Brief Description

This unit looks at the results obtained and goals scored in English league football. It is shown that the proportion of home wins and of other results settles down. These are used to predict results with reasonable success, particularly when compared to newspaper forecasts. The number of goals scored nowadays is compared to the number scored 20 years ago.

Design time: About 3 hours

Aims and Objectives

On completion of this unit pupils should be able to use known relative frequencies to estimate probabilities and to use probabilities in predicting future events.

They will become more aware of the fact that relative frequencies do settle down, the problems of comparing sets of data and the need for additional information to draw more reliable conclusions. They practise calculating the mean, picking out other summary statistics from data and plotting data as a time-series on a graph.

An optional extension of B1 (see notes below) provides practice in completing a two-way table and investigating patterns which emerge.

Prerequisites

Pupils need to be familiar with tally marks and with calculating the arithmetic mean of ungrouped data. They will need to do long division, convert fractions to decimals and work to two significant figures: calculators would help here. A familiarity with the use of random number tables would be useful but is not essential.

Equipment and Planning

This unit needs to be done during the football season, preferably at a time when matches are unlikely to be cancelled (before December or after February).

Results from last Saturday's matches are required for B1, though the given results could be used. Avoid any Saturday when Cup matches are being played. In Section D predictions need to be checked, so there will be a gap between D2d and D2e: the class could continue with Section E and return to D2 later. Football results are published in most Sunday newspapers, special Saturday papers and in some newspapers on Monday.

Random number tables are needed in Section D. Calculators are useful in B2, C1, C2 and Section E. Graph paper is also needed in Section E.

Pupils could work through the whole unit individually, though class discussion of certain points is recommended (chss results are.needed in C1b).

Detailed Notes

Section A

The opening section reminds pupils what is meant by a score draw, etc., and poses the general questions considered by the unit. Recently each team scored an average of just over a goal per match, though Liverpool did do rather better in the 1978/79 season - a statistical freak perhaps. More goals were scored 20 years ago when leading goal-scorers averaged a goal per match: relevant data is considered in Section E. Prediction of results is done in Section D.

Section B

B1
It is recommended that recent results are used as they will have more impact and significance. It is probably easiest to read out the results while pupils complete Table 2, though pupils could bring in results themselves.

If many matches are cancelled because of bad weather, it may be better to postpone starting the unit, though pools panel forecasts could be used. It is also best to avoid Saturdays when Cup matches are played. There are generally more draws than away wins. You may like to discuss why the distinction between a no-score draw and a score draw was introduced: to increase the chances of a higher payout, because of the preponderance of draws.

Reinforcement is possible here and in B2, C1 by using the printed results as well as more recent ones.

The results can also be collated with tally marks on a two-way 7 x 7 matrix with home goals (0-6) along the top and away goals (0-6) on the vertical axis. The diagonal from top left to bottom right will show draws; above are home wins, below are away wins. This does take extra time but throws up interesting patterns and provides practice in completing a two-way table. Questions on reading a two-way table can also be set, for example, the number of matches where the total goals scored is 5 or the number of matches where the home team scores two goals or two more than the away team.

B2
It is not entirely valid to make predictions from just one Saturday. One needs to collect results over a longer period to make a fair comparison of goals scored in different divisions. In 1975/76 there was no clear difference. The relevant data can be found in the three references listed after E2. Pupils may need reminding that one divides by 11 or 12 in g.

Section C

C1
This introduces the idea of probability in a known situation (like balls in a bag) but using real data. Pupils need to make a dot 'at random', though this is hard to achieve in practice. The dot needs to be chosen fairly: perhaps the results page could be put down 'randomly'. It could be done by writing each result on a slip of paper and choosing one from all the slips. Note that class results are needed in C1b.

You could discuss the 'pin' and other methods of doing the pools with the class. The Rothschild report suggests picking from the last teams rather than the first teams for big prizes. This assumes that, although draws occur randomly in matches, fewer people choose draws near the end than near the beginning of the table. If this is true, then one is less likely to have to share a prize with other people if one chooses from the end of the list.

C2
The major difference between this section and C1 is that in C1 all the results are known, and probabilities were assigned by choosing at random from a known population. Here we use past results to estimate probabilities for future unknown results.

The idea here is that the proportions do settle down (an example of the law of large numbers) and can be used for prediction. Pupils may like to copy down Table 4 in answering a. If reinforcement of this idea is needed, the away wins and draws for Table 4 are given below:

In Table 4 the number of matches phyed is the accumulation of matches played during the season.

The advantage of the home team does not matter so much in the League, as two matches are played between each pair of clubs, each club playing once at home. It is more important for Cup matches.

Section D

D1
Here random prediction is contrasted to newspaper forecasting and often does surprisingly well. Pupils may need more explanation on how the random numbers are used in predicting.

There are problems in deciding which prediction method is best, D1b. If one predicts all matches to be home wins, one would expect half to be right; thus simply getting the most right is not a sufficient criterion. There must be discrimination in the prediction.

