[Believe it or not....We do it all the time !!]

"Working with numbers is just part of ordinary life"

We live in a world where facts and figures abound. In our daily lives, at home and at work, we have to constantly assess numerical information and make rapid deductions from that information. In an age of mass communication, we are also bombarded with data from TV, radio and newspapers. How do we judge, assess, sift, weigh or digest this information? It is certain that we cannot be an expert in all things and therefore much of what we are told has to be taken at 'face value'. This also places a large responsibility upon those who collect and work with data (intended for others) to understand how that data was derived, where it came from, it's validity and how it is presented to the listener or reader.

Now we need a definition: essentially STATISTICS is concerned with the collection, presentation, analysis and interpretation on numerical data. The subject is usually divided into two parts, Descriptive or Exploratory Statistics involves describing and summarising data. Then there is Inferential Statistics where we try to go beyond basic logging and 'pidgeon-holeing' and try to draw inferences about a population based upon results collected from a representative sample.

Our brain likes numbers and pictures more than words!

We make mental calculations all the time ...and often subconsciously.... How many bread rolls to buy? How much is a round of drinks going to be? How much petrol do I need? How much money is in my bank account? etc. etc.

All information can be divided into one of two categories: qualitative and quantitative. Qualitative information uses language , particularly descriptive words to convey ideas and meaning and this has been the foundation of all human communication. However, quantitative information is much more specific....how many? how big? how fast? how expensive?

One of the great strengths of quantitative information is that it can be compared with other related information over time and space and, if used properly and with care; with little or no bias.

Compare the following two comparisons of a new sports car.....

The first description is virtually useless, it is highly subjective and is just one person's (highly biased) opinion of the car but tells us very little about the true nature of the machine. Of course, qualitative information does not have to be subjective but can be reasonably objective...

The second description, whilst being very dry and humourless, tells us far more in three lines than the first description did in 6! With the second description, we also have the opportunity to compare this car with the specifications of any other car. This is not a practical proposition if we use the first description method.

There are other problems with qualitative data.....

Task: Describe as ACCURATELY as possible what you see in the picture below:

It may be easy to give a brief descriptive account of what we see, even where numerical information is involved, but to accurately quantify is much more difficult.

Our eyes are able to pass summary information to the brain with incredible speed but with certain limitations.....Small numbers are quite easy (how many people?, how many chairs?) quite hard (how many purple flowers?) almost impossible (how many blades of grass?).

Using statistical methods; small data sets (<25 pieces of data) can be dealt with using simple techniques to 'describe' what the data represents to us (means, increase / decrease in magnitude or simply 'changing') but when datasets grow beyond 25; we have to use other methods to 'reduce' the data down to a component that is manageable for our brains.

The idea of 'Statistical Analysis'

We have already noted that we do many hundreds of minor calculations in our heads each day and they help us to understand the world around us. Our brain has evolved to cope with quantitative information remarkably well. These subconscious calculations can sometimes be quite advanced and use multiplication,division and proportions just as readily as addition and subtraction. They also rely upon previous experience and numerical memory. The outcome will be a deduction and possibly a consequential action......

An example.....

Let us imagine we are preparing for a car journey from Bournemouth to London.

Is simply guessing the amount of petrol needed for the journey such a good idea? No!

Surely it would be better to have some means of estimating the required quantity in order that we don't run out before getting there!! For this to work, we need access to certain prior numerical data. In this case, an accurate measure of the distance involved and an approximation of the number of miles per gallon that the car will do.

The manufacturer of the car will have carried out numerous road tests i.e. Experiments (under many driving conditions) to give an 'average' (be very careful of this word!!) figure for the petrol consumption for that model. You might say that the 'experiment' to establish the average mpg has been replicated many times and so any average mpg figure has been based on a range of results that have been 'condensed' to yield just one figure...the average mpg. Because the figure has been arrived at in a logical and repeatable way, we should have some faith in its accuracy albeit that we are putting our trust in work carried out by others.

Also note that our car is only a sample and so were the few cars actually tested at the factory. The 'population' from which these samples were taken would be every single example of that model built to the same specification. This could be tens of thousands of vehicles and it is clearly impossible to check the fuel consumption of each and every one of them.

Now we are beginning to think "How typical is my car?, How closely does it conform to the specifications of all the examples of this model?, Has it been modified or possibly neglected?" These 'factors' may affect the fuel consumption of just this one sample. Inevitably, we are going to have to make an inference about the car based on values derived from other cars and not actually on the one we are using. So in order to make an inference about the amount of petrol to buy.... we need to do a little mental arithmetic.

The handbook said that this model will achieve 30 mpg.....

So ...120 miles @ 30 mpg = 4 gallons needed.

Whilst the calculation is precise the conclusion may be inaccurate. We may be held up in traffic, the car engine may not be running at peak efficiency or we may just have a heavy right foot!!!! Our brains seem to have an amazing ability to adjust mathematical data with no apparent effort. The likelihood therefore is that we would buy 5 gallons of fuel, "just to be on the safe side". In essence what happened here was that our brain said "I know that 4 gallons should be sufficient but how confident can I be that the result is correct?

In general conversation we might use a phrase such as:

"I am 90% confident that my car would make it on 4 gallons but I can't be 100% sure"

In formal statistics, such use of "90%" would have a far more precise  meaning and   would involve the use of some recognised inferential test  to establish whether or  not we are really justified in saying that something is 90% certain to happen.   In other words, we need to ask "what is the mathematical likelihood that the results   of my estimations are in fact the true state of affairs?" We could set  an arbitrary   'cut-off' point at say, 90%. This would lead us on to say: " If I did this journey 10   times, I am totally confident that the car would complete  the journey successfully   nine times ....I would fail to get there one time in ten.

Probability theory is often difficult to grasp but essentially we are looking at the chance of an event (outcome) taking place. We can measure this as a proportion on a scale of 0 to 1 or as a percentage from 0% to 100%. We are drawing a distinction between something happening by random chance and the same event happening because some agent 'caused' it to happen.

So what is the chance of that one in ten 'disaster' journey happening on this particular trip?

In reality what our brain tells us is that we don't want to run the risk that this journey will be that 'one in ten' and therefore we say... "I am not sufficiently happy with the odds to take the risk and therefore I will buy 5 gallons instead". By doing this we effectively reduce the odds to such a point that we consider the risk of running out of petrol to be negligible.

Our confidence in the likely outcome is approaching 100%!

WE HAVE MANIPULATED RECEIVED AND EXPERIENTIAL DATA TO ACHIEVE A SATISFACTORY OUTPUT!

Statistics and the advertising industry

For the advertising executive, numerical data comes from consumer surveys, fees, budgets, sales figures, market share figures, feedback surveys and competitor data. The analysis may be complex but the end goal is to justify the expenditure made in order to promote the specific goods or services. These documents must then represent the 'snapshot' view of the success (or otherwise!) of the campaign.

The statistics can be also be used as a management tool to aid future decision-taking and to provide the management with accurate information so that they may guide the company's affairs successfully. Quite complicated issues can be helped by accurate collection of data and subsequent analysis. For example: How best to allocate advertising budgets when there is a range of products to promote.

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