Numeracy Skills

1. Introduction

We use maths every day, whether we realise it or not. You'll need maths to work out your UCAS grades and your university finances. As adults, you'll need maths at work, at home, and in every other aspect of your life too, such as going shopping, planning holidays, deciding on financial products such as mortgages and bank accounts, or decorating your house or garden. Good numeracy skills will help you interpret data and statistics, from economics to health information (such as medicine dosages).

Love it or loathe it, maths is a part of our lives, and we need to stay numerate in order to succeed and achieve what we want.

"Good numeracy is the best protection against unemployment, low wages and poor health."
Andreas Schleicher, Organisation for Economic Co-operation and Development (OECD)

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Those of us who didn't enjoy maths at school often look forward to "never having to do it again!", but this is not a good attitude to have towards numeracy. These days, a lot of maths is automatically done for us - calculators, computers, tills, etc. But can we always trust these machines, or the people using them, to get things right? Have you ever got the wrong answer from a calculator, even though you thought you had put the numbers in correctly? A lot of people don't even check the change they're given in shops, or whether they've been charged correctly for food or a household bill, etc. How will you know you're not being ripped off? And if you think you want to be a millionaire one day, or retire early, the only way to plan for that is to start understanding the financial world as soon as possible - which you cannot do without numeracy!

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Even if you have your pass grade for GCSE Maths, you still need to maintain your basic numeracy skills.

This short modular course is adapted from an AS Critical Thinking course (by Oliver McAdoo) and looks at:

• interpreting information,
• percentages and ratios
• averages,
• probability,
• interpreting graphs and charts, and
• patterns and correlations.

These are all numeracy skills which will prove very useful in the 'real world', as well as improving your ability to analyse and evaluate data in college courses and beyond.

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If you're not sure how to navigate this course book, click here.

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Other helpful resources:

A fun way to improve your basic arithmetic skills is to learn to play Dominoes (e.g. Fives and Threes) or Darts!

If you are keen to develop your numeracy skills further than just the above topics, you could take a look at the National Numeracy Challenge and complete their online check-up to see which numeracy skills (if any) you need to brush up on. Check out their video on why numeracy is important and what the challenge is all about here:

If you are struggling with your college Maths course, you should discuss this with your subject tutors. The Maths department have included numerous links to helpful websites on their own GO page (see the right-hand column).

You can also find some helpful video tutorials on the Khan Academy website, on all aspects of mathematics - from the basics of arithmetic, to the more challenging topics for the experienced mathematician. This extensive site is currently free, but requires you to log in to access it.

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A Numerical Skills Prompt Page

2. Interpreting Information

In this short course you will learn how to:

• extract information that is relevant to a given question or task from a range of sources – verbal, graphical, numerical, etc;
  
  
• compare information in different forms, and translate information from one form into another, e.g. from a graph into a verbal statement, from a data into a chart;
  
  
• discuss the advantages and disadvantages (limitations) of different ways of presenting information.

2.1. Mathematical Analysis

Throughtout this chapter you will learn how to understand and use basic methods of numerical and statistical reasoning in order to support any claims you may wish to make. For example, you might want to make the argument that women live longer than men by analysing statistics to try and back this up. 

WARNING: the following maths may hurt your head a bit, but stick with it as it's easier than it may at first appear...

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Mathematical Knowledge

To make things a little more interesting (remember, it is the skill rather than content that is important here!), let us imagine a scenario where, in the distant future (say the year 3010), cats have evolved to the extent that some of them have developed a basic capacity for speech. Let us also imagine that you have transported yourself forward in time in order to study these ‘talking-cats’. There is only one island which is inhabited by such animals: Baal. A set of ‘raw’ data has been compiled by a team of visiting zoologists who have a keen interest in feline abnormality. It runs as follows:

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Table 1 - Baal

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The human inhabitants of the island (‘Baalanders’) are worried about this sudden ‘talking cat’ influx, and are considering a cull in order to protect the native non-speaking cat population, which they believe to be under threat of extinction as a result of interbreeding with their more vocal relatives.

The zoologists, rightly appalled by the idea of a cull, are anxious to avert such an outcome. In order to come to a reasoned decision about what should and should not be done, certain questions need addressing:

• What was the initial percentage rate of talking cats?
• What is the current percentage rate of talking cats?
• What is the overall percentage rate of increase in the talking cat population?

