Brief Description

This unit investigates whether the Equal Pay Act is working, by applying statistical techniques, and considers some of the inherent problems in making comparisons. The median, cumulative percentages and the interquartile range are used.

Design Time: 4-5 hours (C5, E3 optional)

Aims and Objectives

On completion of this unit pupils should be able to convert percentage frequency tables of grouped data into a cumulative form and draw the cumulative graph. They should be able to obtain the median and the interquartile range from this graph. They practise interpreting, reading tables and reading percentiles from the cumulative frequency graph. They become more aware of the interquartile range as a measure of spread, of some sources of published data and of the way statistics can throw light on everyday affairs.

Prerequisites

Pupils need to be able to multiply two 3-figure numbers with decimals (B1d), plot points on a graph, interpret percentages and calculate a simple mean.

Equipment and Planning

Section A introduces the Equal Pay Act and some of its implications.

Section B takes a first look at the April 1978 data to establish whether weekly or hourly pay rates provide a better comparison and to determine the effect of footnotes on two mean pay rates.

Section C introduces cumulative percentages so that median and other intermediate pay rates can be compared. In Section D the interquartile range is used to compare the spread of pay rates. The April 1978 figures are compared with earlier figures in an attempt to measure the effect of the Equal Pay Act.

Section E goes on to compare wage rates in comparable jobs and similar industries; it can be left to a later date for revision.

Detailed Notes

Section A

The 1974 roadside notice is a particularly blatant example of discrimination, others are more subtle. The principle involved is one of equal pay for equal work. Establishing a fair rate of pay for different jobs is much more difficult. It is raised here, and again in E1, since the fact that men and women do different jobs prevents aggregate data from showing whether the Equal Pay Act is working. These problems can emerge from the discussion questions and in question d, which has no clear-cut answer. Supply and demand, union organization and militancy, as well as skill, are important factors in determining the actual rates paid. The whole concept of fair pay is fraught with difficulties, as evidenced by wage negotiations in this country.

The Equal Pay Act 1970 came into force on 29 December, 1975. Its purpose is to eliminate discrimination between men and women in regard to pay. There is a right to equal treatment on broadly similar work or jobs which have been given an equal value to a man's job under job evaluation. The Act applies both to men and women, but does not confer any rights on people to claim equal treatment with other people of the same sex. Further details and a free booklet on the Act may be obtained from local employment offices or from The Equal Opportunities Commission, Overseas House, Quay Street, Manchester, M3 3HN.

Section B

The simplest and crudest measurements of pay rates of men and women are the national averages. There are several sources available (e.g. the Inland Revenue Statistics, annually). We have used the New Earnings Survey. Some of the limitations of these crude measures are mentioned below.

B1
Since men and women work different hours the hourly rate is a better reflection of pay rates than the weekly rates. However, even the hourly rate can be confusing, since longer hours would include a greater proportion of higher overtime rates; but this usually has little overall effect.

B2
Table 1 contains average gross earnings. The average used is the mean (see text below the table). Some implications of this are followed up in C5. The 'pound in your pocket' (net pay) is what people notice more when budgeting or shopping. Nevertheless, it is probably fairer to use gross pay for comparison of earnings. True net pay figures would be much harder to collect. Tax allowances are not equally distributed between husband and wife.

B3
This shows the importance of footnotes in interpreting the data. The fictitious small factory shows that it is possible to keep strictly to a principle of equal pay and yet not finish up with equal mean pay for men and women. This is because of the effect of (i) the clause on age, (ii) men and women doing different jobs, and (iii) including figures for the selfwmployed. Although children have no objective evidence of this last effect, general knowledge may lead them to expect that relatively more men than women are self-employed, and that the self-employed have higher-than-average incomes. Together, these two statements imply that men's earnings are underestimated by excluding the self-employed.

Section C

Means are unduly influenced by extreme values and can become unrepresentative. Medians may be more representative. Percentiles yield a fuller comparative picture between wage distributions of men and women.

