Big. Sometimes theoretical

They have mean, variance and standard deviation

mean(population)
var(population)
sd(population)

Variance is the square of standard deviation

var(population) == sd(population)^2

Standard deviation is the square root of variance

sd(population) == sqrt(var(population))

An outcome is a random element of the population

Any outcome will be similar to the population mean

How similar? It depends on the population variance

The distance between outcome and population mean depends on the confidence level

outcome will probably be between mean(pop)±k*sd(pop)

\[\langle X\rangle - k\,\mathbb{S}(X)\leq\text{outcome}\leq\langle X\rangle + k\,\mathbb{S}(X)\]

The key idea is: different k have different probability

(This is not always true)

When the population is Normal, then

Choose your own alpha

(Chebyshev always works)

A sample is a group of outcomes

It has size, mean, variance, and standard deviation

length(sample)
mean(sample)
var(sample)
sd(sample)

Each sample is different (random)

Each sample mean is random

If the sample size is large, then sample mean has Normal distribution

If sample mean has Normal distribution, we need to know it parameters

The average of sample mean is the population mean

The standard deviation of sample mean is population standard deviation divided by the square root of sample length

The variance of sample mean is population variance divided by sample length

sample mean will probably be between mean(pop)±k*sd(pop)

\[\langle X\rangle - k\frac{\mathbb{S}(X)}{\sqrt{\text{length}}}\leq\text{sample mean}\leq\langle X\rangle + k\frac{\mathbb{S}(X)}{\sqrt{\text{length}}}\]

The key idea is: different k have different probability

Choose your own alpha

In real life we do not know population mean, and we want to know.

We only know sample mean, and sample variance

We can approximate population variance by sample variance

But we have to pay a cost

population mean will probably be between mean(sample)±k*sd(pop)

\[\text{sample mean} - k\frac{\mathbb{S}(X)}{\sqrt{\text{length}}}\leq \langle X\rangle \leq\text{sample mean} + k\frac{\mathbb{S}(X)}{\sqrt{\text{length}}}\]

The key idea is: different k have different probability

Now we have Student’s distribution

This depends on degrees of freedom (sample length-1)

Choose your own alpha