Module Seven: Resources for Data Collection and Presentation - Mathematics Pathways

Qualitative and Quantitative Data

Qualitative date refers to data that is in categories.

It is known as nominal data if there are several categories which have no particular order such as favourite colour or brand of car.

Or ordinal if the categories have an order such as a pain rating (none, mild, moderate, severe)

Quantitative data is numerically measured such as age or weight and the data has a natural order.

Not all numeric data is quantitative eg telephone numbers are not really numbers as you can't do any mathematical analysis of them such as finding the mean

Links

A Powerful Edge PDF Page 78

If you have not already downloaded this book, go to the Australian Bureau of Statistics website

StatTrek - Variables in Statistics

Discrete & Continuous Data

Quantitative data can be discrete or continuous.

If you roll a die, there are six outcomes and they are all discrete, meaning the values can be 1,2,3,4,5,6 but not 1.5 or 5.77 or 3.78346

If you measured the temperature every hour of the day, the value could be anywhere in a range such as 0 degrees to 40 degrees and does not need to be a whole number. This is called a continuous variable because it can take any value in a given range.

Links

A Powerful Edge PDF Page 78

If you have not already downloaded this book, go to the Australian Bureau of Statistics website.

StatTrek - Variables in Statistics

Univariate & Bivariate Data

Univariate data refers to the study of a single variable

Example

The height of each child in a school.

The speed of each car passing a speed camera.

Bivariate data refers to the study of two variables which may be related

Example

The height of each child in a school and their age.

The speed of each car passing a speed camera and the time of day.

In these cases we may be able to find a relationship between the two variables.

Links

A Powerful Edge PDF Page 78

If you have not already downloaded this book, go to the Australian Bureau of Statistics website.

StatTrek - Variables in Statistics

Maths is Fun Univariate and Bivariate Data

Data Collection Methods

Data can be collecting in a number of ways. 

Examples

• A census which includes every item in a population. This is not always practical or affordable
• A sample survey
• An experiment
• A series of observations

Links

Australian Bureau of Statistics - A Powerful Edge PDF Page 25 - Data Collection

StatTrek - Data Collection Methods

Sampling Methods

Data can be sampled in many ways.

Some are random meaning that a sample is chosen in such a way that it has a good chance of representing the whole population. 

Some are non random such as asking for people to phone in with a vote or choose answers on a web site.

StatTrek - Sampling Methods

Australian Bureau of Statistics - A Powerful Edge PDF Page 175 - Sampling

Bias in Surveys

When sample data is collected it is important not to introduce bias. For example if you conduct a survey using home land line telephone numbers during working hours your sample is likely to be biased.

StatTrek - Bias in Surveys

Australian Bureau of Statistics - A Powerful Edge PDF Page 186 - Bias

Summarising Data

The total set of data collected is known as Raw Data and can be difficult to understand especially if there are a large number of data items. Raw data is often displayed in columns or tables.

Amongst the ways of summarising raw data to make it more understandable are;

• Frequency Distribution Tables
• Stem and Leaf Plots (also known as Stemplots)
• Boxplots
• Scatter Plots

Links

StatTrek - Stemplots

StatTrek - One Way Frequency Tables

StatTrek - Two Way Frequency Tables

Australian Bureau of Statistics - A Powerful Edge PDF Page 82 - Frequency Distribution Tables

Australian Bureau of Statistics - A Powerful Edge PDF Page 87 - Stem and Leaf Plots

Presenting Data

There are many ways of representing data graphically to highlight its key features. Some of the most common are;

• Bar Graph (for categorical data)
• Histogram (for ordered numerical data items or groups)
• Pie Chart
• Frequency polygon

Links

StatTrek - Bar Charts and Histograms

StatTrek - Box Plots

StatTrek - Cumulative Frequency Plots

StatTrek - Scatterplots

StatTrek - Comparing Data

Australian Bureau of Statistics - A Powerful Edge PDF Page 103 to 128 - Covers all kinds of Graphs

Correlations

Correlation

Correlation measures the strength of a relationship between two variables. See Module 7 - Topic 3. A correlation does not necessarily show a causal relationship.

Maths is Fun - Scatterplots and Correlation

StatTrak  - Linear Correlation Coefficients

The tutorial includes a 17 minute video.

Spearman's Rank Correlation Coefficient

The Spearman's Rank Correlation Coefficient is used to discover the strength of a link between two sets of data. The relationship does not have to be linear.

Example looks at whether bottled water costs more as you get closer to a famous monument in Barcelona. - Barcelona Field Studies Centre

Laerd Statistics - Spearman's Rank Correlation Coefficient

Pearson Correlation Coefficient

The Pearson Correlation Coefficient is similar to Spearman but is simpler to use and can only find linear relationships.

Online Stats Book - Pearson Correlation Coefficient

StatTrek - Pearson Correlation Coefficient

Khan Academy - Pearson Correlation Coefficient

Z - Score

The  z - score is the measure of how many standard deviations a data item is from the mean.

It is also known as the standard score.

Laerd Statistics - Standard Score

StatTrek - Z Score

Khan Academy - Normal Distribution

Distributions

Sampling Distributions

A sampling distribution or finite-sample distribution is the probability distribution of a given statistic based on a random sample.

StatTrek - Sampling Distributions

Difference between Proportions 

Difference between proportions is a way of comparing two independent sample proportions.

StatTrek - Difference between Proportions

Difference between Means 

This is a way of comparing two independent sample means

StatTrek - Difference between Means

Central Limit Theory

The Central Limit Theory states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough. (StatTrek)

StatTreK - Central Limit Theory

Student's t distribution

The t distribution (aka, Student's t-distribution) is a probability distribution that is used to estimate population parameters when the sample size is small and/or when the population variance is unknown. (StatTrek)

StatTrek - Student's t distribution

Estimation and Inference

Estimation and Inference

In statistics, estimation refers to the process by which one makes inferences about a population, based on information obtained from a sample.

StatTrek - Estimation and Inference

Khan Academy - Inferential Statistics

Point and Interval Estimates

In statistics, estimation refers to the process by which one makes inferences about a population, based on information obtained from a sample. (StatTrek)

StatTrek - Point and Interval Estimates

Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses. (StatTrek)

One and Two Tailed Tests

A One Tailed Test is a test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution. (StatTrek)

A Two Tailed Test is a test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution. (StatTrek)

Links

StatTrek - Hypothesis Testing

Khan Academy - Hypothesis Testing

Khan Academy - Inferential Statistics

All Khan Academy content is available for free.

The Australian Bureau of Statistics Statistics - A Powerful Edge this is a comprehensive Statistics course with explanations, exercises and solutions. The main index for the statistics course is on page 76 in the PDF. (NOTE: that is NOT the page number at the bottom of the page because some pages are not numbered.)

You can download it and keep it on your laptop or tablet as a reference. It can be downloaded as a free PDF from the Australian Bureau of Statistics.

StatTrek - Dictionary

StatTrek - Statistics Tutorial

Khan Academy - Statistical Studies

Maths is Fun - Probability and Statistics Index

Usable Stats - Statistics Tutorials