Unfortunately the data provided, in its raw form, is rather unhelpful, failing to provide us with the information that we need to answer such questions. To do so, it needs processing.

But how do we go about doing this...?

2.2. Percentages and Ratios

Seeing numbers in percentage form considerably simplifies the process of digesting information, comparing and contrasting data, recognising patterns and correlations and subsequently making predictions.

If, for example, you scored 47 out of 100 in one exam and 58 out of 120 in another, it would be difficult to judge in which exam you performed better, until you converted the result into a percentage. A simple way of calculating this figure is to divide the smaller figure by the larger one (which represents 100%) and then multiply by 100:

Exam 1: (47 ÷ 100) x 100 = 47 (%)

Exam 2: (58 ÷ 120) x 100 = 46.6 (%)

and you have your answer.

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In the case of the talking cat question, if we were trying to figure out the initial percentage rate of talking cats then the equation we would be dealing with would be:

(100 ÷ 15000) x 100 = 0.66 (%)

From here, it is easy to work out that the percentage rate of non-talking cats would be 99.33% (100 – 0.66).

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If we were trying to figure out the current percentage rate of talking cats, then the equation would be:

(32,000 ÷ 32,500) x 100 = 98.46 (%)

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Finally, to work out the overall percentage rate of increase in the talking cat population, we need to start with our initial figure, 100, and calculate the % increase from 100 to 32,000.

To do this we first need to work out the initial difference between 100 and 32,000 (i.e. 32,000 – 100) which is 31,900 - this being the raw increase in number of the talking cat population. Then we divide this difference by the original (31,900 ÷ 100) which is 319. Finally, as with our previous equation, we multiply this number by 100 which yields an overall increase rate of 31,900%!

The equation looks like this:

(32,000 – 100) ÷ 100 x 100 = 31, 900 (%)

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It is also sometimes helpful to state values such as these in ratio form (where a relationship between two separate quantities, for example water and cordial, is expressed in number: 4:1 – four parts to one).

Let us say, for the purpose of simplicity, that for every 100 cats, 70 (%) can talk whilst 30 (%) cannot. Here, the ratio of talking to non-talking cats would be 7:3 (ratios are expressed as two numbers separated by a colon: ‘x:y’). The ratio of talking cats to the overall cat population would be 7:10 (for every 10 cats, 7 of them will be talkers – expressed as a fraction, 7/10ths of cats).

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Quick Activity

Bearing the above points in mind, see if you can answer the questions in the Percentages & Ratios Quiz.

2.3. Averages (part 1)

Having gathered the relevant information, let us say that the islanders of Baal, being an inquisitive race, wish to know more.

Most importantly, they wish to know the average yearly rate of increase in the talking cat population, believing this will help them predict the rate at which this species might reasonably be expected to continue reproducing.

There would be three ways of calculating such an increase, only two of which need concern us here.

The first of these is the mean average which is calculated by adding up all the values in a set, and then dividing this new number by the original number of values.

The second is the median average which is achieved by listing the values in order (small to large), and then identifying the middle value in this sequence.

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Each method suits a specific purpose. If we wished to calculate, for example, what an average salary was for a specific company, then before we decided which method to use, we would first need to know about the range of salaries paid out.

Let us say there are ten employees, nine of whom earn £25,000 a year and one, the director, who earns £150,000. Appealing to the mean would be rather unhelpful here as it would generate an average income of £37,500 per person:

(9 x 25,000) + (1 x 150,000) /10 = 37,500

a figure clearly unrepresentative of the majority of employees.

Using the median, however, we can see that a much fairer calculation can be arrived at:

25k, 25k, 25k, 25k, 25k, 25k, 25k, 25k, 25k, 150k

The mid-point here would be £25,000 - values 5 & 6.

If these are different, then the median will be the mid-point between the two values and this would be a reasonable representation of the general level of pay.

If we were to change the example, however, and say that two of the employees were on £5,000; two on £6,000; two on £25,000; one on £70,000; one on £100,000 and the final two on £250,000, here the median would still generate an average wage value of £25,000:

5k, 5k, 6k, 6k, 25k, 25k, 70k, 100k, 250k, 250k

which would clearly be disproportionate as it represents the wage value of only two employees and bears no relation whatsoever to those of the other eight.

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Applying this to the situation that confronts the islanders of Baal, in order to calculate the mean and median yearly rate of increase of the talking-cat population, we first need to extract the raw figures of this increase - i.e. by how much the cat population has increased each year.

