Brief Description

This unit discusses appropriate levels of accuracy in various contexts. It distinguishes between variation from human error and bias from faulty instruments. It shows that using the mean reduces variation and demonstrates how to spot bias. Finally it discusses sensible answers in conversions and calculations.

Design Time: 4 hours

Aims and Objectives

On completion of this unit pupils should appreciate the appropriate level of accuracy to use in measurements and calculations and the distinction between variation and bias in causing errors. An optional section explains why the mean is likely to give a more accurate answer.

Pupils will have practised using the mean and collecting data. They should be more aware of the meaning of bias and variability and see how bias can be useful. They also see an example of using randomness in understanding errors.

Pupils are introduced to the idea of outliers and the range of a distribution.

Prerequisites

Pupils need to understand significant figures and be aware of Imperial units. They should be able to work with decimal and directed numbers and calculate the mean of a set of individual results.

Equipment and Planning

All pupils will need rulers and protractors. Pupils also need cm squared paper, tracing paper and an A4 sheet of paper in Section A2. Class measurements for Section B3 need to be collected, perhaps on the blackboard. Dotty paper with dots at the vertices of 1 cm squares would be useful for Section B5.

Section B5 needs eight matchboxes, eight envelopes and about 100 metal washers or objects of similar weight. The experiment can be done by pupils individually whilst the class is working through Section A. Section D is an option for more able pupils. For Section D2 each pupil or group of pupils requires four cubes or dice, two faces on each to be marked - 1, 0 and l respectively; alternatively coding could be used.

Detailed Notes

Section A

The opening questions are designed to promote a discussion on the level of accuracy appropriate in various situations. In some scientific experiments timing needs to be very precise; in others precision is not so important. Pupils should realize the need to think about the context in order to decide the level of accuracy required.

A2
The difference between 0.56 and 0.57 litres can be demonstrated as two 5ml medicine spoonfuls. Measuring the amount of milk poured into a cup for a cup of tea (in ml) will also help pupils see the futility of quoting five significant figures for the conversion factor in this context.

Other examples where more accuracy is needed are (i) scientific experiments and (ii) timing in modern sports events such as at the Olympic Games. Electronic calculators usually give too many figures for sensible accuracy.

A3
Some explicit examples in which pupils need to assess whether the accuracy is at the right level are given.

A4
These give examples where mathematics teachers may face increasing problems with the advent of calculators for pupils. Pupils may not be familiar with the formulae used. The diameter of a 10p piece is 2.8 cm.

A5
This gives two more examples where figures are given too accurately (a common abuse of statistics). The football figures were presumably the 'gate' figures. Other people also watched the match, e.g. the managers and trainers, policemen, St John's Ambulance men. It is impossible to count the population of a country like India to the nearest one, and in any case the population is changing almost every minute.

Section B

Here pupils make some measurements of their own to provide concrete data for discussion of levels of accuracy.

B1
Pupils are left to find their own way of measuring the diagonal, which is bigger than their rulers. They may need to be reminded of a method of using part squares to find the area of the given region: the region could be traced and cm square graph paper used. The traced map could then be used in Section B5 with dotty paper.

This section shows the degree of variability that can be expected in fairly simple circumstances. It is worthwhile using other data from the science department to amplify this section. An example is the timing of the fall of a ball-bearing through a viscous fluid.

B2
Pupils should see that John's results are not accurate enough, while Ann's are too precise. You could illustrate this practically. Pass round two sheets of paper 25 cm and 30 cm long. Ask pupils to mark one as l (longer), s (shorter) without lining one against the other. Similarly, pass round two sheets 25.71 and 25.72 cm long (as near as possible) for pupils to distinguish.

B3
Pupils are asked to choose a sensible answer, after discarding outliers. They will need the class measurements from Section B1. The 8.2 was measured in inches, not centimetres. Clearly this is a major error and should not be considered in the same light as the other readings. Distinguishing errors and outliers is not easy. Anything well outside three standard deviations from the mean of all the data could be ignored.

B4
This uses the mean to reduce variation and improve accuracy.

B5
Here dotty paper and randomness are used to estimate area. Further verbal instructions may be needed by slower pupils. The rule of counting the dots is equivalent to counting each square if its centre is in the region. The dots can be thought of as being at the centres of the centimetre square grid. A small piece of tracing paper, but large enough to cover the region, should be used.

B6
This experiment shows that, although individuals may have widely varying answers, the class answer can be surprisingly accurate. You need eight identical matchboxes. In seven boxes put in 6, 8, 10, 12, 14, 16 and 18 washers, and in the eighth 'standard' box put in 13 washers. Other identical metal objects such as screws could be used instead. The idea is that it should not be easy to tell by hand the differences in weight. You need seven envelopes and about 100 slips of paper marked l, h, or s. Pupils take it in turn to compare each of the seven boxes with the standard one and put the appropriate slip of paper in the appropriate envelope. At the end, count up the results and give the true order to the class.

Section C

Accuracy depends on being unbiased and on minimizing variation.

C1
Both Ann's and Barry's results are biased to the right, but Ann is more accurate than Barry. This section shows how one might spot bias.

C2
Pupils are given examples of measurements in which they have to decide on bias. They may need reminding that a set of measurements is to be assessed, not each individual measurement in a, b.

C3
Biased estimates are useful in situations where errors in one direction can lead to problems or even disasters. Pupils could be encouraged to give their own examples.

C4
Normal advice for the mean is one significant figure more than the original data, but clearly this rule depends on the size of sample. This could be mentioned to pupils here.

