# Statistics

'A guide for for teaching! Ideas and resources and lessons to look forward to'

*This page is good for both the 'Descriptive Statistics' and 'Statistical Applications' units in the new syllabus and the 'Statistics' unit from the previous one!*

This is a big, key topic in the syllabus that covers a variety of function families, both in the abstract and in context. Its a fabulous opportunity to show where the abstract and contextual meet, using functions as a tool to model the world around us. Its key because of the presumed knowledge involved, the presence of which, or not as the case may be, has a big impact on the way its delivered! This topic is the one that involves the most use of the GDC and as such provides the opportunity to help students become fluent with that tool.

This page is a guide to the topic, ideas for classroom activity that encourage critical thinking, practise exercises and some of the best resources available on the Internet to help! The overview written below aims to help teachers think about the main objectives of the module and potential opportunities and issues there are with teaching it! Please follow links to the pages on Ideas and resources, exercises and the internet guide.

#### Ideas and resources

Classroom experiences are the most important thing that we as teachers are responsible for. This links to a page of activities and ideas for teaching this unit that are aimed to be both engaging and effective in encourage students to enjoy, discover and understand mathematics. The list is a brief outline of an activity that links to a page that gives more detail and the associated resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

#### Exercises

This page has a variety of tasks designed for practise and revision. Practice questions are an important part consolidating students' understanding of a concept. They can be found from numerous sources and the following is not intended as a 'fix all' solution. Most teachers appreciate that questions gathered from numerous different sources make the best diet!

#### The Internet Guide

This page is intended to list and outline some of the best resources available online to support the teaching and learning of this module. These will mostly be videos, virtual manipulatives and self help sites. Where appropriate, there will be a short commentary of what they are and how they might best be employed! It is also intended that this selection grow over time and that users contribute.

## Overview

Statistics are some of the most important tools we have for understanding and studying our world. Without a fundamental grasp of statistics it is not possible to comprehend large amounts of media coverage. Equally it is not possible to discern the validity of those media sources. One can argue extensively about whether statistics is Mathematics or not and this is a valid and useful discussion for ToK studies, but there can be little argument as to whether or not is important to understand statistical principles. Many will make sweeping remarks about statistics but fewer will understand them. A simple exercise with any given newspaper will demonstrate most of these issues. As such this topic could not be more relevant or more interesting. It represents and excellent opportunity to demonstrate how crucially important Mathematics can be as a tool for analysis, description and argument and how it adds a perspective we might not get otherwise. This topic also represents an opportunity to engage students and bring real relevance to their studies. There is, with this module, a tension between using semi-real situations as contexts for teaching statistical skills and using very real data as a starting point from which the needs for different types of analysis arrive! As with most such issues a suitable compromise is probably best, but the latter gives rise to a broader, richer experience.

### Key elements

- discrete and continuous data
- frequency tables and polygons
- grouped data, histograms, stem and leaf diagrams
- cumulative frequency
- measuring central tendency
- measuring dispersion
- scatter diagrams
- linear regression
- chi
^{2}independence tests

Please refer to the subject guide for full details

### Teaching time

This topic is given 24 hours in the syllabus and this is a healthy amount of time given the amount of this topic that is likely to have been seen before by most students. Equally, there is a lot that can be done here that is useful for internal Assessments. Even if students choose non-statistical based projects, there is an element of data handling involved and so there can be a cross over with some of the project hours here as well.

### Sticking points

#### Why and When?

As suggested in the overview there is a tension between a context for learning statistics and a context that gives rise for a need for statistics. If a group of students are given some raw data and asked to comment on it, there instinct is often to start to make summary remarks and judgments about the data and it is this instinct it is important to capitalise on. The danger of the alternative is that statistics can appear as process for process sake. Understanding why a particular process is valid in a particular situation is in many ways more important than the process itself. Too often text book examples will gloss over this and leaves students asking,' why didn't we just.....' and so on. When ultimately we would want students to choose the most appropriate statistical tool for a situation it is important that examples we provide them with have the same quality!