And once we have this, calculating the mean average increase is straightforward:

(100 + 100 + 200 + 500 + 1,000 + 2,000 + 4,000 + 8,000 + 16,000) / 9 = 3,544

And so too the median:

100, 100, 200, 500, 1,000, 2,000, 4,000, 8,000, 16,000

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Quick Activity

Before moving on, have a go at answering the questions in this worksheet. 

2.4. Averages (part 2)

You have probably noted that, whilst offering a partially adequate account of how much the cat population has increased up until now (and here, it would seem that that the mean average is more suitable than the median) neither of the above methods is particularly useful for determining by how much we can expect this trend to continue.

This is because the increase appears to be exponential (growing at an increasing rate) rather than constant. You might have suggested, for example, that a more helpful way of doing this would be to plot the figures on a line or bar graph, or generate a set of percentage figures that would help us see an underlying pattern, and you would be right:

and here, if the current trend were to continue, we would expect the yearly increase to double, assuming there are no other factors such as lack of resources, feline cull, etc. to limit this growth.

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Quick Activity

Complete the Averages (part 2) Quiz to analyse the yearly decrease in the non-talking cats of Baal. 

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Before we move on from averages, here's a useful little rhyme to help you remember the different types:

2.5. Probability

The islanders of Baal, still concerned by the rate at which the talking-cat population is expanding, have decided, against the wishes of the zoologists, to cull a significant proportion of these in order to stem this increase.

The cats, however, hearing of this plan, have other ideas. One of the brightest of these puts forward a suggestion. If all (talking) cats remain silent, he argues, it will be impossible for the islanders to tell apart talking-cats from their mute relatives and this being so, the Baalanders, who are known for their great love of this native creature, will be forced to call off their campaign.

Realising this duplicitous action, the Baalanders have to make a choice. They reason that, since the vast majority of felines will be talking-cats, if they cull a sizeable percentage of the overall cat population - say, 20,000 - then the chances are that only a few non-talkers will be destroyed and this is perhaps a price worth paying.

Nevertheless, if they were to select a single cat entirely at random, they would like to know what the chances or probability would be of its being a native non-talker.

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To draw an analogy here, if you were to select an individual card from a deck, assuming it to be a standard 52 card pack (Jokers removed), the chances of selecting, for example, the 7 of clubs (♣) would be 1/52. Thus the probability of selecting any particular card at random would be 1/52 (or, for every 52 attempts, one should be successful). The probability of selecting any spade would be 1/4, since there are only four suits in a pack. The chance of picking either a red or a black card would be 1/2 and the chances of picking out a numbered or head card (including aces), for similar reasons, would be 36/52 and 16/52 respectively. It is also possible to work out the probability of picking out a succession of cards. The chances of picking out two red cards in a row, for example, would be 1/2 x 1/2 (1/4); the chances of picking out a red card, then a club would be 1/2 x 1/4 (1/8) and so on.

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Equations such as these are particularly helpful when it comes to risk calculation - as we see with the situation that confronts the islanders of Baal, not to mention their cats!

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Optional Activity

Have a go at a few easy probability questions on this Mathopolis site to help improve your probability skills before moving on. This is an optional activity, but well worth a go.

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Applying the above reasoning to the Baalanders case, if there were an equal proportion of talking and non-talking cats (50:50), then the probability of selecting either cat at random would be 1/2. Given that the ratio is 32,000:500, however, the equation is slightly trickier. See if you can work it out!

Since the overall cat population is 32,500 and the number of talking-cats, 32,000, the probability would be 32,000/32,500. But this is rather a large number to handle! We could break it down further:

32,000/32,500 = 16,000/16,250 = 8,000/8,125 etc.,

but in this case, it might be easier to convert such a value to a percentage.

Going back to our earlier method of doing this, we would be left with the following equation:

(32,000 / 32,500) x 100 = 98.46.

The probability of selecting a talking cat at random, then, would be 98.46%.

Fortunately, there is a clause written into the Baalanders’ constitution which states that, for any activity which might endanger a native feline, unless this has a greater than 99% safety rate, it must not be pursued. So, for now at least, the cats are safe...!

3. Reading & Interpreting Graphs and Charts

In this chapter, we will look at how to read and interpret the information from some of the most commonly used types of graphs and charts.

