Study Guide
Introduction
Plato, who thought about things quite a lot, considered that numbers were some of the most beautiful
things in the universe: he thought so because of their ordered form and symmetry. Some of us, as we
ponder over a bank statement, a table of projected sales targets, or a calorie calculation chart, might
well feel that we do not share Plato's enthusiasm for matters numerical. Many students embarking on
a course where quite a high level of numerical competence is expected can feel very daunted. If you
are reading this, you are more likely to be in the daunted category than in the numerical genius
category.
Who does the worrying?
Anyone studying mathematics, physics, engineering, quantity surveying, or any other technical or
scientific subject is not likely to be concerned that numbers and mathematical formulae will be part
and parcel of their everyday academic lives. If, on the other hand, you have just started a course in
business studies, social science, management, information technology, or humanities, you may have a
few anxieties. You have perhaps looked at the course outline and found that statistics, spreadsheets,
or other numerical concepts are something you will have to get to grips with. It is often the fear of the
unknown that undermines our confidence.
Why do we worry?
It does not seem to matter whether you left school 3 months ago, or 30 years ago; there will always
be people who have unpleasant memories of maths lessons. Either the teacher was grumpy, or
everyone else seemed to understand and you did not dare say you did not. Or else, a lot of the
subject simply seemed unrelated to the world around us. Ask yourself; will you use a quadratic
equation or struggle to calculate the internal volume of a sphere? Yet, you can probably calculate your
shopping bill as you walk round the supermarket without too much trouble.
Basic arithmetic
Calling it basic is a bad start, because it makes you feel inadequate if you are not familiar with it.
Being able to handle basic arithmetic means you can add, subtract, multiply, and divide numbers. Of
course, on most occasions it is fine to use a calculator, but it is often good to understand what the
calculator is doing for you, rather than believing it works like magic. Have a look at the explanations
below to remind yourself of the main arithmetical procedures. You are likely to find that most of it is
familiar and this should be an encouraging starting point for you.
2+2
Here is a quick revision of basic arithmetic terms.
Addition (symbol +) is the simplest of all number concepts. It means exactly what it says; you take
one number and add it to another.
Subtraction (symbol -) means taking one number away from another.
Multiplication (symbol x) describes sums like “What is 6 times 7?”, or “What is 12.5 times 1712.2?”.
Division (symbol /) describes sums like “What is 25 divided by 5?” or “What is 130 divided by 260?”.
A few more terms
To square a number means to multiply it by itself, i.e., 3 x 3 is 9 and 9 x 9 is 81.
Conversely, the square root of a number is the number that multiplied by itself becomes the number
whose square root you want to find. It is far easier than the words that explain it. The square root of
16 is 4 because four 4s are 16.
Percentage means per hundred. If 60% of people say they prefer tea to coffee, then 60 in every 100
people prefer tea to coffee. You will come across percentages a lot if you are interpreting data of many
kinds.
Statistics or sadistics?
Statistics is one of the areas of mathematics you are likely to have to deal with even if you did not opt
for it or did not think you had opted for a highly numerate subject. You have probably already been
part of all kinds of statistical data yourself. Statistics is a branch of applied mathematics concerned
with the collection and interpretation of quantitative data. It is simply a tool to make handling data
easier, not to drive you to distraction. Statistics are used by government departments, market
research organisations, managers, sales staff, teachers, medical researchers, and, of course, you:
students.
What are they for?
Statistics is all about working out what people have done and what they will do; it is linked to
questions of probability. How likely is it, for example, that if you eat six bars of chocolate you will feel
sick? How many people do eat six bars of chocolate a day? What kind of chocolate are they most likely
to eat? Of what age group, gender, and social class are they likely to be? How much is their average
weekly chocolate bill etc.; these are the sorts of question that can be answered by statistical
applications.
Terminology
For a mathematical subject, statistics is remarkably full of words, some of them common, everyday
words that take on their own meaning when they have a statistical application.
Variables are those factors that might affect a population you are studying, e.g., age of people, colour
of vehicles, amount of money spent, etc.
Frequencies: this term describes the pattern of responses in a table of data. They allow you to review
how different categories are distributed within a sample.
Correlation refers to how much agreement there is between different sets of data.
There are some special terms associated with averages that need further definition.
Mean
Is the average score of a set of data obtained by adding all the responses together and then dividing
their sum by the number of respondents.
For example: 10 people go to a restaurant
The bill is £250,00
The average cost of the meal is £25.00 a head
Of course, we all know that it never works like that in a restaurant because someone has only had
mineral water, someone else did not have a starter, etc., etc., so let us look at one or two other
figures, which, incidentally, will be of no help in solving restaurant bill disputes.
The median
Think back to the restaurant bill and cast your eyes down all the items. The lowest price is £3.50
(garlic bread), the highest is £18.50 (monkfish). There are lots of others in between, of course. The
midway point between these two items is £11.00 because it is halfway between the lowest figure and
the highest. On a table of statistics this would be the median.
The mode
Look down your bill again. Is there one price that occurs more than any other? Perhaps there are far
more items at £12.99, so this price appears far more times on the bill than any other. This represents
the mode on any table of statistics. If you look down a table and see that the largest number of
people gave the same answer to a particular questions: that is the mode.