D2
Pupils predict results for next week and compare with newspaper forecasts. These are usually published in daily papers on Tuesdays or Wednesdays and for the following week in the Sunday papers. There will be a break after D2d; pupils could return to D2e after completing Section E or waiting until the relevant matches have been played. Squared paper could be used for the forecast results; this makes checking the forecasts easier.

Section E

E1
Data are given on goals scored by the First Division teams in 1957/8 and 1977/8. Initially pupils are asked to summarize the data and make comparisons; calculators may be useful here.

We used a Wilcoxon test to compare goals scored in 1957/8 and 1977/8. The goals scored are put in order (for both seasons), and the ranks for 1957/8 (which include the first 10) are totalled. The sum of ranks here is 314, which is significant at least at the 0.001 level.

E2
Here more data are presented to make a more meaningful comparison. Pupils may need help in choosing scales in a; b can be plotted on the same graph with vertical lines to show the range in each year. E2e can be treated as an open class discussion. Possible reasons are: football is a more tactical game - it is easier to plan to stop goals than to score them; there is more money in football; the result (winning) is more important nowadays than the method; goalkeepers are more valued (and may have improved).

Further data may be found in the Rothman's Football Yearbook (Queen Anne Press), Playfair Football Annual (Queen Anne Press) and the book of the Football League Tables (Collins).

Answers

A	a	Fulham, home win; Bury, no-score draw; Reading, away win
	b	Yes
B1	d	For our results: 22 home wins, 11 away wins
	e	13
	f	43
B2		The answers refer to our results which will probably differ from those used by pupils.
	b	Division 1: 16/14/30 Division 2: 22/10/32 Division 3: 14/11/25 Division 4: 19/16/35
	c	Yes, possibly. There will be variation; one needs to see results over several months to pick out patterns.
	d	No
	e	Results over several months
	f	More games in Divisions 3 and 4 than Divisions 1 and 2
	g	2.73, 2.91, 2.08, 2.92
C1	d	0.48, 2.24, 0.065, 0.28
C2	a	0.41, 0.46, 0.50, 0.49, 0.50, 0.49, 0.50
D1	a	13, 13, 10, 17, 14
	b	Daily Mirror, Sun
E1	c	1721, 1213
	d	78, 56
	e	51, 26
	f	104, 76

Test Questions

John is keen on European football. He collected the results of 50 matches:

35 home wins
5 away wins
10 draws
90 home goals
35 away goals

1. Use John's results to estimate the probability in future European matches of:
   1. A home win
   2. An away win
   3. A draw
2. Find the average number of goals scored in the 50 matches.
3. John wants to guess the results of the next 20 matches to be played. He decides to use random numbers. The result is a home win (H) if the number is 1,2,3,4,5,6,7; an away win (A) if the number is 8; and a draw (D) if the number is 9,0.
   He uses the random number table below, starting at the beginning of the line. Use the table to predict the results of 20 matches, writing H, A or D.
   52 14 49 02 19 31 28 15 51 01 19 09 97 94 52 43
4. John finds the results of the next 50 matches in European football:
   37 home wins
   4 away wins
   9 draws
   Estimate, using all the given results, the probability in future European matches of:
   1. A home win
   2. An away win
   3. A draw

   Would these probabilities be more accurate than those in Question 1? Give a reason.

5. The table below gives the number of goals scored in Division 2 in 1950/1 and in 1970/1. For each season find the mean number of goals scored and the range of goals scored. Write two sentences comparing the number of goals scored.
   

   1950/1	82 74 79 62 73 76 71 64 53 48 81 63 50 74 66 53 63 52 64 53 64 48
   1970/1	57 73 64 65 54 62 60 59 58 54 58 61 52 41 51 46 29 38 46 41 37 35

   How would you make a fairer comparison of the goals scored in Division 2 in the 1950s and the 1970s?

Answers

1	35/50 = 7/10, 5/50 = 1/10, 10/50 = 1/5
2	2.5
3	H, H, H, H, H, D, D, H, H, D, H, H, H, A, H, H, H, H, D, H
4	72/100, 9/100, 19/100; Yes, more results used
5	64.2, 48-82; 51.9, 29-73. More goals scored in 1950 than 1970; smaller range in 1950, even poor teams scored quite a lot of goals. Figures from other seasons are needed to make a better comparison.

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 2)

Authors Anonymous
Opinion Matters
Seeing is Believing
Getting it Right
Fair Play

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

Level 1

Shaking a Six
Probability Games
If at first ...
Leisure for Pleasure

Level 3

Phoney Figures

Level 4

Choice or Chance
Testing Testing
Retail Price Index

This unit is particularly relevant to: Social 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 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.