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Some important concepts and terminology

To understand this better, let's return again to the original data of the feline population of Baal, as used in the previous chapter on Mathematical Analysis:

Introducing some semi-technical language here, of the kind you would be expected to use at A level, we might say that the above data - relating to talking-cats - has a range of 31,900.

Range - broadly speaking the difference between the highest and lowest number in a set - is calculated by subtracting the latter from the former (i.e. 32,000 – 100).

For the same reason, the range of data for non-talking cats is 14,400 (14,900 – 500) and the range for the total number of cats is 17,500.

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Another important concept which (ironically!) often crops up in data analysis is frequency.

‘Frequency’ relates to the rate at which a particular value occurs in any given data set. The number 5,500, for example, in the data compiled for the total number of cats, has a frequency value of 2 (it occurs twice); the value 32,500, on the other hand, only occurs once, so it has a frequency rate of 1.

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In the above table, the data which expresses the yearly increase in the talking-cat population is laid out cumulatively.

In a ‘cumulative’ setting out of data, each successive number includes (i.e. ‘adds on’) the number that precedes it.

If this data were presented so as to include only the specific yearly increases, like so:

then this would be non-cumulative. Each value is given independently of its predecessor.

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Finally, the values (variables) of any given set of data can be regarded as discrete or continuous.

A ‘discrete’ variable is one that takes on only certain values. For example, the above data is restricted to whole (cardinal) numbers. It would not be possible to include half a cat; at least not in any sense that might interest us here!

Continuous variables, in contrast, take on any value. If we were compiling data on weight, rather than amount, this would not be restricted to whole numbers alone since a cat’s mass can be continuous (2.4; 2.5; 2.6 kg etc.).

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It is worth taking some time to digest these terms before continuing.

Quick Activity

Have a go at this very short Data Quiz to make sure you've understood some of the above terminology. 

3.1. Bar Charts

Bar charts (also known as bar graphs) present data in the form of either vertical or horizontal columns.

The x axis (horizontal) of the chart represents the type of data we are dealing with and the y axis (vertical) assigns a value (number/amount) to it.

Translating the information from the grid on the previous page generates the following graph:

Bar charts offer an effective method for comparing two or more sets of data, especially when these have been compiled over a given period of time (a weakness, as we shall see, of pie charts). They generate a visual depiction which can reveal patterns and trends in numerical data, particularly in tabular form.

3.2. Pie Charts

Pie charts are divided into sectors (‘slices’), each of which represent a specific quantity of data or ‘slice of the pie’.

For this reason, they are particularly useful for comparing a range of amounts or a specific amount, with a total one (ratio and proportion).

This is especially true when these are expressed as a percentage (360° = 100%).

Whilst simplicity renders such charts visually attractive, their inability to incorporate a range of data (here it has only been possible to present a particular piece of data from a specific year), means they are unpopular with those wishing to present their findings in a more comprehensive or scientific fashion.

When a more complex representation of data is required, we need look elsewhere.

3.3. Line Graphs

As with bar graphs, line graphs offer an effective way of visually depicting a relationship between a set of two or more variables (for example, amount and type) and this relationship is reflected in the comparative steepness (or gentleness) of the lines gradient:

The range of variables that can be accommodated for in a line graph is clearly an advantage that this type of chart has over others, but when the data you are dealing with does not display any trend or pattern (for example, if it deals with proportionate amounts, rather than an increase or decrease over time) then here you would be better off using a pie chart.

3.4. Scattergraphs

As with line graphs, ‘scattergraphs’ (also known as scatter plots) can be used to depict, where one exists, a proportionate relationship between two or more sets of data.

Such graphs do so by plotting dots (or crosses) against the x and y axes of a chart.

They are particularly useful for comparing quantities of information for which timescale is not a relevant factor.

They would not, for this reason, offer a suitable arrangement for laying out our previous data on cats.

Where a positive trend (i.e. when there is a direct relationship between x and y) can be identified, the pattern will look something like this:

Where a negative trend (i.e. when there is an inverse relationship between x and y) can be identified, the pattern will look something like this:

Where no trend is present, the pattern of dots will appear fairly random:

3.5. Flow Charts and Tree Diagrams

Further graphs/charts you might expect to come across include flow charts:

where the steps of a process (sometimes referred to as an ‘algorithm’) are laid out in order of the priority in which they are to be carried out.

A good way of understanding this is to think about trying different keys in a lock, in turn, in order to establish which the correct one is. The above diagram gives us a flowchart for a non-functioning lamp.

