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Print UKCAT Magic Method: Speeding up the Averages

I’ve come across a lot of UKCAT questions that require you to calculate the average or mean of a set of data. This is not a difficult task but it is time consuming and time is a luxury you may find in short supply when you sit you Quantitative Reasoning section in the UKCAT.

How to Save Time

Any technique that will save you a few seconds could make the difference between finishing and not. But more than this, having a set of slick techniques can give you a confidence to help you ace the tests in a way you never thought possible.

(When I was at school a couple of years ago – OK so I’m using the word ‘couple’ a little loosely here… it was in fact 40 years ago but that’s not the point. The point is, I learned this a long time ago, I have used it hundreds of times and I still use it so it has proved its worth. Now, where was I...?)

The Magic Calculation Trick

When I was at school some time ago, I was taught a little ‘trick’ to speed up the calculation of averages. Now, to be fair, it doesn’t help in every situation but when the data items are reasonably similar it really comes into its own.

Let’s try a little example. What is the average of these 10 numbers:

507, 488, 520, 487, 499, 511, 479, 514, 512, 503

I’ve made this relatively easy by selecting 10 numbers but only to demonstrate a point. Now the time-honoured way of calculating the average is to add all the numbers together and then divide by the number of items.

This will give you 5020/10=502. Now I did this on my calculator and it took me 32 seconds (actually it took longer because the first time I did it I keyed something wrongly and I got the wrong answer) but when I did it in my head I did it in 16 seconds. 

How?

When you have a set of data such as those given which are reasonably similar then use this technique. Look at the data items and select a convenient value that you think will be close to the average.

With this set then the number 500 leaps out as being both reasonable and convenient. Next add up in your head the differences between the data items and your guess.

In this case they would be:

7-12+20-13-1+11-21+14+12+3=20

1. When you do this and just keep the cumulative total in your head.

2. If you have selected a reasonable guess as your average then the number will remain quite small. If the data items vary too much then some of these numbers can get quite large and it becomes more difficult to do the sum in your head.

3. If the sum of all the differences was 0 then your guess was spot on. Well done!

Here, we have a total of 20 from our sum of differences. Divide this by the number of data items:

20 / 10 = 2

And add the answer to your guess and you will have the correct answer. Note that if you make your initial guess too high then the number you will get from the sum of difference will be negative but this works just as well. You will just subtract it from your original guess. Practise with the numbers above using, say, 503 as your guess or 505.

Easy? Maybe not, but with a little practice, your UKCAT quantitative score will certainly improve significantly.

Wish to practice more? Check our UKCAT practice tests here!

 

ECY 16 Aug 2014

Hello!

Thanks for your method :) For your list of number, when I choose a number that is smaller, say 495, I seem to get decimals when I divide it by 10. Am I doing something wrong?

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