Page R1

77  04 01  09 73  89 84  35 77  76 12  39 43  64 97  40 83  99 18  26
39  00 29  43 44  23 01  92 63  88 89  61 91  67 90  04 22  34 19  93
63  78 56  92 64  87 82  73 33  53 25  36 40  91 19  52 36  40 91  19
52  67 36  19 67  84 34  55 97  37 92  30 27  26 71  04 71  78 38  15
58  21 59  06 07  57 57  99 40  43 47  18 03  62 91  41 60  90 45  13
24  65 06  55 72  04 87  31 29  39 56  29 93  95 65  90 95  99 87  46
66  36 07  93 49  20 02  59 48  54 35  73 34  68 72  44 28  87 44  81
09  77 10  52 52  52 65  29 15  82 81  23 56  99 82  21 01  62 81  98
14  56 32  69 71  27 29  74 87  24 79  42 66  10 50  75 47  87 08  26
35  84 64  56 47  54 11  22 93  84 75  65 06  91 47  47 67  25 97  25
08  35 58  94 06  04 02  41 56  90 12  38 09  87 20  22 20  30 72  51
39  84 92  69 36  47 42  09 72  28 20  63 90  67 24  56 54  27 12  89
16  20 61  32 75  91 50  16 53  51 83  14 30  93 83  74 59  31 70  81
54  35 42  49 55  57 13  50 70  03 72  39 48  67 94  73 37  67 13  39
66  29 74  71 55  60 88  08 10  62 08  10 55  28 51  86 52  75 00  14
59  00 51  60 44  72 59  53 94  22 10  74 38  54 43  43 45  29 91  74
43  45 29  91 74  43 58  08 72  99 89  09 38  66 75  45 49  00 47  42
75  47 88  59 25  21 04  61 07  14 40  73 42  68 67  25 68  76 98  45
28  80 46  57 74  80 62  57 51  32 33  42 06  56 17  81 94  25 05  63
58  62 21  99 86  58 90  78 87  05 96  57 38  14 37  35 05  51 87  25
87  71 56  03 65  03 11  69 23  98 78  64 52  19 04  99 04  73 90  48
41  21 95  96 34  83 03  16 31  72 11  50 65  47 58  80 68  92 79  82
77  93 27  40 49  08 05  83 42  49 80  95 99  46 24  51 85  74 13  83
81  27 96  24 42  13 33  55 25  65 91  39 43  36 83  32 40  32 48  71
93  44 83  25 03  62 06  48 98  74 38  18 76  63 58  44 87  58 91  26
47  04 95  29 28  67 85  59 17  41 49  89 23  35 50  90 28  97 55  86
20  52 82  47 00  24 00  46 69  91 07  37 21  93 54  92 73  09 06  08
36  67 47  47 03  16 69  50 48  41 70  97 26  43 30  52 10  16 85  03
35  60 74  94 29  84 89  72 57  65 49  30 11  61 54  88 18  85 68  32
37  80 42  50 20  09 57  58 41  58 42  62 17  11 94  98 81  98 04  49
10  91 74  06 38  02 57  04 25  67 52  47 72  59 62  22 42  44 98  26
10  17 59  75 76  74 67  12 19  68 34  28 32  54 11  80 14  51 42  07
42  45 57  52 07  84 44  43 01  65 20  56 64  01 46  39 26  73 83  92
01  61 18  96 23  36 41  01 57  70 20  29 64  90 49  77 41  32 85  93
74  91 20  66 07  62 81  51 40  58 26  21 96  98 14  57 69  96 99  86
30  25 71  25 27  20 69  11 38  51 41  67 45  95 22  35 55  75 36  20
84  64 38  27 68  61 01  90 31  58 18  77 70  79 15  29 55  10 20  18
28  69 32  14 56  22 86  70 48  24 83  87 16  63 66  62 21  74 98  04
38  40 21  06 72  81 04  57 41  98 12  60 98  24 11  51 34  27 02  49
06  36 38  42 84  53 41  95 37  29 48  68 72  86 22  22 71  76 85  09
30  36 31  16 12  35 75  25 20  31 83  50 84  83 34  07 37  45 09  73
18  87 76  43 56  63 19  65 36  86 14  47 86  86 30  97 48  08 80  49
32  70 17  68 75  98 52  05 67  68 22  94 80  18 05  90 28  45 40  52
66  60 69  56 87  43 72  87 76  43 40  66 08  77 50  43 70  91 86  54
32  60 71  47 28  06 21  63 63  16 25  32 21  35 62  47 20  42 08  87
43  89 32  54 85  23 87  60 87  38 11  47 76  85 83  97 89  52 11  56
49  55 09  63 51  15 26  48 22  99 40  82 75  31 19  71 87  57 58  67
00  04 13  23 93  86 64  21 15  55 69  21 19  54 22  57 61  46 85  70
99  50 06  22 15  92 33  21 68  45 25  97 27  21 06  67 93  15 96  29
80  62 34  15 07  51 34  99 93  37 31  96 54  85 39  37 94  10 91  51

Table 8 - Random Numbers

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