Statistics and data

mean(a, b, …)

The mean of all the numbers given.

mean([1; 2; 3; 6]) = 3

median(a, b, …)

The median of all the numbers given.

median([1; 2; 3; 6]) = 2.5

var(a, b, …)

The variance of the numbers as a whole population (dividing by n).

var([1; 2; 3; 6]) = 3.5

Var(a, b, …)

The variance of the numbers as a sample (dividing by n − 1).

Var([1; 2; 3; 6]) = 4.667

stdev(a, b, …)

The standard deviation of a whole population.

stdev([1; 2; 3; 6]) = 1.871

Stdev(a, b, …)

The standard deviation of a sample.

Stdev([1; 2; 3; 6]) = 2.16

corr(vx, vy)

The correlation coefficient of the two vectors.

corr([1; 2; 3], [2; 4; 7]) = 0.993

slope(vx, vy)

The slope of the least-squares straight line through the points.

slope([1; 2; 3], [2; 4; 7]) = 2.5

intercept(vx, vy)

Where the least-squares straight line crosses x = 0.

intercept([1; 2; 3], [2; 4; 7]) = -0.667

linterp(vx, vy, x)

The value at x of the straight lines between the points (vx, vy); beyond the ends, the end line carries on.

linterp([0 m; 2 m], [10 kN; 20 kN], 0.5 m) = 12.5 kN

cspline(vx, vy)

A cubic spline through the points, with cubic ends, for interp.

vx := [0; 1; 2; 3]
vy := vx^3
interp(cspline(vx, vy), vx, vy, 1.5) = 3.375

pspline(vx, vy)

A cubic spline with parabolic ends, for interp.

vx := [0; 1; 2; 3]
vy := [0; 1; 4; 9]
interp(pspline(vx, vy), vx, vy, 1.5) = 2.25

lspline(vx, vy)

A cubic spline with straight ends (the natural spline), for interp.

vx := [0; 1; 2; 3]
vy := [0; 1; 4; 9]
interp(lspline(vx, vy), vx, vy, 1.5) = 2.2

interp(vs, vx, vy, x)

The value at x of a spline from cspline, pspline or lspline, or of a polynomial from regress.

vx := [0; 1; 2; 3]
vy := [1; 3; 5; 7]
interp(regress(vx, vy, 1), vx, vy, 10) = 21

regress(vx, vy, k)

The least-squares polynomial of degree k through the points, for interp.

vx := [0; 1; 2; 3]
vy := [1; 2; 5; 10]
interp(regress(vx, vy, 2), vx, vy, 4) = 17

linfit(vx, vy, F)

The least-squares coefficients of a sum of terms: F(x) returns a vector of the terms.

F(x) := [1; x]
linfit([0; 1; 2], [1; 3; 5], F) = [1; 2]

dnorm(x, μ, σ)

The probability density of the normal distribution at x.

dnorm(0, 0, 1) = 0.399

pnorm(x, μ, σ)

The probability that a normal variable is below x.

pnorm(1.96, 0, 1) = 0.975

cnorm(x)

The probability that a standard normal variable is below x.

cnorm(1.645) = 0.95

qnorm(p, μ, σ)

The x below which a normal variable falls with probability p.

qnorm(0.95, 0, 1) = 1.645

rnd(x)

A random number between 0 and x, the same at every recalculation.

rnd(1) = 0.731

runif(m, a, b)

m random numbers between a and b, as a column.

runif(3, 0, 10) = [7.309; 4.101; 2.077]

rnorm(m, μ, σ)

m random numbers from the normal distribution, as a column.

rnorm(3, 0, 1) = [1.562; -0.608; -1.091]

hist(intervals, data)

How many of the data fall in each interval: intervals is a vector of their ends, or the number of equal intervals.

hist([0; 1; 2; 3], [0.5; 1.5; 1.7; 2.2]) = [1; 2; 1]

histogram(intervals, data)

As hist, beside the middle of each interval: a column of middles, then one of counts.

histogram([0; 1; 2; 3], [0.5; 1.5; 1.7; 2.2]) = [0.5, 1; 1.5, 2; 2.5, 1]

fft(v)

The Fourier transform of the real vector v (2ⁿ elements), its first half and the middle, scaled by 1/√n.

fft([1; 0; 0; 0]) = [0.5; 0.5; 0.5]

ifft(c)

The inverse of fft: the real vector whose transform is c.

ifft(fft([1; 2; 3; 4])) = [1; 2; 3; 4]

FFT(v)

As fft, scaled by 1/n and with the opposite sign in the exponent.

FFT([1; 0; 0; 0]) = [0.25; 0.25; 0.25]

IFFT(c)

The inverse of FFT.

IFFT(FFT([1; 2; 3; 4])) = [1; 2; 3; 4]

cfft(A)

The Fourier transform of a complex vector or matrix of any size, scaled by 1/√n.

cfft([1; 1i]) = [0.707 + 0.707i; 0.707 - 0.707i]

icfft(A)

The inverse of cfft.

icfft(cfft([1; 1i])) = [1; 1i]

CFFT(A)

As cfft, scaled by 1/n and with the opposite sign in the exponent.

CFFT([1; 1i]) = [0.5 + 0.5i; 0.5 - 0.5i]

ICFFT(A)

The inverse of CFFT.

ICFFT(CFFT([1; 1i])) = [1; 1i]

READPRN("file")

The numbers in a text file, in columns separated by spaces, tabs or commas; lines that aren’t numbers are skipped.

READCSV("file")

A comma-separated file as a matrix, numbers and text kept apart.

READFILE("file", type)

A data file, type being “delimited” (spaces, tabs or commas) or “csv”.

READEXCEL("file"[, "Sheet1!A1:C10"])

The cells of an Excel workbook (.xlsx or .xls), all of its first sheet or a range; in the desktop app.