Branch of mathematics

Statistics and probability

Mean, median and deviation, counting and classical probability — the mathematics of data and chance.

Topics in this branch

Basic
Mean, median and mode
Three answers to one question: which single number best describes a whole data set. The mean shares the total out evenly, the median stands exactly in the middle, and the mode points at the most frequent value — and which one you pick can change the conclusion.
Intermediate
Variance and deviation
Two data sets can share the same mean and look nothing alike. The range, the variance and the standard deviation measure what the mean cannot see: how widely the data is spread around it.
Advanced
Combinatorics
The art of counting without listing. The rule of product, factorials, permutations, arrangements and combinations all answer one question — in how many ways can this be done — and the whole difficulty comes down to two decisions: does order matter, and may items repeat.
Advanced
Classical probability
When every outcome of an experiment is equally possible, the probability of an event is an ordinary ratio: how many outcomes favour it, over how many there are. Everything else is being able to count both numbers — and knowing when the opposite event is easier to count.

Branch formulas

Branch: Statistics and probability

Mean, median and mode

  • Arithmetic mean

    xˉ=x1+x2+⋯+xnn\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}

    the sum of all the data divided by how many there are

  • Total from the mean

    x1+x2+⋯+xn=n⋅xˉx_1 + x_2 + \dots + x_n = n \cdot \bar{x}

    this is how a missing value is recovered when the mean is known

  • Median — an odd sample

    Me=x(n+12)(n odd)\mathrm{Me} = x_{\left(\frac{n+1}{2}\right)} \quad (n \ \text{odd})

    once sorted, it is exactly the middle number

  • Median — an even sample

    Me=x(n2)+x(n2+1)2(n even)\mathrm{Me} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2} \quad (n \ \text{even})

    the mean of the two middle values

  • Mode

    Mo=the value with the highest frequency\mathrm{Mo} = \text{the value with the highest frequency}

    the number that occurs most often — there may be none at all

  • Weighted mean

    xˉw=w1x1+w2x2+⋯+wkxkw1+w2+⋯+wk\bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \dots + w_k x_k}{w_1 + w_2 + \dots + w_k}

    every value counts as many times as its weight says

Variance and deviation

  • Range

    R=xmax⁡−xmin⁡R = x_{\max} - x_{\min}

    the simplest measure of spread — it looks only at the two extremes

  • Deviation from the mean

    di=xi−xˉ,d1+d2+⋯+dn=0d_i = x_i - \bar{x}, \qquad d_1 + d_2 + \dots + d_n = 0

    the deviations always sum to zero, which is why they cannot be averaged as they are

  • Variance

    σ2=(x1−xˉ)2+(x2−xˉ)2+⋯+(xn−xˉ)2n\sigma^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n}

    the mean of the squared deviations from the mean

  • Standard deviation

    σ=σ2\sigma = \sqrt{\sigma^2}

    the square root of the variance — back in the units of the data

  • Shortcut formula

    σ2=x12+x22+⋯+xn2n−xˉ2\sigma^2 = \frac{x_1^2 + x_2^2 + \dots + x_n^2}{n} - \bar{x}^2

    the mean of the squares minus the square of the mean — quicker to compute

  • Sample variance

    s2=(x1−xˉ)2+⋯+(xn−xˉ)2n−1s^2 = \frac{(x_1 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n-1}

    divisor n − 1, when the data is a sample drawn from a larger population

Combinatorics

  • Rule of product

    N=k1⋅k2⋅…⋅kmN = k_1 \cdot k_2 \cdot \ldots \cdot k_m

    a choice made in several independent steps — multiply the counts

  • Factorial

    n!=1⋅2⋅3⋅…⋅n,0!=1n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n, \qquad 0! = 1

    the product of the first n naturals; 0! equals 1 by definition

  • Permutations

    Pn=n!P_n = n!

    the number of orderings of all n items

  • Arrangements without repetition

    n!(n−k)!=n(n−1)…(n−k+1)\frac{n!}{(n-k)!} = n(n-1)\ldots(n-k+1)

    choose k out of n, order matters, nothing repeats

  • Arrangements with repetition

    nkn^k

    k times over, we choose from the same n options

  • Combinations

    (nk)=n!k! (n−k)!\binom{n}{k} = \frac{n!}{k! \, (n-k)!}

    choose k out of n, order does NOT matter

  • Symmetry of combinations

    (nk)=(nn−k)\binom{n}{k} = \binom{n}{n-k}

    choosing k items is the same as discarding the other n − k

Classical probability

  • Classical probability

    P(A)=∣A∣∣Ω∣P(A) = \frac{|A|}{|\Omega|}

    favourable outcomes over the number of all possible ones

  • Range of a probability

    0≤P(A)≤10 \leq P(A) \leq 1

    never below 0 and never above 1

  • Impossible and certain events

    P(∅)=0,P(Ω)=1P(\varnothing) = 0, \qquad P(\Omega) = 1

    edge cases of the definition, not results of a computation

  • Complementary event

    P(A′)=1−P(A)P(A') = 1 - P(A)

    counting "not A" is often easier than counting A

  • As a percentage

    P(A)⋅100%P(A) \cdot 100\%

    the same probability expressed in percent

  • Independent events

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

    multiply the probabilities — only when one does not affect the other

Related articles

Luminous histogram bars settling into the soft bell of a distribution, one outlying point brighter than the rest, against a cosmic nebula in violet and magenta
Statistics and probability

The average person does not exist

All of descriptive statistics rests on a single trade: you give up information about a crowd and receive one number in return. The trade is often worth making, but it is never free — and it pays to know exactly what gets lost in it. From the mortality tables of 1662, through Quetelet's average man, to the aircraft nobody ever saw.

16 min