Almost certainly related to the above, students do often find it difficult to choose an appropriate statistical tool. When, for example, is a 5 figure summary, or box and whisker diagram or ogive, appropriate for making comparisons? When is the standard deviation of a set of data relevant? These are questions that become more relevant during project work but should be addressed during the study of statistics. If students can answer them then they have a true understanding of the concept and are, as such more likely to apply it correctly.

#### Process vs meaning

Ask a random group of Maths Students to estimate the mean of a set of grouped data and at least one (and probably more) will add the numbers in the frequency column together and divide by the number of groups! This is a classic example of a student latching on to a process and applying it to the most likely looking set of numbers without any thought about what is meant by either the table or the data in question. The pessimistic view here is that this is a students who had learned that this is what Mathematics is. It is abstract and can therefore be performed without a need for meaning. Whilst there is an element of truth in this the more optimistic view is that this is just one of the pitfalls to be pointed out.

The good news is that it can be solved by simply thinking about the problem,

- What is the data being collected?
- How many data items are there?
- Since we know only the 'group' what shall we consider the value of the data item?
- How many were there in that group and each of the others?
- How do we calculate the mean value of a set of data?

Of the five questions above on the last refers to a process, the rest pertain to understanding the problem. These are important questions.

#### Sense and convention

There are many 'rules' about how statistical diagrams should be drawn and whilst many of them can be reasoned, a number of them are also conventions and some are open to debate. For example, should the points on a cumulative frequency curve be joined by a straight line or a curve? Since is a worthwhile discussion if it can be kept relatively brief, but it can contribute to confusion. We plot at the end of intervals on cumulative frequency diagrams but in the middle for frequency polygons and grouped data. This can, of course, be reasoned. The point here is that when these things can be reasoned, they should be so that students are not trying to remember conventions but are able to reason their way out of a problem. This can be achieved by regular brief focussed questions on these points. For example, pick a point on a cumulative frequency diagram and ask 'what is meant by this point?' Reason and memory are equally powerful but best in combination.

#### First principles

Considering the origins of a correlation coefficient, the least squares regression line or the chi^{2} statistic very quickly moves Maths Studies students on to a whole new plane of understanding. This is a fabulous opportunity to show what Mathematics can do but it can be pretty daunting. It is important just to be sensitive to this. It can be powerful to build the columns of one of these calculations in a spreadsheet or GDC just so that students understand the complex structure behind and labour saving power of some of the functions on their calculators.

#### Raw vs grouped

In the age of computer software for statistical analysis there is less of an argument for grouped data and students will often be surprisingly astute at pointing this out. For example, why when you have a set of raw data would you group it and estimate the mean from the grouped data when the raw data will give you an accurate calculation. Statistical packages will often ask if you want to use the raw or grouped data to plot a cumulative frequency diagram and this question had big implications. What is difference? What impact does it have on a 5 figure summary?

If you change the group intervals on a histogram then you change the histogram, if you use the raw data, then you no longer have a histogram. Grouping data can make it easy to represent but makes it less accurate in doing so! This is a fascinating tension that it is worth exploring.

#### Causation

This is about understanding when there is reason to conclude that two data sets correlate with each other.

#### Validity

### Teaching Opportunities

#### Application

talk about the importance and list examples

#### Cross-discipline

science, geography

#### ICT

#### Project link

a summary paragraph that should link to the internal assessment section

### Ideas and resources

This links to a page of activities and ideas for teaching this unit. The list is a brief outline of an idea; some are just ideas and others link to a page that gives more detail and some resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

## Selected Pages

### Statistics Teaching Ideas 26 September 2016

Here you should find a wealth of resources that can be used by you and your students for teaching this topic. The resources...

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### Statistics Internet Guide 29 May 2011

This page is intended to offer a selection of some excellent sources for videos, virtual manipulatives and weblinks that...

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### Focus on correlation and regression 7 November 2018

The search for meaningful relationships between two variables. Here we take the leap from 1 variable to 2 variable data...

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### Statistics Practice 18 August 2018

This page has a variety of tasks designed for practice and revision. Practice questions are an important part consolidating...

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### The Normal Distribution 2 May 2018

The normal distribution is a fascinating, naturally occurring phenomenon that has very relevant applications to understanding...

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### Focus - Cumulative Frequency 26 September 2016

This topic is all about these two related tools for helping us look at how a data set is spread out. Learn about filling...

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