Whether you are dealing with statistics or other large numbers, you don’t want to be landed with too
many digits.
Rounding
Rounding up has nothing to do with chasing cattle. It has to do with making numbers manageable and
more meaningful. All it means is adjusting figures to the nearest large (consider using the term ‘whole’
or another term instead of ‘large’) number. If, for example, 7,725 people were questioned about
something, you would round the number down to 7,700. If 7,782 were questioned, you would round it
up to 7,800 if you were describing the data. Rounding also avoids dealing with very long figures where
you have used decimals. Instead of a figure like 5.23798 you could round that up to .238. You are
often told how many decimal points to work to, so you do need to understand rounding.
Charts and diagrams
One of the good things about mathematics, especially statistics, is that there are many different ways
of presenting data. Some are far more pictorial and diagrammatic rather than just being based
entirely on figures. As your knowledge and experience grows you will learn in which sorts of situation
to apply these different methods of explaining and presenting data. You may also find that even
though you would not describe yourself as “good with numbers”, as soon as you can convert these
into diagrams things start to fall into place. Conversely, if you hate diagrams, once you learn to
extrapolate the numbers you might feel much happier.
Pie charts
This is a popular way of representing proportional data. It is called a pie chart for the simple reason
that it looks like slices of a pie.
Pie charts are fine, so long as you are not dealing with too many figures. If you have a large number
of categories and lots of very small percentages, the slices of pie become too tiny and meaningless.
Pie charts are ideal for broad representations. They are also ideal if you are explaining things to
people who are not all that comfortable with numbers, because they are such a good visual
representation.
Line graphs
You probably remember drawing graphs; perhaps it is even a happy memory. Graphs normally show
the relationship between two variables, X and Y, drawn as two lines at right angles with the right
angle at the bottom left corner. X and Y obviously stand for something, the variables might be stress
level and work efficiency, or number of cigarettes smoked and number of chest infections.
There are many things you can do with graphs apart from just looking at them. You can use them to
find the break-even point between, say, advantageous and disadvantageous situations and you can
compare more than just two relationships with graphs.
Bar charts
Bar charts are useful when you want to measure variables that can’t necessarily be converted into
numbers. Imagine you are plotting a chart of average rainfall and temperatures in different countries.
A country can’t be measured by numbers, or in mathematical terms, it can’t be measured on an
ordinal scale.
A bar chart allows you to categorise data in various ways; it could be alphabetical, by size, age, or
country of origin. Many tables of data can be more meaningfully presented as bar charts or pie charts.
Calculators and computers
Calculators were mentioned under 7.4 (should chapter references be included as they aren’t used to
organise content in the CWS? Consider rewording). You will always be told in test situations whether
you can or can’t use a calculator, and fortunately for most of us, you are nearly always allowed to use
one. Just in case though, buy a really basic maths book and remind yourself of how to work with
percentages, decimals, and fractions. You can also make better use of any data management software
on your PC if you understand why you are creating a pie chart or a bar chart, rather than just knowing
how to make it appealing.
The thought that counts
When you are given a rather tasteless present, you have to console yourself with the fact that it is the
thought that counts. When you are being assessed on your work with numbers, it really is true that it
is the thought, or at least the working out, that counts. Examiners who can see that you have followed
the right methods, even though your final answer may not be 100% correct, will still award you marks
for taking the right approach. This is encouraging if you are doing assessed assignments and you are
not confident about your final answers.
Numbers in your job search
You often find yourself having to deal with numbers when you start your job search, and not simply to
count how many times you have sent off your CV. Many employers, especially large organisations in
the private and public sectors, use selection tests as part of their recruitment procedure. Often, one of
these tests is a numerical reasoning test. It might be anything from simple arithmetic to being asked
to derive information from various sets of data. If you have conquered any maths anxieties that you
had early on in your course, you won't be daunted by such tests.
Numbers at work
When you consider it, you’ll find it difficult to think of jobs where numbers do not play at least some
part. We are not just talking about accountants and actuaries here either. The further you progress,
the more likely you are to be in charge of a budget. This is true whether you are a marketing
manager, a management consultant, a teacher, or a hospital administrator. You may think that it is
mainly financial forecasters and market researchers who have to produce or interpret statistics, but
social workers or librarians might find they have to do this too. In short, numbers are everywhere.
Never be afraid to ask for help
With practice you will improve, but some people still find maths difficult. If you are really having
difficulty with your course, just because of the numerical content, talk to your tutor. Your university
may well run short maths courses and if they do not, a local college may have a suitable evening class
or other part-time course that fits in with your timetable. Have a look on the Internet too! You may
find there are useful courses you can do. With experience, you get better at understanding what you
do not understand and at being able to ask for the right sort of advice.
Enjoy the world of numbers
Numbers are not all bad. If you are at ease with making basic calculations, you are far less likely to be
conned by people offering you bad financial advice. You may be much better at managing your budget
and you might buy the right quantity of paint for your kitchen or tiles for your bathroom. It is probably
best to keep your talents to yourself when it comes to looking for someone to work out a restaurant
bill, but other than this your flair and ease with numbers will always impress. Start doing a few
number puzzles and quizzes.