Carlisle used methods that parallel those described long ago by the biologist JBS Haldane (son of the Oxford physiologist <a href="http://www.selleckchem.com/products/r428.html
</a> JS Haldane) in two letters to Nature, describing his analysis of suspicious data [1, 4, 5]. Haldane, like Carlisle, drew back from an accusation of fraud, but likened the chance to a monkey typing out Hamlet by sheer luck. The p values found by Carlisle and Haldane are similar. There are broadly two types of data in question: categorical, grouped into distinct types (e.g. male/female or headache/no headache); and continuous, having any value within a scale (e.g. BP, in mmHg). We can try to understand the expected-by-chance distributions of categorical data by using much simpler analogies of tossing coins or throwing dice. For both, the results can only have fixed <a href="http://www.selleck.cn/products/3-methyladenine.html
">3-Methyladenine</a> values of heads/tails or the numbers on the dice, but no value in between. Unsurprisingly, the probability of obtaining a certain value when throwing a single six-sided die is ?16% (Fig.?1) but this is only the average expectation. The variance (V), i.e. the degree of departure from expected (or SD, which is ��V), becomes smaller as the number of throws increases (Fig.?1). This is described by a mathematical function known as the binomial probability distribution (which applies to any case of independent events where there are only two possible outcomes; here, throwing a six on a die vs not throwing a six). If the number of throws is n and the probability of the event is p, then the mean rate (��) of the event (in this case throwing a six) happening is given by: (1) In this case, �� is 16/100 throws, 32/200 throws, etc. The SD of this (which can be proved mathematically for the binomial distribution) is given by: (2) Therefore <a href="http://www.selleckchem.com/products/PF-2341066.html
</a> for 60 throws, the mean (SD) number of sixes should be ?10 (3), for 100 throws it is ?16 (4), for 1000 throws, it will be ?160 (12), and so on. Readers should see that, because we now have a variance (or SD), we can use this to assess statistically the departures of any actual data from what is expected (I will not detail the calculations here). Thus if a friend offers a die that results in 30 sixes in 100 throws, we can use statistical testing (using the principles of variation above) to assess its fairness (the actual chance of this is p?<?0.005; Fig.?1). Incidentally, another approach that can be applied to all the numbers thrown is to use the chi-squared test; this yields the same result. When throwing two dice, the plot of possible totals now resembles something readily recognisable as a normal (Gaussian) distribution for continuous data, with 7 being the most likely total as it arises from most combinations (Fig.?2). If our friend��s dice deviate from this overall pattern (Fig.