Singing Star

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Purpose

This is a level 4 statistics activity from the Figure It Out series.
A PDF of the student activity is included.

Achievement Objectives
S4-1: Plan and conduct investigations using the statistical enquiry cycle: determining appropriate variables and data collection methods; gathering, sorting, and displaying multivariate category, measurement, and time-series data to detect patterns, variations, relationships, and trends; comparing distributions visually; communicating findings, using appropriate displays.
Student Activity

    

Click on the image to enlarge it. Click again to close. Download PDF (1259 KB)

Specific Learning Outcomes

create a bar graph

interpret a graph

calculate mean and median

sort data usng mean. median, or totals

Description of Mathematics

This diagram shows the areas of Statistics involved in this activity.

 stats diagram.

The bottom half of the diagram represents the 5 stages of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) statistics investigation cycle.

Statistical Ideas

Singing Star involves the following statistical ideas: using the PPDAC cycle, multivariate data sets, bar graphs, medians, and means and using statistics for decision making.

Required Resource Materials
A copy of the judges' scores table (see copymaster)

A computer spreadsheet/graphing program (if available)

FIO, Levels 4 -4+, Statistics in the Media, Singing Star, pages 5-7

Classmates

Activity

Activity One

Before the students begin this activity, you could discuss with them a current or past TV show that has a talent quest format. The students may have noticed that different judges have different judging personalities or varying consistency with their scoring. You could also discuss the relative merits of using a total, the median, or the mean (see the definitions on page 6 of the students’ book) of the judges’ scores to decide the winner and which would be fairer in the case of the TV show they are discussing. Alternatively, you could have the second part of this discussion after the students have completed Singing Star.
For question 1, discuss with the students whether the bar graph gives any more information than the raw data. The purpose of a graph is to make data more meaningful and to give a picture that may not be apparent when numbers alone are reported. If a graph does not provide a better picture, it shouldn’t be used. In this
case, although the graph is difficult to read in terms of the contestants, it does show patterns in terms of the judges’ scoring that are not so apparent from the table of raw data. (Question 2b shows how the bar graph can be enhanced to be more useful in terms of the contestants. This is a useful learning point for the students to keep in mind before they decide not to use a certain sort of graph.) Perhaps you and your students could create graphs that reveal the story better. There are several possibilities.
For question 2b, make sure that the students understand the defi nitions given for median and mean. The example showing the use of colour and a bar on the student page is one way to record this information. Students may decide to show both the mean and the median as lines on the graph, using different colours, or to place symbols on the graph to represent the median and mean figures.
Question 3 provides an opportunity to explore the functions of the median and the mean. They are both measures of central tendency (numbers used to give an idea of “middle”). In instances such as this, where each contestant has their own data set, a convenient first step when comparing them is to compare their “middles”. This would also be true for comparing large data sets.
The median of the 3 scores may be the best measure of a contestant’s skill because it minimises the effect of outlier judges. However, a disadvantage of the median is that it is blind to all numbers except for the one in the middle: the median of {1, 1, 3, 3, 3, 4, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9} is exactly the same as the median of {1, 1, 3, 3, 3, 4, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 21}, yet the range of numbers is quite different because of the difference between the two numbers at the higher end. The advantage of the mean is that it does take account of every number in a data set (generally, if any number in a data set is changed, this will affect the mean). Its main disadvantage is that it is unduly infl uenced by outliers (as in the second range above).
The students may come up with some novel methods for ranking the students in question 3b. Methods using the median and the mean are given in the answers. Discuss with your students the advantages and disadvantages of mean and median as a basis for ranking the contestants. They may want to change their minds about their chosen ranking system!
Questions 3c, 4, and 5 focus on elimination of contestants – a hotly contested part of any show of this type!
Various choices and reasons for them are provided in the answers. You could give the students the following situation to consider: What if Ethan’s scores were 3, 4, 5? His median would still be 4, but his mean would be 4 as well. Therefore, Ariana and Ethan would both have the lowest medians and means. Who should be eliminated? (In this situation, it might be better to ditch each person’s worst score and find the mean of the other two scores.)

Extension

Students could investigate the judges’ scoring rather than the contestants’ results. The students will need to make the judges the subject of a graph (see the answers for Activity One) or rank contestants by each judge and compare lists. Have them look at how the judges’ opinions differ – is there a “nasty” judge? Any crossover in the graph may show that a judge dislikes a particular person.

Activity Two

Students may be interested in exploring whether the producers of the show are more interested in finding talent or making money. High viewer numbers mean more advertising revenue, so a contestant who gets a bigger audience is more valuable than the others, regardless of talent or judges’ scores. Data is important in decision making – but the right data needs to be collected! (You may need to suggest to the students that they look again at the performance order of contestants given on page 5.)
Question 2 should provide a good discussion point about what makes a “good” graph. Encourage the students to look critically at their own graphs to see if they can fi nd better ways to tell the stories involved.

