Biometry: Measurement, Standard Deviation and Standard Error

Let us make an in-depth study of the biometry. After reading this article you will learn about: 1. Measurement of Variation 2. Standard Deviation and 3. Standard Error.

Biometry or Quantitative Biology is the application of statistical methods to biological facts. The science was begun by Sir Francis Galton (1822-1911) and was developed by Karl Pearson (1857-1936) and his disciples. Through the researches of R. A. Fisher, S. Wright, J. B. S. Haldane, K. Mather and others, Biometry has today become a very exact tool in the hands of geneticists. While some such work is too mathematical to be well understood by the non-mathematical scientists, the formulae usually employed are so useful that nowadays very few experimental evidences are much relied upon unless the results are statistically tested. This is true not only for genetics but for most branches of science where living organisms are concerned or, where chance plays a role.

The scope of the present work is too limited to give any real idea of the different aspects of Biometry. Only a few topics of immediate importance are being taken up.

Measurement of Variation:

Biometry developed with measurement of variations. It deals with masses and not with individuals. Thus, in stating that the percentage of sugar in a variety of sugar beet  if 15.5, one has got to examine a whole population of beets or a suitable number (called a sample) of them, and to get the mean or average value.

Fig. 822 shows a variation curve plotting the percentage of sugar in forty thousand beets as obtained by de Vries. The curve shows how variations are evenly distributed about a mean in case of fluctuating variations. It should be noted that while considering the data of quantitative variations, the individuals are not considered separately but are grouped under different classes.

Thus, in this case the classes were 12% (i.e., 11.75 to 12.25%), 12.5% (12.25-12.75%), 13 % (12.75 —13.25%), etc. The number of individuals showing the measurement of a class is known as the frequency of the class. Thus, 635 beets showed the sugar content of the 12.5% class (i.e., 12.25—12.75%).

In case of ordinary fluctuating variations, it is found that deviations from the average lie equally distributed on both sides of the average. A curve for this (as the curve in Fig. 822) is a normal curve which is symmetrical about a vertical axis called the mode which represents the highest frequency.

In a perfectly normal curve the mode should also show the mean value. The mean or average may be obtained by adding up all the values and dividing by the total number of by the formula.

If, instead of taking one sample, several samples are taken it will be found that the standard deviations of the different samples also vary. This variation is measured by the standard error (or, S.E. of a mean) which is given by the formula σ/√n which gives by the formula an idea as to how any mean obtained from a sample may differ from the true hypo­thetical mean of the population. Sometimes, instead of the Standard Error, Probable Error is used which is equal to 0.6745 times the standard error.

In making a statement on some measurement one should state the average ± the standard error. Thus, one may say that the leaves of a particular plant measure 2.5 ± 0.15 cm.

Correlation Coefficient (r) is another measure which shows how one type of variation may be related with another type of variation. If X and Y be the two vari­ables, their correlation r is given by the formula

Which means perfect positive correlation, i.e., mutation rate varies directly with intensity of X- rays.

The ‘t test’ for significance:

If the measurements of two different samples (e.g., the yields of two different crop varieties) vary there is a method of testing whether the difference is significant enough to enable one to assert that the two samples are really different or if the difference is merely due to some ordinary cause of fluctuation or some error in the experiment. This is known as the ‘t test’.

The statistic t = mean difference/standard error of differences

Let us take up a concrete example to understand this method.

The table below shows some hypothetical data on the yield trials of rice grown in a year at 10 different farms in different localities:

Now let us look at the table oft. For 9 degrees of freedom (or d.f. which is one less than the number of trials or comparisons) t=2.26 vor P of 0 05 and 3 25 for P of 0 01. So in this case P is much < 0.01. Stated in words, this means that chances are much less than 0.01(or 1 % or 1:99) that this difference is due to mere chance and not due to some real difference between the two varieties. In other words, the yield difference observed is highly significant which fact may be noted by two asterisks. In t tests P < 0 .05 only is considered significant.

Analysis of Variance:

Finally, there is an elaborate method of Analysis of Variance which enables one to measure critically the effect of every source of variation. This is rather compli­cated and pot discussed in this book.