Brief Description

This unit deals primarily with quality control, but the notion of variability is central throughout. The unit concentrates on one particular aspect of the quality control process - that concerned with whether a stated minimum content is exceeded.

The quality control process

Quality control charts are employed to record information obtained when data are obtained sequentially. The sample mean is used as a representative measure of a distribution, while the range is used to measure spread.

Design Time: About 5 hours

Aims and Objectives

On completion of this unit, pupils should be able to construct quality control charts and interpret them in a simple way. They will have practised calculating means, drawing histograms, interpreting graphs and tables, using the range and obtaining and recording data. They will have been introduced to calculating the mean, using a change of origin.

The pupils should be more aware of the variability of weights of packeted goods, the implications of the stated 'Minimum contents' or 'Average contents' and some of the constraints governing quality control. They meet examples of difficulties arising in interpretation and comparison of statistics, of thinking quantitatively in a disciplined way about everyday affairs and of coming to decisions on the basis of statistical arguments.

Prerequisites

Pupils should be able to draw axes for graphs, plot points and perform simple operations with directed numbers. Subsection D2j, which is optional, requires an awareness of percentages.

Familiarity with the mean and bar charts would make progress easier but is not essential.

Equipment and Planning

Section A invites pupils to think about the notions of 'Average contents', 'Minimum contents'and 'Overweight'and provides a context for the work of the next two sections.

In Section B the pupils play two games to develop the concepts of (i) overweights and (ii) the constraints governing quality control, i.e. that production of underweights can lead to legal action while that of overweights can lead to financial problems.

About 250 (or at least 200) small cubes will be required by each pair of pupils for the games. Larger groups can be used if necessary. Beads, counters, small stones, plastic coins, small squares of card will suffice instead of cubes but each group of 250 must be homogeneous. Some pupils can work on Section C while others work on Section B to cut down the number of cubes required.

In Section C, pupils find the mean overweight of a sample of loaves (data supplied in text) and use page R1. They progress to construct a quality control chart for eight such samples. Graph paper is required. C3 is optional to reinforce the ideas of this section.

The flow of the unit would not be unduly interrupted if Section D is not continued until a later date.

In Section D bags of crisps are weighed. The overweights are analysed by means of a quality control chart and a histogram. D4 is an optional section in which the mean for ratio data with class intervals is calculated using a change of origin. This section and D2j may not be suitable for weaker pupils.

Twenty-five bags of crisps and a fast weighing balance accurate to 1g will be required for each group. The number of groups may be determined by the amount of equipment available. If the collection of data is done as a class activity, it may be started as pupils are approaching the end of Section C.

Graph paper is required, preferably of large size (to fit the histogram next to the quality control chart.)

Detailed Notes

Section A

A1
This section lends itself to class discussion. Reference may be made to packets of food at home, and some of these may be used in the classroom. It may be worth emphasizing that as well as expecting a lower mean for a packet marked 'Average contents 100 grams' than for a packet marked 'Minimum contents 100 grams', this system is less easily checked by the customers, since they will have no idea of what might constitute acceptable variation. European practice incorporates tests which check that not more than 1 in 40 of the packets weigh below a certain minimum. Price may be lower for packets marked 'Average' rather than 'Minimum'.

A2
The precis of The Grocer article helps to set a realistic context. The notion of the British practice of goods sold by 'Minimum contents'is reinforced.

Section B

B1
The purpose of the game is to help the participants develop a concept of overweight. The variation in weights during a production process is simulated by the pupils' estimates of piles of 16 cubes made within a time limit. Their record of 'weights' (asked for in a) is referred to in B2b. You may need to demonstrate the game and explain the rules. Sufficient counters should be made available to avoid undue constraint.

B2
The introduction of a fine for 'underweight' loaves shouia help to make the pupils aware of the two directional nature of the constraints which govern quality control and discourage the production of underweights. From the firm's viewpoint, while underweight goods may be more profitable in the short term, possible legal action and loss of reputation offset this.

Consistently overweight goods may be costly or uncompetitive if accounted for in the price. These points can be brought out in a class discussion.

