Index >> Biometry and Statistical Applications in Genetics >>Measures of Dispersion Or Spread

	Measures of Dispersion Or Spread While the arithmetic mean is the most commonly used measure of central tendency, the variance, or its square root the standard deviation, is the most commonly used measure of spread in biological statistics. Where statistics are used in the physical sciences, the range (difference between largest and smallest values of the variable) and the IDUD deviation are used reasonably often. In fact, the range is now used considerably more in biological statistics than it was in the past. It may be thought that an indication of spread or distribution by (Xi-µ) or IXi- x bar). However both these are zero since in each case the sum of the positive deviations cancel with the sum of the negative deviations.
To avoid this cancelling, and an infrequently used measure of dispersion is the arithmetic mean of the absolute deviations  ∑ |Xi – µ| / N for a finite population of size N, or ∑ |Xi – X| / n for a sample of size n. This quantity is called the mean deviation. For a finite population of size N, the variance is denoted b σ2 (σ = the small Greek letter sigma and this is read as sigma squared) and is defined to be σ2 = (Xi- µ)2/(N-1)
The positive square root of the population variance is called the standard deviation. For a sample of size n, the sample variance is defined to be S2 = (Xi - x bar)2/(n-1) and the positive square root of the sample variance is called the sample standard deviation. The importance to be attached to the size of the stranded deviation clearly depends on the values of the variables. Thus a standard deviation (s.d) of one foot for a population of heights of trees is of less significance than an equal s.d for a population of heights of wheat plants. The ratio of the standard deviation to the population mean, or the sample standard deviation to the sample mean, expressed as a percentage is called the coefficient of variation. It is an absolute measure of dispersion in the sense that it is independent of the unit employed.
Using this coefficient, a comparison of the variabilities of populations or samples having different means may be made. For a biological data, coefficient of variation of the order of 10% to 15% are common. For very homogeneous material this figure may be reduced to 5%, while a coefficient of variation of 25% would indicate very considerable variability. Population and sample In statistical nomenclature population is used to mean the aggregate of all possible observations on the characteristic under consideration. In other words, the population is the total set of actual or possible values of the variable. A population may be finite or infinite, and the variable continuous or discrete. Where a sample is any finite set of items drawn from a population. A sample which is the only fraction of the population is drawn from a population to obtain information about that population. For this information to have real value, the sample must be drawn in such a way that the results obtained are unbiassed. This is ensured by drawing the sample at random, i.e., in marking up the sample, each individual in the population has an equal chance of being included. A random sample from a given population is a sample, chosen in such. a manner that each possible sample has an 'equal chance' of being drawn. Quantities which characterize populations are known u parameters while those which characterize samples are called statistics.
	A parameter is a fixed quantity, not subject to variation, whereas a statistic is a variate, since in general different samples from the same population give different values of the statistic. Conventionally, parameters are designated by letters from the Greek alphabet, while statistics are represented by letters from the English alphabet. Standard error A random sample, precludes any personal bias 'but it nee-d not necessarily be quite representative of the population. The mean calculated from the observations in the sample need not be exactly the same as the parameter of the population. This can be easily seen if a number of random samples of the same population have been studied. The difference between the estimate from a sample and the actual population parameter is arising out of sampling and as such it is called sampling error.

As the average of the means of several samples is an estimate of the population average if k samples are studied we can expect k different sample means and then the standard deviation of these sample means is on estimate of the sampling error; this standard deviation of the means is termed the 'standard error' (S. E., of the mean, which is the mean of the means. The magnitude of the standard error tells how precise the mean of the sample is. Usually, an experimenter has to be satisfied with a single sample. In such cases the mean of the single sample 'studied is that statistic and its standard error is calculated by dividing the standard deviation of the sample-observations with the square root of the number of observations in the sample.

i.e., S.E.(x bar) = s/
because S.E.(x bar) = = = = =
3. Probability
Probability is a measure of the relative chance of the occurrence of an event from among a set of alternatives. If there are N equally likely ways in which an event can happen if n is the number of ways in which the, desired can occur, then the probability of the occurrence of the desired event is given by n/N and is generally denoted by p. Since p is a measure of the relative chance to the totality of equally likely events, its value does not exceed 1. All the values of p lie between 0 and 1. both limits inclusive.
The rolling of an ordinary, cubical die may result in any one of the six different faces facing upward. The group of six possible results is said to be exhaustive. The possibility of die standing on its edge is not considered! The six possible results are said to be mutually exclusive since the occurrence of one result precludes the happening of the other. The rolling of the die is called a trail.
Suppose that the die is a perfect cube made of homogeneous material and that there is no reason to expect that any particular face will come upper most more frequently: than any other. The six possible results are said to be equally likely or equally probable.
If a trial may result in anyone of n exhaustive, mutually exclusive and equally likely outcomes, and if m of these outcomes entail  the occurrence of an event E, then the probabilities that E wi. happen as the result of the trial is given by P(E)-p=m/n.
Since 0< m< n, p is a positive number less than, or equal to unity. Since the number of outcomes involving the failure of E is n-m, the probability of the failure of E, denoted E, is given by
 P(E) =q=(n - m)/fl=1-m/n= 1-p
further, p + q= 1.
Example; Sex determination in bisexual organism is a chance event. For a normal fertilization of an ovum, there are only two equally likely ways. The zygote may develop into a male or female depending on the type of gamete the male parent provided in animals (man, Drosophila, etc.) but in poultry it depends on the type of gamete the egg contains because in poultry female is the heterogametic sex. The gamete produced by the heterogametic sex may or may not contain the X-chromosome and these are equally likely too. So the chance that a fertilized egg will develop into particular sex is 1/2.

Mathematical probability and statistical probability
In the case of above example of sex determination, the probabilities have been calculated on deductive reasoning even before any trial or experiment is conducted. So these probabilities are known as mathematical or apriori probabilities.
But, in practice, the actual probability in trials conducted may not be coinciding with the apriori probability. For instance, suppose a coin is tossed and it falls with the face E up. Though the mathematical probability of its occurrence is only 1/2, in this case P(E) = 1 and P(E )=0.
But if the coin is tossed 10 times, the number of times E appears can be 0, or 1 or 2 , or 10, the extreme cases being very rare with an unbiased coin. Suppose E was shown up in 4 out 10 trials. Considering the occurrence of E as the favourable events the 4 occurrences out of the 10 equally likely cases gives the relative frequency 4/10 for the occurrence of E. (The operiori probability is 1/2). But if the number of trials is increased from 10 to 20 it is likely that the ratio of number of times E comes up out of 20 trials becomes more close to 1/2. In general, if there are n actual occurrences of the favourable event, say, E out of N trials of equally likely ways as far as E is concerned, then the relative frequency of the event is n/N. The limit of this relative frequency as N becomes indefinitely large is known as the statistical probability.
i.e., P(E) = Lt/N→ n/N.