Statistics - mathematical content

Year 8 - Data investigation and interpretation#

TIMES Module

See: mat081c-2024

General notes#

Information from data#

obtain data to obtain information
census aims to get total information for a population
obtaining sample data aims to obtain representative data
representative data enables us to obtain representative information
representative data relies on obtaining a random sample either by
- choosing randomly if we're dealing with a population, or
- taking observations randomly if we're dealing with observational/experimental situation
practicalities means we have to assume data is representative of a more general situation
very important to consider how data were obtained or collected
data can be representative for some questions but not others

Generalising from data samples: statistical inference#

but care must be taken to infer from data
statistical inference provides principles/methods for inferring from data to take account of variation
these methods are studied at university level
but examples comparing datasets collected under similar conditions (or simulation) can hep in understanding the need and being cautious in generalising from data

Assumed pre-requisite knowledge#

From F-7 students have some familiarity/understanding with concepts and usage

statistical data investigative processes
types of data and variables
types of investigations
some types of graphical and summary presentations of data

Arising from which is a need for representative data and for describing/allowing for variability within and across data sets

learning outcomes

Understand the effects of sampling variability
Challenges of obtaining representative data
Emphasising the importance of clear reporting of how, when and where data are obtained or collected
Identifying the issues/questions around what makes data representative
Understanding of how sample proportions and averages vary across samples
familiar with categorical, count and continuous data ??
learning experience in recording, classifying and exploring individual datasets of each type
seen and used tables, picture graphs and column graphs for categorical data and count data with small number of different counts treated as categories, dotplots and steam-and-leaf-plots for continuous and count data
familiar with use of
frequencies and relative frequencies of categories (categorical data)
counts (for count data)
intervals of values (for continous data)
students have used and interpreted averages (or means) and meadians of quantitaite
familiar with concept of statistical variables and subjects of data investigation

Motivation#

Societal calls for more information/evidence
A natural environment for learning statistical thinking is experiencing the process of real statistical data investigation: first thoughts; planning; collecting and exploring data, to reporting on its features
conditions for active learning, hands-on experience and problem solving

Content#

Focus on whether we can use data to comment on a more general situation?
yes because random representativeness of data our data are/considered to be a random set of observations that are representative of a general situation or population
Consider
- How data is collected
- How to obtain random representative data
- What can take our data to be representative
Compare the nature of censuses, surveys and observational investigations
If not census data are often samples of data
Introduce notions of sampling to obtain random representative data
Sample - A portion/segment/piece that is representative of a whole |
Samples of data can be representative for some questions, but not others
Only a random selection of data can be considered representative
Variation comes in multiple forms

Concept	Definition
Population	The entire group of subjects of interest
Representative data	Set of observations obtained randomly in circumstances that are representative of a more general situation or larger population of interest
Statistical variable	Description of an entity that is being observed

Revision: Types of data and statistics variables#

Variable type	Description	Examples
Continuous	Can take any values in a given interval. Never exact and correspond to little intervals. Can't be any value.	Height, age
Count (aka discrete)	# of items in a set	# of students in a class
Categorical	Each observation fits into one of a number of categories	Pet preference between cat and dog

Initial questions#

A long list given summarised as

school planning or wanting opinions from students or parents about issues
other organisations wanting to promote/enquire about usage/preference of outcomes
investigating features of objects of interest (e.g. length of most popular songs on JJJ over the years)
interesting features of people (e.g. observable human traits)

Identifying issues and planning to obtain representative data#

Planning a data investigation through the interaction of three questions

What do we want to find out about?
What data can we get?
How do we get the data?

Planning involves identifying

the variables we want
the subjects the variables it will be about
how to collect representative data

Census#

Examples include

survey to all school families
census of top JJJ songs over 25 years | Accuracy

Questions

Do all families get the survey or respond?
Misunderstandings about the survey questions?
Changes between the design of the survey and the night of the concert?
A census is not a random sample and can't be considered represenative more broadly (depending on the question)

Sample survey#

Examples include

Interviewing all parents of a school

Random sample by various means - e.g. stratified random random in each year level.

Challenging and complex

Asking people to register an opinion is considered very unreliable - relying on their motivation

Linking to Australian Idol etc. type approaches. - Random contact is better, but not without issues (people hang up, unrecognised phone numbers) - Importance on non-respondees should change depending on the nature of questions (e.g. opinion generating questions require more attention to ensure randomness to weed out influence of individual motivations/biases) - Formal answers to the make up of a sample are difficult/university level
- Students can get some idea of effects through investigation and experimentation
- Depend on the aims of the data investigation nature of variable
  - How much might a continuous variable range

Survey questions

Avoid leading questions
Statements that are difficult to agree with increase chance of distortion
open and closed questions each have their advantages/disadvantages
Pilot surveys can be useful

Observational investigation#

e.g. which thumb is on top, estimating a length of time, usage of a facility

Not a survey, observing people. But still sampling. Questions of how to be random. This trait may be linked to other traits may need to gather other observations, but arbitrary sample may be appropriate

Variation within and across data sets#

Statistics is teh science of variation and uncertainty of investigating, identifying, measuring, estimating, describing, modelling, attributing, interpreting, minimising and allowing for, variation and uncertainty.

Census data#

Collection information from whole population - questions of representativeness do not arise
important to identify, describe, investigate and attribute variation are import
But mistakes, omissions, non-responses must be considered, estimated, allowed for

Using dot plot and stem-and-leaf plots to visualise the data

Sample data: categorical dat#

Relative frequencies or proportions of the different categories - census data we report % or proportions.

From sample data we're seeking to estimate proportions for a more general situation. Hence variation across different samples can be useful.

Are our samples representative of the general situation/what are our samples representative of?
Can the variation observed be attributed/explained by other variables?

e.g. since which thumb on top has a genetically dominant trait we don't need entirely random, just need to avoid all close relatives

Random samples will always have sampling variability. Shows an example of 100 different samples showing variability from 30 to 80%. Goes onto show that the variability is reduced as the sample size increases, but it has to increase quite high.

Argument is we need a lot of data to accurately estimate proportions. Must always report how many observations collected, how collected, and can say only what the % was in our data

Sample data: continuous data#

Interested in variation both

within the sample
between samples

Students estimating time - variation occurs for multiple reasons, including conditions.

Explain variability#

This can be of an interest. For multiple reasons. But beyond the scope here??