We may not be aware of it, but statistics (or, ‘sadistics’, as my undergraduate and postgraduate students used to call it) rule most of our life. This month, we provide you with basic knowledge regarding ‘population’ data which confronts us every day of our life.
One mention of the subject and the average person will run for cover … and you do not have to be an arithmophobic to do so. The subject of ‘Statistics’ is not just the collection of data but also their analyses using statistical test (both parametric and non-parametric). The fact is that one does not need to have a high level of arithmetical knowledge to acquire an understanding and ability to use statistics.
Recently, there was a programme on TV called ‘Child Genius’. As suggested by this name, a number of children were tested in a number of attributes including arithmetic, general knowledge, logic and reasoning, vocabulary, etc. Most of these children had very high IQs (Intelligent Quotient) but one could be left wondering what IQ is considered high and which is low. More to the point, is there such a thing as ‘normal’ IQ? And if so, what is it? One can ask similar questions about this month’s GCSE and A level results. How does one determine how our students performed in relation to the ‘normal’ population?
If we were to plot a population’s IQs (or for that matter, many other attributes) on a graph, we would most likely find that the results form a bell shape (see below), or normal distribution. The good Lord has designed much of His world to follow a particular pattern and design. Phenomena such as height, weight, growth rates, exam score, temperature, etc. have, what is referred to, as a probability distribution that are bell curves. This statistical concept allows us to quantify and qualify attributes. It can also help us make certain predictions based on probabilities.
But let us continue with the concept of IQ as an example. Researchers found out when plotting IQ scores of a population (or sample of a population) that they are distributed in a bell shape. An arbitrary score of 100 was established by some statisticians as the average (also called the ‘mean’), though this figure as an average, depends on the particular IQ test used. A cursory look at the bell shape below shows the following. The distribution of IQ scores shown on this particular graph, goes from 55 to 145, but lower and higher IQs are measurable, as we shall see later.
In this case, the difference between one point and the neighbouring one is 15 – this is referred to as the standard deviation (s.d.). These points are referred to as ‘z’ scores and are used to show how far any IQ score is from the average (i.e. 100). So anyone with an IQ of 115, has a ‘z’ value of +1 (see graph). Whereas ‘z’ scores are constant in a symmetric bell shaped graph, s.ds are likely to be different for different measures, genders, populations, countries, etc. So for example, the average height of adult males in England is reported to be 1.753 metres (5ft 9 in) and the s.d. is about 7 cm. (approx. 2.76 in); the average weight for English adult females was found to be 70.2 kg (11 st) and the s.d. is 13 kg (29 lbs or 2.07 st). In practical terms, one might conclude that a 64.26% of males in England are between 1.683 m (5ft 6 in) tall to 1.823 m (5ft 11 in) tall. Of course, these figures change according to countries, ages and so on.
The frequent challenge is ‘what is normal’. In a sense, there is no exact point suggesting who or what is normal, but one can state that there is a range of ‘normality’. In this example, IQs between 85 and 115 (‘z’ scores between -1 and +1) are considered by some psychologists to be ‘within normal limits’. As can be seen from the graph, 68.26% of the population lie within normal limits. 13.59% of the population has an IQ between 70 and 85, and so on. These individuals usually need some remedial help though one has to qualify this by stating that IQ on its own does not determine academic achievement. With the right environmental, family and educational support, such individuals could do much better academically than their IQ may predict.
Anyone with an IQ between 115 and 130 (on the right hand side of the average) is considered to be quite clever, but again, other factors such as an individual’s motivation, the amount of work (s)he puts in can compromise the level of achievement suggested by their IQ.
There are organisations which are dedicated to a membership of individuals with high IQs. ‘Mensa’ (meaning ‘table’ in Latin) is such an international organisation. Mensa is a round-table society, where race, colour, creed, national origin, age, politics, educational or social background are irrelevant. Members of Mensa have succeeded in acquiring an IQ score in a Mensa approved test in the 98th percentile i.e. top 2% of the population. This could range between an IQ of 132 or 148 depending on the IQ test used. In practical terms, an IQ of 130 makes one eligible for membership. Individuals with an IQ of 145 or higher are sometimes thought of as ‘geniuses’ and form less than 0.13% of the population. Mensa International consists of around 134,000 members in 100 countries, in 51 national groups. The largest national groups are:[America, with more than 57,000 members, Britain, with over 21,000 members, and Germany, with more than 13,000 members. Some members even live in Vatican City.
It is difficult to state how accurate this information is as many of these people have never had their IQ formally tested, but these claims are often speculative and are based on factors such as whether they have gone to university, the success they have achieved in life, and so on.
The phenomenon of ‘normal distribution’ is not just used in the field of IQ. By using this statistical reality, manufacturers, of say clothes or shoes, can calculate how many in each size they should make. Using the normal distribution, they will determine that, for example, 2% of the population will need either the smallest sizes or the largest, with 34.12% below the ‘average’ size and 34.12% just above the average size. They will thus, determine how many items should be manufactured in any one size.
There are some types of data that do not follow a normal distribution pattern. These data sets should not be forced to try to fit a bell curve. A classic example would be student grades, which often have two modes. Other types of data that do not follow the curve include income, population growth, and mechanical failures.
In order to be able to make inferences and conclusions about data, one has to often subject them to statistical analyses to determine how probable it is that the data were obtained by chance as opposed to the result of the influence of a factor or factors. Tossing a coin, or betting on black Vs red on a roulette table, are basic examples which can be used to illustrate the law of probability. One would expect that if the coin is tossed 100 times, heads and tails (black and red) have an equal chance (i.e. 50-50) of landing. But this is unlikely to happen on 100 tosses or spins of the roulette – it may well take many more tosses for heads and tails, or roulette spins, to be equal. We therefore, know that we can get 55 of one and 45 of the other and we accept this as ‘within normal limits’. But at what point can we conclude that the results are due to some variable such as the person tossing the coin using a particular technique, or the coin, or roulette wheel/ball, being weighted in some way which favours one of the two possible results? Would we accept that 60-40 is within the reasonable realms of possibility? What about 70-30? This is what statistical tests are designed to determine.
In the behavioural sciences, the probability or significance level accepted as the results being ‘within normal limits’ (i.e. due to a chance factor) is 5 in 100 or less. Anything above this (e.g. 5.1, or above, in 100) is considered to be due to some variable affecting the result. Significance levels are not necessarily the same for all subjects. For example, in the medical sciences, the significance level is more rigorous for obvious reasons, and could be set as low as 1 in 1000.
Statistical tests have been designed mathematically to determine the probability of differences within and between variables. Other tests will measure correlations and yet others, trends. One has to select the appropriateness of a test according to set criteria. Basically, the main ones are: what is one measuring (i.e. the differences, correlations or trends); also the number of samples of data, the type of data, and whether the samples are related or not.
Without such tests, and the correct choice of test, data can be rendered meaningless and without much practical value.
BY PROF. DR. SAMUEL ABUDARHAM