Contents
Study guide 4
Introduction 5
1 Presenting and summarizing data: the silver darlings 5
1.1 Presenting data 5
1.2 Describing samples of data 10
2 Introducing SPSS: North Sea cod 14
2.1 Navigating SPSS 14
2.2 Printing and pasting output 20
2.3 Line plots and scatterplots 21
2.4 Histograms and numerical summaries 24
3 Populations and models: health e铿?ects of
air pollution 27
3.1 Samples and populations 28
3.2 Probability models for continuous random variables 32
3.3 Probability models for discrete random variables 35
4 From samples to populations: asthma and air quality 40
4.1 Samples and estimates 41
4.2 Con铿乨ence intervals 42
4.3 Testing hypotheses 45
5 Related variables: pollutants and people 49
5.1 Association between two continuous variables 49
5.2 Association between two discrete variables 53
6 Statistical modelling in SPSS: the air we breathe 57
6.1 Transforming variables 57
6.2 Con铿乨ence intervals and correlations 60
7 Modelling exercises 63
Summary of Unit 65
Learning outcomes 65
Solutions to Activities 66
Solutions to Exercises 70
Index 74
3
Introduction
In M249 Practical Modern Statistics you will be introduced to four topics in
statistical modelling: medical statistics, time series, multivariate analysis and
Bayesian statistics. Each of these topics is largely self-contained, and most of the
statistical methods required will be taught where they are needed. This
introductory unit includes a review of the basic statistical techniques that form
the common background to the more advanced topics to be covered later. SPSS,
the main statistical package used in M249, is also introduced.
The emphasis throughout this unit is on statistical modelling as an approach to
deriving information on a particular topic of interest. Two topics with an
environmental theme are used to motivate and link the material: levels of 铿乻h
stocks in the North Sea and the Irish Sea, and air quality and asthma in
Nottingham. In Section 1, methods for presenting data using graphs and
numerical summaries are described. An introduction to SPSS is given in Section 2,
where you will learn how to obtain graphs and numerical summaries. Some
commonly used probability models are described in Section 3, while approaches to
statistical inference are discussed in Section 4, including con铿乨ence intervals and
signi铿乧ance tests. Methods for describing and analysing related variables are
described in Section 5. In Section 6, you will learn how to implement some of the
techniques described in Sections 3, 4 and 5 using SPSS. Finally, Section 7 consists
of computer-based exercises on the material covered in Sections 1 to 6.
1 Presenting and summarizing data: the silver
darlings
There are many ways of presenting data, and which method to use depends
entirely on the type and amount of data available, and the purpose of the
presentation. In this section, three ways of presenting data are reviewed: tables,
graphs and numerical summaries. This is done in the context of several data sets
relating to 铿乻h stocks around the British Isles. The data used in this section and
in Section 2 were obtained in October 2004 from the website of the Department
for the Environment, Food and Rural A铿?airs (http://www.defra.gov.uk).
In Subsection 1.1, tables, bar charts, line plots and scatterplots are discussed.
Numerical summaries and histograms are reviewed in Subsection 1.2.
1.1 Presenting data
Fishing for herring, the 鈥榮ilver darlings鈥? of the title of this section, was once a
mainstay of the economy of the east coast of Britain, from Great Yarmouth in
East Anglia to Peterhead in Scotland (see Figure 1.1). The herring industry has
now largely disappeared, and has been replaced by more intensive forms of 铿乻hing,
Figure 1.1 Drifters used for
which are threatening 铿乻h stocks in many sea areas. Fish stocks are now carefully
monitored. This provides information that can be used to set 铿乻hing quotas, and herring 铿乻hing, Great
also to assess the impact of environmental pollution and conservation measures. Yarmouth, 1932 c Empics
5
Introduction to statistical modelling
Example 1.1 Annual 铿乻h catch 1999
Table 1.1
Table 1.1 shows the total annual 铿乻h catch in the North Sea, for seven 铿乻h Total catch
(thousand tonnes) for seven
species, measured in thousands of tonnes, for the year 1999. The key features of
铿乻h species, North Sea, 1999
this table, in addition to the data, are a title describing the contents of the table
(with the relevant units 鈥? in this case thousands of tonnes, which is abbreviated Fish species Catch
as 鈥榯housand tonnes鈥?), and short column headings. Note that the data have been Cod 96
rounded to the nearest thousand tonnes. Herring 372
Haddock 112
Tables are ideal for conveying detailed numerical information. (Large tables are Whiting 59
usually stored on a computer as databases or spreadsheets.) However, to illustrate Sole 23
Plaice 81
a particular point, a graph might be better than a table. For example, it is clear
Saithe 114
from Table 1.1 that the herring catch in 1999 was much greater than that for sole.
