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Counting: permutations and combinations

Factorials, ordered and unordered selections.

n! counts the orderings of n things. Choosing k of them in order is n!/(n − k)!; ignoring the order divides by k! again, giving the binomial coefficient “n choose k”. These numbers are Pascal's triangle and the coefficients of (a + b)ⁿ.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 5 choose 2

Discrete-Math · Counting

5 choose 2

5,\ 2

Àwọn ìṣàmúlò-ètò

\binom{5}{2} = \frac{5!}{2!\,(5-2)!}
Unordered selections: n! / (k! (n−k)!).
= \frac{120}{2 \times 6} = 10

Fi àwọn àgbèwọlé hàn

\binom{5}{2} = 10

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Bii o ṣe le: Counting: permutations and combinations

Unordered selections: n! / (k! (n−k)!).

Àwọn Àtòjọ-ẹ̀yàn

Mean or median, which should I use?

Median when the data have outliers or a long tail (incomes, house prices); mean when the data are roughly symmetric and you want every value to count. Report both if they disagree. The gap is itself information.

What does a p-value actually say?

The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true.

Why divide by n − 1 for the sample variance?

The sample mean sits closer to the sample than the true mean does, so squared deviations from it are slightly too small on average; dividing by n − 1 instead of n corrects the bias.

Diẹ̀ nínú Statistics & Probability

Sampling and data Describing data with graphs Mean, median and mode Probability Discrete random variables Continuous random variables The normal distribution The central limit theorem Confidence intervals Hypothesis testing Comparing two samples Chi-square tests Linear regression and correlation ANOVA and the F distribution