Lesson 3 Working with Outliers

Teacher Advice

Goals of Lesson

- To understand how data about past trends may help predict future performances.

- To understand the concept of how real data is likely to contain ‘outliers’ and hence that graphical representations should be lines of best fit, not connecting every point.

Features of the Lesson

- Pupils consider how they may use the pattern of progression of performances to predict what may happen in the future.

- Pupils discover the difficulties in modelling real (imperfect data) and the concept of outliers.

- Pupils consider outliers and produce a line of best fit for their graph and use this to make predictions about future performances.

Summary

Part 1: Making predictions requires graphical data Pupils consider how they may use the pattern of progression of performances in their chosen event to predict what may happen in the future. (15 minutes).
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Part 2: Using trend lines of best fit Teacher introduces concepts of lines of best fit, extrapolation and outlier data. (15 minutes).
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Part 3: Constructing and using scatter plots and trendlines Pupils use the notion of scatter plots, and lines of best fit and extrapolation to address their questions about the future. (20 minutes).

Preparation

- Copies of the 800m track data plotted as a scatter plot.

- Graph paper for pupils

- Preparation of assessment rubric from information collected in lesson two.

- Picture of the 1896 stadium

Prior Learning for Lesson 3

Pupils should be familiar with the following concepts:

- Scatter plot

- Linear relationships represented in tabular and graphical forms

- Interpolation and extrapolation on graphs

Part 1: Making predictions requires graphical data (15 minutes)

Pupils frame questions about predicting future events and build a need for graphical representation of their data in order to do this (15 minutes).

Activities | Teaching and Learning Notes
Pupils consider how they may use the pattern of progression of performances to predict what may happen in the future. Key question: What other questions may we pose from this data? | Now that pupils have a more conclusive understanding about whether or not women have improved faster than men, pose the question to them, “What other questions may we ask from this data?” Direct their attention to using the data to predict future performances. As a class, generate a list of possible questions. (see: TL11: Using data to predict future events for a possible list). Allow the pupils to select the question of most interest to them. TL11: Using data to predict future events Having just compared men’s and women’s past improvements in performance, pupils are now challenged to consider what questions about future performances in particular events they may pose using this data. Possible questions pupils might pose include – If/when might women overtake men in a particular event What might be the winning performance at the London (or other) Olympic Games in a particular event? When might a significant milestone be broken (eg 10s for 100m for women)? Or any other pupils may think of. Giving pupils some time to attempt to answer their question using the data in tabular form should help them understand that analysing future trends using this form of data is very difficult. To assist them in understanding the benefits of a graph, it is likely to be helpful to demonstrate how it is easier to see a trend in graphical form and to identify potential outliers. Print TL11 Close Box
Pupils realise the limitations of tabular data Key question What are the limitations of tabular data here? | Provide pupils with the opportunity to return to their groups and use the tabular data, to try to answer this question. Discuss with pupils the point that up to now, data has been represented in tabular form and help pupils to see the limitations of tabular data when analysing trends and extrapolating beyond the data in the table. If time is a problem, you could delete asking pupils to try and answer their question using tabular data and just point out why this is hard and how graphical representation can help.
Teacher introduces the need for a graphical representation of the data Key question How can graphical representations help? | It is worth asking what other ways data can be represented, but be prepared to direct pupils' attention to the need to represent the data in graphical form. A quick whiteboard sketch showing how graphical lines can be extrapolated will be helpful see TL11.

Part 2 Using trend lines of best fit

Pupils work with data on the 800m to build understandings of lines of best fit, extrapolation and outlier data (15 minutes)