However, the relative size of the di铿?erent catches may be conveyed more
e铿?ectively using a suitable diagram.
For the data in Table 1.1, a suitable diagram is a bar chart, in which the 1999
catch for each species is represented by a bar, the length of the bar indicating the
size of the catch. A bar chart with vertical bars is shown in Figure 1.2(a).
Figure 1.2 Total catch for seven 铿乻h species, North Sea, 1999
The bar chart in Figure 1.2(a) shows at a glance that in 1999 the herring catch far
outstripped the catches for the other 铿乻h species.
Bar charts may also be drawn with horizontal bars. A horizontal bar chart of the
data in Table 1.1 is shown in Figure 1.2(b). Horizontal bar charts are sometimes
more convenient than vertical bar charts, when the labels for the bars are long, or
when there is a large number of bars, as the bar labels may be easier to read.
Bar charts can be used to represent changes over time when there are only a few
time points. This is illustrated in Example 1.2.
6
Section 1 Presenting and summarizing data: the silver darlings
Example 1.2 Variation in the 铿乻h catch, 1979鈥?99
Table 1.2 shows the annual North Sea catch for the seven species of 铿乻h listed in
Table 1.1, for the years 1979, 1989 and 1999.
Table 1.2 Annual North Sea catch (thousand tonnes)
1979 1989 1999
Cod 270 140 96
Herring 25 788 372
Haddock 146 109 112
Whiting 244 124 59
Sole 23 22 23
Plaice 145 170 81
Saithe 136 118 114
An issue of interest, particularly to biologists and to people involved in the 铿乻hing
industry, is the variation in the catch over time, for di铿?erent species. This
variation can be conveyed using a comparative bar chart, such as that shown
in Figure 1.3.
Figure 1.3 Comparative bar chart for annual catch of seven 铿乻h species
This bar chart is similar to the one in Figure 1.2(a), except that now three bars
are drawn side-by-side for each 铿乻h species, representing the catches for 1979,
1989 and 1999.
Activity 1.1 Trends in 铿乻h catches
(a) Use Figure 1.3 to identify a general trend in the 铿乻h catch over time for the
铿乻h species represented.
(b) Are there any exceptions to this general trend?
When there are only a few time points, a bar chart is 铿乶e for showing trends.
However, to obtain a more complete picture of changes over time, more time Statistical techniques for the
analysis of data consisting of
points must be used, but then a bar chart will be too cluttered to be of much use.
observations collected at regular
In such circumstances, a line plot is used.
time intervals are described in
Book 2 Time series.
7
Introduction to statistical modelling
Example 1.3 Annual herring catch
In Activity 1.1, you saw that the North Sea herring catch increased by a very
large amount between 1979 and 1989. A line plot of the total North Sea herring
catch (in thousands of tonnes) for each year between 1963 and 1999 is shown in
Figure 1.4.
Figure 1.4 Annual catch of North Sea herring
This line plot gives a more complete picture of the variation in the herring catch
than does the bar chart in Figure 1.3. In particular, it shows that there was a big
drop in the annual herring catch in the late 1970s, followed by a peak in the
late 1980s.
A measure of mature 铿乻h stocks 鈥? that is, of the quantity of mature 铿乻h in the
sea 鈥? is given by the biomass. The biomass is the total mass of mature 铿乻h, and
is measured in thousands of tonnes. A line plot of the estimated herring biomass
in the North Sea, between 1963 and 2003, is shown in Figure 1.5, together with
the line plot of the annual herring catch from Figure 1.4.
Figure 1.5 Biomass and annual catch of North Sea herring
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