*Section D

This optional section introduces a very simple probability model to try to help explain why using the mean of a number of readings is likely to be more accurate than one reading.

D1
Here the improvement from one measurement to the mean of two measurements is investigated theoretically. Two measurements give:

Clearly this is an improvement.

D2
The distribution of the mean of four measurements is simulated using cubes or dice. The numbers -1, 0, I should be stuck on the faces of the cubes. The theoretical possibilities are:

The line graph of the simulation should show the greater concentration near 0 and hence the greater likelihood of being near the accurate answer. If you don't have any plain cubes, then ordinary dice can be used and coded as described in the pupil unit.

Answers

A2	a	No. Housewives don't measure so accurately in cooking
A3	a	Too accurate; 100 metres
	b	Too accurate; 114 g (or even 100 g)
	c	Sensible
	d	Not accurate enough; about 22.7 (or 23) litres
	e	Not accurate enough; about $13.91 with the exchange rate quoted
	f	Too accurate; 36.9oC
	g	Too accurate; 4047 sq metres (or 4000 m2)
	h	Not accurate enough; 21/2d
	i	Not accurate enough; 2.27 (or 21/4) kg
A4	a	Not accurate enough; 8.8 cm
	b	Sensible
	c	Too accurate; 6.16 sq cm
A5	a	People may be there illegally; 39600 or 40000
	b	The population is not static each hour, never mind per day! 534 300 000
B1	a	About 36.4 cm
	b	About 43o; 134o
	c	About 18 sq cm
B2	a	Yes; no
	b	No; not accurate enough; no
	c	No; no
	d	No; too accurate
	e	Probably measured the acute angle
B3	a	8.2; measured in inches not centimetres
	b	Variation due to human (or instrument) error; also all sheets of paper may not be identical in size
	c	20.8 cm
	d	20.6 to 20.9 cm
C1	b	Rifle biased, or possibly bad aiming
	c	Bad aiming; rifle biased and they didn't allow for it
	f	Yes
C2	a	1, unbiased; 2, biased below; 3, biased above; 4, unbiased
	b	1, unbiased; 2, biased above; probably read from the end of the ruler, not the zero mark
	c	Unbiased
	d	Biased above
	e	Biased below
C3		Answers here are open to discussion
	a	Over 120 feet
	b	About 250-270, depending on evenness and exact dimensions of walls (certainly more than 242)
	c	Slightly smaller and use putty (it is impossible to cut off, say,0 3 cm, if too large.)
	d	About 60p (more than 50p)
	e	Probably three rolls, depending on walls, windows, doorways and pattern 'drop'
	f	About 170 cm
	g	About 48 mph
	h	Just less; easier to add more
D1	b	Second measurement -1 0 1 First measurement -1 -1 - 1/2 0 0 - 1/2 0 1/2 1 0 1/2 1 Table 2
	c	3/9 = 1/3
	d	See detailed notes.

Test Questions

1. Your friend says the classroom is 4.12312 metres wide. Write down a more sensible answer.
2. The official population of England is 46417600. Your friend wants to know about how many people live in England. What will you tell him?
3. David and John roll balls at a hole. The pictures show where the balls finished.
   
   Describe David's results. How do they differ from John's results?
4. Jane weighed a piece of rubber three times. Her results were: 8.4 g, 8.3 g and 8.4 g. The true weight was 9.1 g. What can you say about her results?
5. A boy estimates how long it takes to cycle to school. Why is a biased estimate useful?
6. Six boys look at their watches as the school bus leaves school one afternoon. They record the time as 3.29, 3.33, 3.28, 6.17, 3.27 and 3.30.
   1. What is a sensible time to give?
   2. What would you do about the 6.17 result?
   3. How do you think the 6.17 result happened?
7. Ignore the odd boy out in the previous question. Use the other five results to get a more accurate estimate of the time the bus left the school.
8. Charles weighs a conker once. Ann weighs it four times and takes the average weight. Write down which of the next three statements is correct.
   1. Ann's answer is more accurate.
   2. Ann's answer is probably more accurate.
   3. Charles's answer is more accurate.
9. How can 'dotty' tracing paper be used to estimate the area of a difficult shape?
10. How can you use the same paper to get a better estimate of the area?

Answers

1		4 metres or 4.1 metres
2		46 or 461/2 million
3		David's results are close together but biased above the hole. They are different from John's results, which are not biased but more varied.
4		Biased results
5		To avoid being late, overestimate.
6	a	3.30
	b	Ignore it.
	c	Misread (hour and minute) hands, or the watch had stopped.
7		Attempt at 1/5 (3.29 + 3.33 + 3.28 + 3.27 + 3.30); answer 3.29
8		b is correct.
9		Count the dots.
10		Throw at random; take mean.

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 2)

Authors Anonymous
On the Ball
Seeing is Believing
Fair Play
Opinion Matter

Units at Other Levels in the Same or Allied Areas of the Curriculum

Level 1

Practice makes Perfect
If at first...

Level 3

Cutting it Fine
Multiplying People
Net Catch

Level 4

Smoking and Health

This unit is particularly relevant to: Science, Mathematics.

Interconnections between Concepts and Techniques Used in these Units

These are detailed in the following table. The code number in the left-hand column refers to the items spelled out in more detail in Chapter 5 of Teaching Statistics 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teaching the unit, but rnore practice can be obtained before or afterwards by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

Page R1

		Second measurement
		-1	0	1
First measurement	-1	-1	- 1/2
	0
	1		1/2

Table 2 - Mean error in two measurements

Table 3 - Proportions of each mean error with two measurements

Table 4 - Four measurements

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