C1
These data give the distribution of hourly pay rates. The differences between men's and women's rates can be seen in the table; these are illustrated in questions a-d.

A histogram could be drawn, but the unequal intervals and the last open interval cause problems. Strictly speaking, the plotting should be done so that the area of the rectangle is proportional to the frequency and the vertical axis labelled accordingly. One could use intervals of 60p, starting with 60p, and adjust the last two or three intervals only.

C2
Questions a, b show how the cumulative frequency table is built up. Questions d to f give simple pracice in reading the table. Question c is intended for discussion, the assumption is probably not fair.

C3
Since any algebraic formulae for interpolation would be too difficult at this age, the cumulative frequency table is of limited use. Drawing a graph of the cumulative percentages shows the distribution more clearly and allows us to read intermediate values. The successive points are joined by straight lines for the following reasons:

1. It matches the assumptions of uniform distribution within each class made in:
   1. calculating percentiles using linear interpolation,
   2. calculating means.
2. Straight lines are always easy to draw.
3. We can make straight lines pass through all the known points of the distribution.
4. Drawing a curve through all the known points is often very difficult (as in this case) and may be quite subjective.
5. Fitting a curve to the frequency distribution is fitting a theoretical model. This theoretical 'curve' would not be expected to pass through the top mid-points of the histogram bars. Curve fitting to the cumulative frequency is also model fitting, and the curve need not pass through the known points. Yet here we are not fitting a model, we have true population (not sample) percentiles and these must lie on the cumulative frequency graph.
6. Great precision is neither needed nor available, fine detail would be specious. The straight-line method is quick and unambiguous between different people working on the same data.

   On the other hand, it is argued that a curve through the known points can be thought of as taking greater account of the data to estimate the true (i.e. original, before any classification of data was carried out) distribution within the class. For this reason some people do prefer to draw a curve through the plotted points.

   All questions in C3 involve reading from the x-axis to the y-axis only. More similar questions could be asked (e.g. up to £2.75) if required. We suggest that, because of problems of scale, pupils should plot the points for the graph at 60p, 80p, etc. (not at 59¹/₂p, etc.). The answers given for quartiles are from original data; other answers connected with the graphs are estimated by linear interpolation. Pupils'answers from their own graphs may differ slightly.

C4
Reading the cumulative frequency graph from the y-axis to the x-axis gives us percentiles. The median is the 50th percentile.

The questions give practice in reading percentiles to prepare for the interquartile range in the next section. Notice that the figure for the 20% lowest paid men is the same as that for the 80% highest paid men (optional j). Questions k and l emphasize the difference between men's and women's pay rates.

C5
Here the different possible averages are compared. The mode is unsatisfactory as it depends on the choice of intervals. The mean gives undue weight to high wage earners, so the median is probably the most appropriate average to use here.

Section D

D1
The interquartile range can be thought of as the wage rates of the middle 50%, removing anomalies in the extremes. The semi-interquartile range ,¹/₂(Q₃-Q₁) is an alternative measure. Comparison of the interquartile ranges from 1975 and 1978 shows some changes. There is a larger overlap between men and women. The absolute difference between the median rates is about the same, even though the overall pay rates are much higher. This may have been due to the Equal Pay Act, but we can't be sure because of other influences on the figures.

D2
When studying change it is best to look at data over a number of years. The interquartile ranges for men and women have both increased; so has the overlap between men's and women's wages. This is to be expected since wages are higher overall, and the Equal Pay Act seems to have made no significant impact in increasing women's pay (although that is not its main purpose).

Section E

One problem which has not been dealt with is comparability of jobs. It is clearly important, but it is rather difficult to make a fair comparison. Here we have used the broad job classifications devised by the government; we also compare wages in different industries.

