^exponent mod modulus using exponentiation by squaring with exact BigInt integer arithmetic.

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