Activities | Teaching and Learning Notes
Teacher introduces the need for a line of best fit. Key questions How will these graphical representations need to be different from most other graphs that we plot in maths? Why can we ignore some data points with real data? | Hand out the scatter plot of the time series data for the 800m and ask why connecting every point is unlikely to be helpful. Call for comments about points that do not seem to fit in the men’s or women’s data and use this to introduce the concept of outliers. See: TL12: Outliers for further detail. TL7 contains background information (and a photo), explaining why the 1896 event was a huge outlier. TL5 includes discussion on some reasons for outliers in other events. TL12: Outliers Real-life data is rarely perfect and the data being considered in this Case is no different. As pupils attempt to sketch lines of best fit they will find points which sit considerably off any predicted line. These are called outliers (we are using this term a little loosely here as technically an outlier is defined as one and half times the interquartile range above the upper quartile or below the lower quartile). A range of factors could contribute to these performances including the introduction of new technology, drugs, environment of competition etc. Upon identifying the outliers in their data, pupils may take educated guesses at potential reasons for these performances or may choose to go away and research this performance following lesson three. Several reasons were listed in TL7 as to why the 1896 Olympic results are outliers in almost all events and reasons in some other events were given in TL5. However you may let pupils try to find some of these. 800m model When examining the scatter plot for the men’s 800m, the first performance in 1896 stands out as an outlier for the reasons about the track given in TL7. Interestingly, the first women’s 800m (1928), although much slower than the next one in 1960, is seen on the scatter plot to not be a significant outlier; the 32 year gap means it is close to any line of best fit. This is not easy to see without using a graph. Key principles for reflection Data rarely fits a perfect pattern and will usually contain outliers. A range of factors may cause an outlier and these should be considered when considering the point in the context of a trend. Print TL12 Close Box Show or have the pupils draw trend lines of best fit that ignore outlier points. Discuss with pupils how these points may be considered and/or ignored in generating a line of best fit. (See TL13: Linear vs non-linear graphing for advice about modelling a non-linear graph). TL13: Linear vs non-linear graphing For the data that this Case deals with, the model of improvement would be asymptotic. Given the scope of the Mathematics to be covered in this Case and the age group of the target audience, it is likely that you will assume that the progression of performances fits a linear model. However in many track and field events the changes in performances have been flattening over the last 20-30 years. If time allows and for more capable pupils/classes, the idea of asymptotic relationships may be explored and the final activity of the Case may involve pupils hand sketching such a relationship on their scatter plot. This is likely to change the answer to predictions about future performances. While using a linear model, in many events, it will appear that women are still improving faster than men and will therefore one day overtake them. As was pointed out in TL5, the data analysis in lesson 2 will address the question “Have women improved faster than men?” However “Are women still improving faster than men?” is a different question. Using a (more accurate) asymptotic relationship, the progression of men’s and women’s performances have generally become reasonably parallel, with the women approximately 10% behind the men. This would indicate the women will not one day overtake men (see Women of the future are on track to run faster than men). If technology such as Excel is available to assist with this activity, pupils could quickly and easily explore modelling with a range of relationship types. Print TL13 Close Box
Pupils use trend lines in the 800m event Key question: What can trend lines tell us about the 800m event? | Have pupils use their trendlines for the 800m event to answer prediction questions about that event – such as whether it seems women will eventually overtake men.

Part 3: Constructing and using scatter plots and trend lines

Pupils use the notion of scatter plots, and lines of best fit and extrapolation to address their questions about the future. (20 minutes)

Activities | Teaching and Learning Notes
Pupils construct scatter plots of the data for their event, draw lines of best fit on their graphs and identify outliers. Key Question: What can trend lines tell us about other events? | Allow pupils the opportunity to generate lines of best fit for their scatter plots. Visit each group and ask about any outlier performances within their event and what reasons they may have for these performances. TL5 and TL7 provide teachers with some information on outliers.
Pupils extrapolate their lines of best fit to estimate results to their question. Key Question: How might we use our graph to predict future performances? | Some groups may want to use non linear trend lines (see TL13 for discussion of this issue). There is scope for using EXCEL here. (See TL14: Graphing with computer based programs) TL14: Graphing with computer based programs There are a number of computer based programs which can be used for graphing data. The lesson plans outline the need for pupils to draw a scatter plot by hand and use it to estimate a line of best fit. This task may be replaced by a computer based program such as “Excel” or graphics calculators if the resources are available. Some of these programs produce their own lines of best fit and apply equations to them. An extension activity for pupils may involve simultaneously solving the equations for the trend line for men and women and using the result to accurately predict cross-over points (points of intersection). At the start of lesson three pupils may identify data points they wish to discount due to extraneous factors affecting that performance. This could be done quickly on a computer based program and allow pupils to see how removing that point from the graph may immediately alter the position of the line of best fit. Print TL14 Close Box
Share results as a class |

Homework:

For homework pupils produce a final draft of the work they have completed in the Case using the ‘e nquiry process’. As part of this, they should research their event for some interesting data about outlier performances that might explain why they had outliers.