Section C

C1
From the sample of 10 loaves, the mean weight and the mean overweight are calculated separately, underweights being considered as negative overweights. The pupils are encouraged in part f to deduce that:

Mean overweight = Mean weight - 400

This method is used later to find mean overweights. This idea is used in D4 as a method for calculating means using a change of origin, i.e.

Mean weight = Mean overweight + 400
where 400 has been subtracted to change the origin.

Questions j to m indicate that a change of origin does not give a change of range; 428 - 394 = 34 = 28 - (- 6)

C2
Pupils use the method from C1f to find the mean overweights for the eight samples and then construct the quality control chart. The sequence of the points shows two basic trends - downwards for samples 1-3 and upwards for samples 4-7. The inference is that adjustments were made after sample 3 and maybe sample 7.

The three charts shown below are suitable for reinforcement of ideas on interpretation of charts and what action should be taken.

1

The general trend is for the overweight to increase. Action should be taken to decrease the mean weight. Random variation exists about the trend.

2

The goods are consistently overweight. Again random variation exists. Action should be taken to decrease the mean weight.

3

The variability in overweights is progressively and systematically increasing. Action should be taken to reduce this variability.

C3
This section is optional and may be used as a reinforcement exercise, perhaps for homework. It is similar in nature to C2. The quality control chart, however, indicates a downward trend in the number of matches being put into boxes. The production manager will be alerted to a possible fault by the non-random pattern emerging and should take action to increase the number of matches being packed into a box. Despite this, the overall mean for the eight samples is 40 matches per box.

Section D

The school tuck shop may lend the crisps, suitable balances may be obtained from the science department.

Ideally all the bags of crisps should be (i) made by the same firm, (ii) of the same flavour and (iii) stamped with a net weight. Some aren't marked with net weight, but this information may be obtained by telephoning the manufacturers. (Different flavours have different weights, not because of the flavour, which is only in the added salt, but because they come from different production runs.)

D1
The pupils obtain the data themselves by weighing 25 bags of crisps and then calculate the overweights. If the actual weighing proves impossible, the following data may be used:

Twenty-five bags of Walkers plain crisps (minimum contents 24.5 grams) were weighed. The mean net weight was 29.4 grams. The bag weighed 2 grams (to the nearest gram), and the following are the weights of bag and contents, to the nearest gram.

36 31 33 33 27
36 32 31 30 28
30 29 30 33 30
31 31 28 35 34
33 34 31 31 28

Alternatively, the above data could be used for a reinforcement exercise.

D2
The work in this section assumes a minimum weight of 24¹/₂ grams. However, if the only bags obtainable are marked with a net weight equal to a whole number of grams, the scale on the chart in Figure 2 will have to be offset by ¹/₂g (i.e., to -3¹/₂g, -1¹/₂g, ¹/₂g, 2¹/₂g, 4¹/₂g, etc.). This is because many of the weights will fall on the boundaries between classes, thereby causing problems with the histogram. The overweights calculated in the last section are now analysed in a quality control chart and the variation in weight of the middle 60% of the bags is measured using its range.

D3
The histogram is constructed alongside the quality control chart to show the distribution of overweights. In c the pupils are encouraged to find the mean overweight of the distribution shown on the histogram. Technically, this is an estimate of the mean as:

2 bags are assumed to be overweight by 1 gram,
7 bags are assumed to be overweight by 3 grams,
6 bags are assumed to be overweight by 5 grams, etc.

This may only be fully appreciated by the more able pupils.

D4
Parts a, b and c, of this optional section serve to reinforce Section D3 in calculating the mean give-away weight. In d the mean weight of a bag is deduced, i.e.

Mean weight of a bag = 24.5 + mean overweight

Thus the mean of the distribution of weights has been calculated using a change of origin to 24.5. The large give-away weight in f is to avoid any possibility of prosecution.

This section is suitable for the more able pupils.

References

A. Huitson and J. Keen, Essentials of Quality Control(Heinemann, 1965).
M.J. Moroney, Facts from Figures, Chapter 11 (Pelican, 1951)
F. Mosteller et al, Statistics by Example; Book 3 Detecting Patterns, pages 29-32 (Addison-Wesley, 1973)

Page R1

Weight of loaf (grams)	Overweight (grams)
415	15
394	-6
408
422
428
397
425
406
414
421
Total overweight

Table 4 - Overweights of ten 400-gram loaves.