E1
This is intended to illustrate the effect of one high wage-earner on the average wage. Men are usually paid more because longer experience usually leads to promotion. There are many other factors too, possibly including discrimination. Whether or not it is fair can be discussed in class: we reserve our judgement.

E2
Here the wage rates of different types of job in the food industry are given. There is still quite a big difference in the wage rates: men get consistently higher rates than women - the men's median is appreciably higher than the women's third quartile. One would compare occupations in other industries to make a more informed answer to d.

*E3
This optional section is intended for reinforcement. The wages of manual workers in various industries are given: pupils may be interested in the range of wages in these industries. Again men's wages are universally higher than women's wages, particularly at the top end.

Additional Data

If you think your pupils need reinforcement of these techniques with different data, here are some additional data and possible questions.

Ages

Age distribution in England and Wales in 1821 and in 1975
(Sources: (i) P. Las ett, The world we have lost (Methuen, 1971)
(ii) Registrar General's Quarterly Return)

1. Draw the two cumulative percentage graphs using the same set of axes.
2. Find the median and interquartile range for each set.
3. What changes have taken place between 1821 and 1975? Can you explain them?

Tax

Personal income (1972/3)

1. Find Q₁ and Q₃ before and after tax.
2. Did income tax reduce the spread of incomes in 1972/3?

Cycles and Cars

Cylinder capacity of motorcycles (% of cycles)
*Excluding 104000 sidecar combinations registered separately.

Cylinder capacity of cars (% of cars)
Excluding older vehicles first classified by horsepower.
(Source: Transport Statistics Great Britain, 1964-74, Table 32.)

1. Use the interquartile range to find whether the variation in engine sizes has risen or fallen in the 10 years - for motorcycles and for cars.
2. Do the figures suggest a trend towards smaller engines in all vehicles? Check this by calculating the median of each distribution.

Answers

Test Questions

1. You have to compare men's pay with women's pay. Which of the following would you use?
   1. Gross weekly pay packet
   2. Net weekly pay packet
   3. Gross hourly rates
   4. Weekly basic rate without overtime
2. Which of the following is correct?
   1. Gross hourly rate is less than net hourly rate.
   2. Gross hourly rate is more than net hourly rate.
   3. Gross hourly rate is the same as net hourly rate.
3. A firm has 100 men and 100 women on the payroll. Table T1 shows their gross hourly pay rates.
   

   	Men	Women
   less than £1.00	5	12
   £1.00 to £1.99	20	43
   £2.00 to £2.99	55	35
   £3.00 to £3.99	14	8
   £4.00 to £4.99	6	2

   Table T1 - Hourly wage rates

   1. Complete a cumulative frequency table for the men and for the women.
   2. Plot cumulative frequency graphs for the men and women on the same axes. (put cumulative frequency along the y-axis, highest amount in each group along the x-axis).
4. Use your graph to answer the following:
   1. What is the men's median pay rate?
   2. What is the women's median pay rate?
   3. What are the lower (1st) and upper (3rd) quartiles of men's pay rates?
   4. What are the lower (1st) and upper (3rd) quartiles of women's pay rates?
   5. What is the interquartile range of the men's pay rates?
   6. What is the interquartile range of the women's pay rates?
   7. e is less than f. Explain in your own words what this tells you about the pay rates for men and women.
5. The total hourly wage bill for the 100 men is £270, and the total hourly wage bill for the 100 women is £180.
   1. Find the men's mean hourly wage rate.
   2. Find the women's mean hourly wage rate.
   3. Find the mean hourly wage rate for the men and women combined.

Answers

Connections with Other Pubilshed Units from the Project

Other Units at the Same Level (Level 4)

Choice or Chance
Testing Testing
Figuring the Future
Retail Price Index
Sampling the Census
Smoking and Health

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

Level 1

Leisure for Pleasure
Tidy Tables

Level 2

Opinion Matters

Level 3

Car Careers

This unit is particularly relevant to: Social Sciences, Humanities, Mathematics, Economics.

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.

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