# Slides for Probability and Statistics module, 2015-2016
# Matt Watkins, University of Lincoln

Populations and Samples¶

Learning outcomes

what is a population?
what is a sample?
how are they related

definition

A population is a complete set of measurements or counts, either existing or conceptial.

Everything we have done so far has dealt with populations

In principle we have all the information available - for instance when we looked at the distribution of your name lengths. You are the complete population (those that are here anyway) of students in the School of Mathematics and Physics at the University of Lincoln in 2015-16.

definition

A sample is a subset of measurements on a population. For instance we could instead think of the 'students in the School of Mathematics and Physics at the University of Lincoln in 2015-16' as a sample from the population that is 'students starting undergraduate degrees in the UK 2015-16'.

The question is then, if we only have information about the sample, can we deduce information about the population?

definition

A statistic is a numerical descriptive measure of a sample.

A parameter is a numerical descriptive measure of a population.

A statistic should be derived only from the observations - there should be no assumptions made about the underlying distribtuion etc.

definition

A random sample of size $n$ taken from some distribution is a set of $n$ random variables $X_1, X_2 \ldots X_n$, satisfying the following two conditions:

(a) Each $X_i$ where $i=1,2,\ldots n$.

(b) The set $X_1, X_2 \ldots X_n$ are mutually independent.

Example

This is the crux, distringuishing between sample and population. Lets look at your heights.

In this case our population is those of you that have turned up this afternoon.

We can take a sample from that population by picking groups of you at random.

definition

Let $X_1, X_2 \ldots X_n$ be a random sample of size $n$ from some distribution. We diefine a statistic (of the sample!) as any function of the set

(a) Each $X_i$ where $i=1,2,\ldots n$.

(b) The set $X_1, X_2 \ldots X_n$ are mutually independent.