Table 5 - Mean overweights of eight samples of loaves.

Weight of empty packet _____ grams.
Weight printed on packet _____ grams.

Table 6 - Weights of 25 bags of crisps.

Total no. of bags = 1000
Total give-away weight = _____.

¹Zero means that the packet contains exactly 24 5 grams.
Table 7 - Overweights of 1000 bags of crisps.

Answers

A1	a and b	See detailed notes.
A2	a	more than 400 grams
	b	4/5
	c	Mean is likely to be more than 400 grams.
B2	d and e	See detailed notes.
C1	a	4130 grams
	b	413 grams
	c	15, -6, 8, 22, 28, -3, 25, 6, 14, 21 Negative means it is underweight.
	d	130 grams
	e	13 grams
	f	See detailed notes.
	g	428 grams
	h	394 grams
	i	34 grams
	j	28 grams
	k	-6 grams
	l	34 grams
	m	They are the same.
C2	a	No
	b	13, 5, -4, 2, 18, 22, 28, 28
	e	Too heavy
	f	Samples 1, 2, 3
	g	Sample 3. They increased the amount of dough being used.
C3	b	Sample 1
	c, d, e, f	See detailed notes.
D4	a	Column 3 Column 4 13 6 x 13 = 78 11 12 x 11 = 132 9 86 x 9 = 774 7 221 x 7 = 1547 5 320 x 5 = 1600 3 290 x 3 = 870 1 63 x 1 = 63 -1 1 x -1 = -1 -3 1 x -3 = -3 Total give-away weight = 5060
	b	5060 grams
	c	5.06 grams
	d	29.56 grams
	f	See detailed notes.

Test Questions

1. Bacon can be bought for the same price in packets marked either 'Minimum weight 500 grams' or 'Average weight 500 grams'.
   1. What is the advantage to the customer of buying the packet marked 'Minimum weight 500 grams'?
   2. What is probably true about the mean weight of bacon in packets marked 'Minimum weight 500 grams'?
2. A firm sells cheese in packets marked 'Minimum contents 250 grams'. Give one reason each why the firm does not want to produce:
   1. many underweight packets,
   2. many greatly overweight packets.
3. Tea is put into packets marked 'Minimum contents 125 grams'. A sample of five packets is taken from the first batch of the day. The results of weighing each bag to the nearest gram are: 128 grams, 126 grams, 127 grams, 124 grams, 130 grams
   1. What is the mean weight of tea in a packet?
   2. What is the mean overweight of tea in a packet?
   3. What is the range of these packets?
4. The quality control chart in Figure T1 is for eight samples taken during a day's production of biscuits. The biscuits are in packets labelled 'Minimum contents 300 grams'.
   
   Figure T1 - Quality control chart
   1. Which samples, if any, had a mean weight per packet of less than 300 grams?
   2. What was the mean weight per packet in sample 6?
   3. In one sample, the packets could be accurately labelled 'Average contents 300 grams'. Which sample is this?
   4. What action might the production manager want to take to correct any fault in this process?
5. Soap powder is packed in cartons marked 'Minimum contents 870 grams'. A sample of 10 boxes was taken every hour during the day. The mean net weight of powder was calculated. Table T1 shows the results.
   

   Sample number	Mean net weight per packet (grams)	Mean overweight per packet (grams)
   1	895	25
   2	885
   3	863
   4	852	-18
   5	875
   6	877	7
   7	865
   8	872

   Table T1

   1. The mean overweight of a packet in each sample is listed in column 3 of Table T1. Complete this column.
   2. Draw a quality control chart for these figures on the axes in Figure T2 below.
      
      Figure T2 - Quality control
      
      Figure T3 - Histogram chart
   3. Calculate the mean overweight per packet for all eight samples.
   4. The results from the first four samples show a trend. Describe this trend.
   5. When samples 5, 6, 7 and 8 were taken, was the process producing: cartons that were too heavy, or cartons that were too light, or cartons that were about right?
   6. The production manager took action during the day. When do you think this was done?
   7. Draw a histogram on the right of the quality control chart to show the distribution of the mean overweights. Put the numbers on the axes clearly.

