# SL Scheme - Statistics

The new Mathematics: Applications and Interpretation syllabus is out and published on the IB website. Log in to your MyIB account and head to this link . It is a new course with a new name and renewed focus on understanding the relationship between mathematics and its application and interpretation. It is still intended very much that the SL course will cater for the same group of students that might currently opt for Mathematical Studies. It is hoped that some of the current SL students will opt for the applications HL. Clearly all sorts of permutations are possible. Read more about the development of this course through the links in the New Syllabus section of the website. This page will focus on the Statistics and Probability unit and evolve over time as the new applications site is built.

### Statistics

So on this page we will just have a quick comparison between the two syllabi for statistics

Current Syllabus

- Discrete and Continuous data
- Frequency tables
- Dealing with grouped data
- Cumulative Frequency and box plots
- Measures of central tendency
- Measures of dispersion
- The Normal Distribution
- Bivariate Data
- Linear Regression
- Chi Squared test of independence

Syllabus 2019

- Population, Sampling, discrete and continuous data, outliers.
- Presentation of data - frequency histograms (equal class intervals) Cumulative frequency, box plots and 5 figure summaries.
- Measures of central tendency and dispersion
- Linear correlation and its measures.
- Basic single event probability and relative frequency.
- Venn diagrams, tree diagrams and sample space for combined events. Including the laws of probability.
- Discrete random variables and their probability distributions.
- Binomial distribution
- Normal distribution
- Spearman's rank correlation coefficient
- Chi squared test of independence both 2 way and goodness of fit.
- T-Tests

### Points of Observation

- Effectively, the Sets Logic and Probability unit has been taken out, with probability and sets being subsumed in to the statistics unit.
- Some set theory is also listed in Prior knowledge and the notation is still used for probability as are Venn diagrams
- The logic unit is no longer in the course.
- Distributions as a concept gets a much bigger focus and extends to the Binomial distribution.
- Likewise, Hypothesis testing is also a bigger section with the addition of Goodness of fit tests and T-Tests.

## An SL Scheme of work

Based on these guidelines, here is an idea about how we might spend the time. Remember that the Toolkit hours can come anywhere we want them. Clearly there are a number of ways in which this can be done, but this is just a suggestion to get us started.

I have included 6 hours of the toolkit time, but this could easily be more if you feel it gets a bit too tight. I have kept 12 hours of toolkit time up my sleeve for the Internal Assessment, BUT some of that could easily be weaved in here to good effect.

I have a feeling that this unit is a bit tight. Like with the modelling unit we get to 42 hours so I propose having it in **2 sections. I** have set this plan out in terms of 'weeks' where we might consider that a week consists of 3, 1 hour lessons. Of course, the 2 units could be independent of each other.

A note about assessment - Each week shows a suggested assignment and there are a number of ASSESSMENT points through the unit. Many teachers like to use regular quizzes and clearly, all of this can be weaved in and out as the teacher chooses.

**Week 1 - Introduction and Statistical ideas**

Over these first three weeks I propose that we look at the first 3 items on the syllabus, but in the context of a statistical investigation that can be the beginnings of an example of statistical investigation for the exploration. The point is to see some statistics in context where we really want to process the numbers to find out any patterns. Something like the UK Number ones exercise can be a lovely way in, or indeed the more serious data in the Box plots - why? activity.

- Statistical Ideas
- Presentation of 1 variable data (Histograms, cumulative frequency and box plots)
- Central Tendency and dispersion

In this first week....

- Introduce students to a data set (eg UK Number ones but many other data sets would do) and lead a discussion where you ask questions about the data set that will lead to students making hypotheses. Having made these hypotheses, discuss issues related to Sampling, data sources and outliers. Guide students in planning some activity to test those hypotheses. (Guided towards measures of central tendency and dispersion)
- Have a formal lesson where you teach and practice sampling techniques
- Have a formal lesson where you teach about identifying outliers and practice

* Assignment* - Part 1 of a written report on the exercises we do in class. This would involve some of the introductory parts of a written report.

#### Week 2 - Measures of central tendency and dispersion

This week, the focus would be on the types of measures we can calculate to describe the data base initially. The week will involve learning about and practicing the calculation and interpretation of those key statistics in a variety of contexts. The week would culminate in students applying this to the data base they worked with last week.

- Teach a formal lesson about calculating mean, standard deviation with and without frequency tables on the GDC. Extend to include some grouped frequency.
- Have lesson where students are asked to apply some of these calculations to a variety of data and interpret the results.
- Following from last week, students should apply what they have learned this week to the data set you introduced last week so that this can contribute to a write up.

* Assignment* - Part 2 of a written report on the exercises we do in class. This would involve students documenting, presenting and writing about how measures of central tendency apply to the data set they have been working with.

**Week 3 - Histograms,Cumulative Frequency and box plots **

- It might be an idea to do an activity like Box plots - why? so that students get a feel for the key ideas and what they mean in context. I think it is really important to see the distribution of individual data items first so that a greater understanding can be made of the idea that a box plot is a summary of that distribution.
- Then I suggest a formal lesson where students work with cumulative frequency data to create and interpret comparative box plots in context. This activity Comparing Data Distributions might be useful here.
- Finally, students should use the ideas from this week to take their investigation of the original data set one stage further.

...

* Assignment* - Complete a written report based on the data that you started with

**Week 4 - Bivariate Data**

The distinction between 1 and 2 variable data is a key conceptual one that should be the beginning of this weeks activity. This week should take us through a review of scatter graphs, correlation coefficient (PMCC) as a measure of correlation and the y on x regression line as a summary of that relationship (if there is one!)