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And tree-diagrams:

which systematically lay out all the possible outcomes of a given process. The above diagram describes the first three available possible combinations of moves for a computer chess program.

3.6. Venn and Carroll Diagrams

Finally we come to Venn and Carroll diagrams, both of which offer an effective, if not dissimilar method for categorising information.

Suppose we were to say of the cats of Baal that they fall in to the category of being either tabby, or non-tabby, then we have a set of four possibilities into which each of the entire cat-population must fall. These are:

1. Non-tabby, can’t talk
2. Non-tabby, can talk
3. Tabby, can’t talk
4. Tabby, can talk

Each of these combinations can be presented in a Venn or Carroll diagram, where the overall population (universal set) is represented by the rectangle/square, and the different subsets (being tabby, non-tabby, talking or not talking), by the spaces enclosed within this area.

Note how Venn diagrams are helpful in being able to depict where these qualities can overlap, whilst in the Carroll diagram, each numbered space corresponds to the type of variable we are dealing with (this can be made clearer by inserting the amount/type etc. of entity adjacent to each number).

More complex data can be accommodated for by more complex diagrams.

Note, however, that beyond a range of eight variables, such diagrams become too complicated for practical purposes.

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Interesting sub-notes:

Carroll diagrams are also known as Lewis Carroll's squares, since the author of the Alice in Wonderland novels was also a gifted mathematician and the inventor of these diagrams.

For his 180th birthday on August 4th 2014, Google made a 'Doodle' to celebrate John Venn and his diagrams.

3.7. Recognising Patterns and Correlations

Patterns and correlations may well offer an indication of some causal relationship common to two sets of variables (suggesting that one data set directly created or affected another data set), but this point should never be taken for granted. (See for example cause/correlation fallacy and plausible and causal explanations)

Nevertheless, we tend to notice the apparently obvious patterns first, before properly judging their significance. Such observations can be worded along the following lines:

‘according to the figures, cases of … have fallen off steadily in line with the increasing incidence of …’ (spec.)

or,

‘according to the figures, cases of … have risen directly in line with the decreasing incidence of…’ etc.

Some examples will help us here:

In the above line graph, a fairly clear pattern can be seen to exist between the rise in heroin addiction and recorded crime.

However, that is not to say, without further investigation, that the former causes the latter, although obviously this might offer a fairly plausible explanation. Rather, all we see at this stage is that the increase in rate of addiction coincides with that of crime.

Such an observation may well go on to support an inference (that there is a direct causal relationship; or that some third factor, for example poverty, is responsible for both rises, etc.) but at the time it is made, it is just an observation, and nothing more.

This type of correlation is said to be positive; it indicates that an increase in one variable coincides with that of another.

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Correlations can also be negative, where an increase in one variable coincides with a decrease in another:

And of course there will be times when no correlation can be observed at all. Compare, for example, the following figures for the comparative international suicide rates of men and women:

Even when the 'maths' looks like it may offer us a clear relationship, and therefore a clear explanation for the data being shown, we still need to be careful before we make any definite judgements that there are no other causes. We can only ever make plausible explanations from data correlation without definitive evidence that our conclusion is absolutely correct. Whenever you see data used in newspapers, magazines, online or in TV programs, you should listen very carefully to how that data is being used and whether the interpreter is really giving you all the facts, or just speculating or summarising the points that interest them.

4. Summary and Review Quizzes

Working through this course has hopefully given you an idea of how numerical information can be used (and misused!).

You've now used averages, probability, fractions and ratios and analysed how they can be both useful and unhelpful.

You've also seen the different ways that graphics can be used to display numerical information, and hopefully noticed the limitations of each of these too.

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Numerical skills in the real world are not just about doing sums. They're actually skills you're likely to need to use a lot in order to:

• carefully consider what the best methods are for finding the answers you need in difficult situations, and
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• critically analyse and evaluate the information other people or organisations are giving you - pretty charts are not always credible evidence for the claims made around them!

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Now that you've taken a look through a good selection of the types of averages, graphs and charts you are likely to come into contact with, both in your A level or BTEC studies, and in every day life, it's time to review your knowledge and skills.

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Review Quiz

Complete the following two review quizzes to finish this course. Part one tests your new Numerical Skills and part two quickly checks your learning and gives you the chance to offer feedback.

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Well done for completing this course!