There are two parts to the statistics you might be asked to work with in a numerical aptitude test.
These are calculating some relatively simple descriptive statistics (mean, mode, median, range) and interpreting some information with statistical values that you are given.
Although calculators are becoming more and more commonly used a general numeracy test should not expect you to know how to calculate the tests without a formula because they are not the level of knowledge we expect at GCSE level and a standard test works at that level, not higher. If you are applying for a job which specifies statistics knowledge as part of the criteria then this is not the place for you to brush up your knowledge I’m afraid: find your notes and text books.
Calculating descriptive statistics:
Most people are familiar with the mean, mode, median and range, but just in case let’s review them:
Mean is calculated by adding together all the data values and dividing by the number of values. A mathematician would write this as Σx/n. Σ simply means “the sum of”, x represents the values of the data, and n is the number of values in the data. I have yet to see a test that demands you understand this, but a question about it is easy to write and could crop up at any time.
Mode is the most commonly occurring value. It is possible (common in some sorts of data) to see multiple modes. If, for example, you plot adult heights on a graph you get two modes, one for women and one for men: this reflects our experience that men are generally taller than women statistically, and also that both men and women are more or less evenly distributed for height around the mean for their sex.
Median is the middle value if you arrange the values in order. Mathematicians would write this as the (n+1)/2
Range is the highest value - the lowest value: it is a crude measure of how spread out the data values are.
The interquartile range (IQR) is based around a similar idea to the median and the range. The quartiles are the values at ¼ and ¾ through the data set when it is in order (n+1)/4 and 3(n+1)/4, again taking half way points if required. The interquartile range is the upper quartile value - lower quartile value. It takes longer to work out than the range, but is less affected by the extreme values.
Why are there so many measures for the "middle" value?
Let’s consider a small company, employing 5 people. There are 4 people who do the actual work and who are paid £5/h. Their boss pays himself £40/h. If we calculate the mean, mode, median and range of these data we have for the mean 5+5+5+5+40=60, 60/5=12, or the mean = £12/h. For the mode 5 occurs most often, so the mode = £5/h and the middle value in the set 5, 5, 5, 5, 40 is 5, so the median is £5/h. Finally the range is calculated as 40 - 5 = 35, so the range is £35/h.
Forgetting the ethics of this pay structure, it highlights one of the reasons we have all these different ways for calculating a representative value. The mean is affected strongly by what we call outliers, that is values a long way from the rest of the numbers. If the company all takes a pay rise, to £6/h for most and £60/h for the boss, the mean increases to £16.80/h, an increase of £4.80/h despite the fact that most of the people only got a £1/h increase because the outlying value changed so much. The mode and median are less strongly affected by this - in fact they both increase to £6/h. Mode, as mentioned above, is not always a unique value, and that can cause problems when we are trying to use a value to represent or describe the data. The median has issues too, including being relatively harder to work out as the size of the data set increases.
The BBC ran a story titled "Most of us have more than the average number of feet" to highlight the same principle of outliers.Interpreting statistics:
The range of interpretative statistics you can be given is limited, simply because the underlying knowledge about them is not assumed. However, since more and more aptitude tests are permitting use of calculators and expecting you to understand your calculator and to follow formulae is reasonable you can be talked through working out such tests and then asked to interpret your results. You can also be given questionnaire settings and results to consider, although neither of these is currently in the test for this course.
The first statistic you should have some awareness of is the standard deviation. Calculating the standard deviation is not fun at all, but most calculators have statistics on them and so knowing what the buttons tell you could be important. There are generally two such buttons, σnand σn-1. These refer in order to the population standard deviation and the sample standard deviation. A population is either the whole of the possible data, or by convention the value used when the sample size is 30 or more (the calculated values become sufficiently close at this point that it isn’t worth the extra step in the calculation for samples). Fully understanding standard deviation is not required, but what it does is allow you to understand the spread of the values - it is a more useful description but rather similar to the range. The larger the standard deviation the more spread out the data within the set is. For example, returning to our small company before and after the pay rises, the calculated standard deviations are £14 and £21.60 respectively.
If you are given a test to work through and evaluate there is a number you need to remember, and that is 5% or 0.05. Unless you are told otherwise this is value at which, by convention, we say there is a significant difference between the samples. Again a full understanding of this is not necessary, but to illustrate what we mean, let’s consider adult women’s and adult men’s heights once again. We generally accept that men, overall, tend to be taller than women, overall. But, is this a significant difference? Significant in this situation does not mean large, although large differences are more likely to be significant, it means is it a difference that suggests there is actually a difference rather than we happen to have selected smaller women and larger men in our survey. In fact, on average, the difference between men’s heights and women’s heights is quite small: about 10 cm in about 165 cm as average adult height. The fact that we’ve seen thousands and thousands of adults in our life and we still see this difference makes it real, and the statistics bear this out.
We could, however, go out and measure female international basketball and netball players (who tend to be tall) and male elite marathon runners (who tend to be short as well as wiry) and get a perfectly correct significant difference where women are, on average, taller than men. It doesn’t reflect the general expectation, but the statistics will tell you there is such a difference and it is meaningful. This illustrates nicely the other key point to interpreting statistics: use your brain and think hard about what has been done. Obviously selecting tall women and short men is unreasonable if not downright perverse. Collecting your data is a key point - something like measuring the heights of every undergraduate at the University of York will give us reasonable values for young adults, because we’ve got a population of about 10,000 people who are mostly (but not all) 18-21. Is this a good measure for the whole population? No. The trend is starting to die down, but most daughters are still (at least a little) taller than their mothers, most sons are taller than their fathers. This will die out as the impact of better nutrition on height dies down, but certainly applies to your parents and grandparents. Saying that the other way round: 18 year olds are, on average, taller than 60 year olds. A sample of 18-21 year olds is therefore good for that age group, but not the whole population. (The difference is about 12 cm in fact, for both men and women.)
Size of the data set is also important. If 8 out 10 owners say their cat prefers it - how many people did they ask? If they asked 10 people, is that really important? If they asked 10,000 how much more strongly do you feel about it? You can actually see on some of these adverts information about the data set now. There is an advert that says something like 80% of midwives use this product on their own children, and that 250 midwives responded. (With apologies to the original advert, it is not currently being broadcast, nor available that I can find. I know these are not the right numbers, but they are indicative.) The point of indicating the size of the data set is it gives you an indication of how trustworthy the results are likely to be. 200 midwives said they use it on their own children (80% of 250) is a far stronger indication of worth than 4 (80% of 5) midwives said they used it.
The final element is rather more fun, and particularly arises when asking people questions. People will answer them and lie. The two classics for this are questions about sex and drug use - people just outright lie about it. When asking about smoking people tend to be more, but not wholly, honest - light smokers tend to lie and say they smoke less than they actually do. Anything that has a social pressure, actual or perceived, or may be illegal will produce such reactions, but any question can produce them and will bias the results.
Summary:
There are only a few statistics you can be expected to calculate from memory, mean, mode, median, range and interquartile range. You can, if you are allowed to use a calculator, be expected to be taken through calculating other statistics and using them, remember the 5% value for statistical significance. Whenever you are asked to interpret data think about the collection and what the data are telling you. It’s not likely to be as extreme as men are shorter than women, but if it seems silly it probably is: it’s up to you to sort out what’s gone wrong.
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