 

Answers

1	a	The packet is very likely to contain more than 500 grams.
	b	The mean weight is likely to be more than 500 grams.
2	a	May be taken to court and fined or suffer loss of reputation.
	b	Causes financial loss or is uncompetitive if accounted for in the price.
3	a	127 grams
	b	2 grams
	c	6 grams
4	a	Sample 1
	b	320 grams
	c	Sample 3
	d	He would want to eliminate the systematic increase in overweights, and reduce the mean overweight.
5	a	(25), 15, -7, (-18), 5, (7), -5, 2
	b	Figure T2 Figure T3
	c	3 grams
	d	The trend shows that the mean net weight per packet was decreasing over the period when the first four samples were taken.
	e	About right
	f	After sample 4
	g	See Figure T3.

 

 

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 3)

Car Careers
Phoney Figures
Net Catch
Pupil Poll
Multiplying People

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

Level 1

Getting a Six
Probability Games
If at first...
Being Fair to Ernie
Practice makes Perfect
Leisure for Pleasure

Level 2

Seeing is Believing
Fair Play
Getting it Right

Level 4

Figuring the Future
Retail Price Index
Sampling the Census
Smoking and Health

This unit is particularly relevant to: Science, Mathematics, Commerce, Social Sciences.

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 more 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.

Code No.	Statistical Prerequisites
	None
	Idea or Technique Used	Introduced in	Also Used in
1.1a	Data collection from small population simple data	Sampling the Census	Practice makes Perfect Getting it Right Leisure for Pleasure
1.2a	Using discrete data	Seeing is Believing	Shaking a Six Probability Games Leisure for Pleasure Getting it Right Net Catch Phoney Figures Sampling the Census Being Fair to Ernie If at first... Fair Play Car Careers Multiplying People Figuring the Future Retail Price Index
1.2b	Using continuous data	Seeing is Believing	Practice makes Perfect Getting it Right Leisure for Pleasure
1.3c	Sampling from distributions and infinite populations	Getting it Right	Being Fair to Ernie Fair Play If at first...
1.4a	Data by direct counting and measurement	Shaking a Six Leisure for Pleasure Sampling the Census	Being Fair to Ernie Net Catch Fair Play Retail Price Index
2.2a	Bar charts	Shaking a Six Leisure for Pleasure	Being Fair to Ernie Seeing is Believing
5b	Reading bar charts histograms and pie charts	Being Fair to Ernie Leisure for Pleasure Car Careers Multiplying People Phoney Figures Smoking and Health	Seeing is Believing
5d	Spotting possible errors (outliers) as not fitting general pattern	Getting it Right Multiplying People Smoking and Health
5v	Inference from tables	Leisure for Pleasure Net Catch Multiplying People Phoney Figures Figuring the Future Sampling the Census Retail Price Index Smoking and Health	Shaking a Six Seeing is Believing Practice makes Perfect
	Idea or Technique Introduced	Also Used in
1.3e	Variability in samples	Being Fair to Ernie Fair Play Net Catch Probability Games Getting it Right Pupil Poll If at first... Car Careers Smoking and Health
2.2f	Histogram for grouped data	Leisure for Pleasure
2.2j	Plotting time series	Car Careers Figuring the Future Multiplying People Smoking and Health Phoney Figures
3.1c	Mean for small data set	Practice makes Perfect Fair Play Net Catch Retail Price Index If at first... Getting it Right Phoney Figures Smoking and Health Seeing is Believing Car Careers Figuring the Future
3.1d	Mean for small data set change of scale
3.1f	Mean for frequency distribution	Seeing is Believing Sampling the Census Fair Play Car Careers
3.2a	Range	Practice makes Perfect Figuring the Future If at first... Phoney Figures
3.2b	Fractiles
5c	Reading time series	Practice makes Perfect Multiplying People Leisure for Pleasure Phoney Figures Car Careers Figuring the Future
5e	Comparing directly comparable data	Practice makes Perfect Retail Price Index Figuring the Future Smoking and Health Sampling the Census
5g	Looking for sources of non-comparability	Sampling the Census Retail Price Index Smoking and Health
5t	Costs and risks in decision making
5z	Detecting trends	Practice makes Perfect Phoney Figures Car Careers Sampling the Census Multiplying People Smoking and Health

 

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