- A good way in might by this 15 Countries and 200 years activity which is based on bivariate data. The activity should bring out the key points with interpretation.
- Introduce the PMCC as a measure of correlation and explain how it can be calculated along with a the regression line. Students should then be given the chance to practice this calculation in a variety of contexts.
- Students can then be invited to see how these ideas can be applied to the data set you started with in week 1.

**ASSESSMENT POINT** - Students should submit the written assignment that they have been working on for assessment. This is a sort of 'mini' IA activity that brings together hiw the work of the last 4 weeks can culminate in a study of a data set.

#### Week 5 - Spearman's Rank

This will be a challenging conceptual shift as well. Understanding the key differences between Pearson's and Spearman's is something that many may struggle with.

- An activity that highlights how 'ranking' can be used to look for correlation.
- A formal lesson on the process and interpretation of Spearman's rank, followed by the opportunity to practice using it in a variety of different contexts. (Note - one of the major reasons for including Spearman's rank is because it is used in Geography - a good source of relevant examples)
- Some examples that highlight the important differences between Spearman's and Pearson's with a focus on which one you might choose and why

* Assignment* - A set of exam style questions to practice application and interpretation of Spearman's rank correlation coefficient and the choice between that and Pearsons.

#### Week 6 - Probability Part 1

This week, the focus should be on reviewing the probability that students are likely to have seen before. Much will depend on the classes previous experiences with probability.

- Key definitions, notation and terminology
- Probability of an event and its complement
- Probability of combined events
- Use of different diagrams to represent probabilities

Introductions could be done with activities like Fairground Games and/or Probability Trees to bring out the key concepts. Then a good deal of practice using some of the resources here on the Focus on Probability page for some work on application and interpretation.

* Assignment* - A set of exam style questions to practice application and interpretation of probability (see Focus on Probability )

#### Week 7 - Probability Part 2

This week should focus more on the less intuitive results related to conditional probability, dependence and the laws of probability. Use of the following activities False Positives , Nerf gun roulette , and Monty Hall are all good for provoking the kind of thought and activity that brings these issues out. Then, more practice activity from the Focus on Probability page.

**ASSESSMENT POINT** - Students are set an independent assignment with exam style questions to work on as part of a formal assessment of the first part of this unit. This could be a Test if yuo have time for it.

#### Week 8 - Distributions Part 1

This section brings is in to some new territory with a more sustained focus on the idea of a probability distribution.This week can be devoted to the definition and exposition of the concept of a discrete random variable and different types of distributions, before beginning a specific focus on the Normal Distribution.

- The Frequency Distribution Match is a great activity and context for discussing and defining these ideas.
- A formal lesson of the definitions and use of the key terminology.
- Start an investigation in to The Normal Distribution where we might define the key mathematical structure.

* Assignment* - This week might be an opportunity for a review assignment of something from a previous unit. It can be really useful to squeeze these in from time to time to keep ideas current.

#### Week 9 - Distributions part 2

This week would continue to focus on The Normal Distribution with a focus on application and interpretation of results in different contexts.

- Begin to solve problems using the symmetry of the distribution of the kind 'What is the probability of....' This is a key bridge that builds on intuitive understanding of the shape to use of the GDC
- This lesson might lead us to inverse normal calculations and associated practice.
- A third lesson this week could be focussed on general problem solving with the Normal distribution

* Assignment* - An assignment designed to hep with the practice of questions on the Normal distribution.

#### Week 10 - Distributions part 3

This week, then, is left for an exploration of the Binomial distribution.

- Exploration and Definition
- Practice of application and interpretation
- Broad range of examples.

* Assignment* - An assignment designed to hep with the practice of questions on the Binomial distribution.

#### Week 11 - Hypothesis testing 1

Again, with the introduction of the goodness of fit and T tests, we now have a broader topic under the heading Hypothesis testing. More work, but a greater global understanding of the idea. I suggest that the first 2 weeks here be devoted to the idea of Chi Squared tests, both two way and goodness of fit.

- The activity Independence Day is really designed to build - again - on intuitive ideas (related to stratified sampling) towards a more complex process done on the GDC.
- A more formal lesson can be done on learning and understanding thee process of carrying out a hypothesis test, with some key definitions and terminology.
- Real independence tests can be used for some contexts with which to practice.

* Assignment* - An assignment designed to hep with the practice of questions on the Chi Squared 2 way test.

#### Week 12 - Hypothesis testing 2

It is possible/likely, that more time will be needed to build on work from last week before moving on to Goodness of Fit tests. Again, another key reason for inclusion here was because of the use of these tests in Biology and so this should be a useful source of problems.

- Focus on the difference between two way and goodness of fit.
- Formal exposition of goodness of fit test
- Practice in context.

* Assignment* - An assignment designed to hep with the practice of questions on the Chi squared goodness of fit test.

#### Week 13 - Hypothesis testing 3

This week then, the focus would be on the T-test.

- Exploration and definition
- Formal exposition
- Practice in context

* Assignment* - An assignment designed to hep with the practice of questions on the T-test.

#### Week 14 - Review and test

Following 13 weeks of work on Probability and Statistics where students have experienced a wide range of problems and contexts, this week will be devoted to the review and practice of these key ideas ahead of another big assessment point.

**ASSESSMENT POINT** - The end of this week will be an end of unit assessment for the Statistics and Probability Unit unit of the course.

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