Extension

Data could be collected from your school speech competition (or other judged event) and analysed in a similar way. Students could do a food-tasting experiment – judging three different fl avours of a particular food. Data could then be analysed similarly.

Answers to Activities

Activity one
1. a. A suitable graph would be:

graph.

b. Very little. (You can’t tell how good the contestants really are because the judges’ scores vary so much.)
c. The graph tells you quite a lot about the judges. For example:
• Judge 2 generally gives higher scores than the other two.
• Judge 1 consistently gives the lowest scores.
• For 7 contestants, judge 2’s score is at least double judge 1’s score.
• Half of judge 1’s scores are much lower than those of the other two judges, and the other half are equal to or 1 point less than judge 3’s.
• Judge 2’s scores vary most (6 are higher than those of the other judges, and 4 of these are 3–4 points higher than the next score).
• Ethan is the only contestant that all 3 judges agree on, with scores within 1 point.
• The judges’ scores don’t seem to be very consistent across the contestants.

There are other possible graphs, including a multiple line graph like this:

graph.

2a.

table.

b. In the graph below, the highlighted bar shows the median and the line drawn across each set of three bars shows the mean.

graph.

c. There are a number of comments you can now make about contestants, including:
• Based on the median, the top 3 contestants are Alexia, Lesieli, and Aoife.
• The median clearly identifi es the lowest 4 contestants.
• The median does not help sort out the middle of the fi eld because 3 contestants have a median score of 5.
• The mean puts Alexia and Lesieli in top-equal position and Whetù and Simon in third-equal position.
• Both median and mean identify Alexia as the top contestant.
• The median identifies Lesieli as second equal, and the mean identifies her as second.
3. a. The median looks at the middle judge’s score and is not affected by the other two judges’ scores. The mean considers all three judges’ scores and can be
strongly influenced by a particularly high or low score. Where the highest score is a lot higher than the other two scores (for example, Hone’s), the mean score is higher than the median score. Where the lowest score is a lot lower than the other two scores (for example, Ding’s), the mean score is lower than the median score. The median is always one of the judges’ scores, but this may not be the case for the
mean score.
b. Two possible rankings are:

rankings.
The first table is sorted by the median, and the second is sorted by mean scores. (You might also sort by the total score. This would have the same effect as sorting by the mean, so Alexia would still be top.)
c. Ariana because she has one of the lowest median scores and the lowest mean score.

4a.

table.
b. Comments could include:
• The public’s top 4 contestants (Simon, Alexia, Whetù, and Lesieli) are also the top
4 contestants according to the means of the judges’ scores, but in a different order.
• The public’s top 4 contestants are all in the group of the top 6 contestants according to the medians of the judges’ scores, but in a different order.
• Of the public’s bottom 3 contestants, Ethan is equal bottom according to the judges’ median rankings. He and Ding are second bottom according to the judges’ mean ranking, but Aoife doesn’t show in the bottom 4 in either ranking.
• Ariana is the lowest ranked by the judges’ mean score and equal bottom according to the medians, but she is ranked 6th by the public.
5. Choices and reasons may vary. A good place to start would be the 3 lowest-scoring contestants in the public poll: Ethan, Ding, and Aoife. Of these, Aoife scored best with the judges (second-equal median and fifth-equal mean). That leaves Ethan and Ding.
Ethan got the lowest public vote and a lower median judges’ score than Ding, which suggests Ethan could fairly be the one to go. Although Ariana scored bottom
with the judges, she came sixth in the public vote, so she should probably remain in the competition at this stage. Alternatively, you could start with the bottom 3 according to the judges and then compare them with the public vote. From both perspectives, Ethan should be eliminated.
 

Activity Two

1. The graph shows that there is a peak when the fifth performer is on stage. The fi fth performer is Tàne, who may be the contestant whom people love to hate
(one member of the audience says he is “so cool”, but that doesn’t mean he can perform well!). Although he scores in the bottom 4 with both judges and public, he
generates big audiences, so the producer may not even look at eliminating him at this stage.
2. Ideas may vary, but the most important improvement would be to replace the “performance order” numbers on the horizontal axis of the graph with the names of the performers. This would not only save you having to refer back to the table on page 6, it would also mean that the graph showed all the information needed to tell its story.

Key Competencies

Singing Star can be used to develop these key competencies:
• thinking: discerning if answers are reasonable, hypothesising, thinking critically, and making decisions
• using language, symbols, and texts: interpreting statistical information, capturing thought processes, interpreting visual representations such as graphs and diagrams, and demonstrating statistical literacy
• relating to others: sharing ideas and understanding others’ thinking.

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Level Four