Three Experiments
Here we take a closer look at the first three illusions. We shall see how effective they are. You will need the three experimental models and page R3. Do the experiments in Getting Results. Your teacher will give you the equipment.

Getting the Results
Experiment 1

Experiment 2

Experiment 3

Your teacher will record the class results and give thern to you. You will need them for the next section.

Finding the Median
The experiments gave lengths in centimetres. The lengths can be put in order. We can use this order when drawing tables or diagrams. We can also use it to calculate a median.

The 29 children in Class 2J did experiment 1. Their results are given in Table 5.

Table 5 - Class 2J: Results of experiment 1

We can put the pupils in order bv looking at their answers. The two pupils who said 11.4 cm are numbered 1 and 2. No pupils said 11.6 cm, so the eight pupils who said 1 1.8 cm are numbered 3 to 10. and so on.

These figures are shown in the third column of Table 6.

Table 6 - Class 2J: Results of experiment 1

Pupil number 15 (the 'middle' pupil) said 12.0 cm. The median is 12.0 cm.

In Class 2J there is an odd number of pupils (29), so there is one middle pupil (the 15th). With an even number of pupils there will be two in the middle. The median is then taken as the mean of their two answers. For example:

Suppose another pupil joined Class 2J, and his answer to experiment 1 was 12.8 cm. Then, of the 30 pupils, numbers 15 and 16 are the middle ones. Their answers are both 12.0 cm, so the median is ¹/₂( 12.0 + 12.0) cm = 12.0 cm.

A Diagram
A diagram can help to interpret the figures in Table 4. Look at Figure 3.

Figure 3 - Class 2J: Results of experiment 1

Notice that:

The bars touch.

The bar for 11.8 cm goes from 11.7 to 11.9 cm. This is because any length between 11.7 and 11.9, read to the nearest 2 mm, is 11.8 cm.

The Mean
In your experiments everyone recorded lengths. There were probably several different answers. We can find the 'average' length estimated by the class. This is called the MEAN length.

One way to find the mean is to add all the individual pupil readings and then divide by the number of pupls. For example:

If five pupils had separate readings of 11.0 cm, 11.4 cm, 12.2 cm, 11.6 cm and 11.4 cm, the mean would be:

^{(11.0 + 11.4 + 12.2 + 11.6 + 11.4)} / ₅ (the number of pupils)

So in this case the mean would be ^57.6/₅ cm = 11.52 cm.

(This is 11.5 cm to 1 decimal place.)

This method will give the correct answer, but takes quite a long time if there are many pupils. An easier and quicker way is to use multiplication to help the addition. Look at Table 7.

The two pupils' readings of 11.4 cm add up to 22.8 cm (11.4 cm x 2).

The eight pupils' readings of 11.8 cm add up to 94.4 cm (11.8 cm x 8); and so on.

The total of all the class readings is 350.8 cm.

So the mean is ^350.8/₂₉ cm = 12.1 cm to 1 decimal place.

Table 7 - Class 2J: Experiment 1

Summary

How Right Are You?
It is unlikely, that everybody, had the correct answer to any of the experiments. If they did the experment was not very effective! But how near no the correct answer did you get?

Class 2J found the accurate answer to their experiment 1. It was 12.4 cm. Only five pupils had the correct answer. Twenty-five pupils had answers which were too low.

The class median was 12.0 cm. This was not exactly right.

Using the median as the class answer, we can sav that the class error was:

12.4 cm - 12.0 cm = 0.4 cm

The class mean was 12.1 cm. This was not exactly right. Using the mean as the class answer, we can say the class error was: 12.4 cm - 12.1 cm = 0.3 cm

Ask your teacher for the correct answer to each of the three experiments. Answer questions a to e for each experiment.

*A Closer Look
Table 8 shows the errors in Class 2J's answers to experiment 1.

Errors are measured from the true length of 12.4 cm. They are found by taking 12.4 cm from each answer.

Answers below 12.4 cm give negative errors.
Answers above 12.4 cm give positive errors.

Correct answer = 12.4 cm

Table 8 - Class 2J: Errors in results of experiment 1

An error of 0 means the answer was correct, i.e. no error.

An error of - 4 mm is twice the error of - 2 mm.

We can find the mode, median and mean of these errors.

We can also draw diagrams.

For Class 2J:

The median error was -0.4 cm (the error of the 'middle' pupil number 15).

The mean error was ^{- 8.8}/₂₉ cm= - 0.3 cm

Draw a blank table like Table 8. Put in the results of one of your experiments.

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