Appendix 2 Statistics Useful in Experimental Designs

APPENDIX 2

STATISTICS USEFUL IN EXPERIMENTAL DESIGNS

1.

NORMAL DISTRIBUTION The continuous random variable x is distributed normally with a mean X and a standard

deviation o if the density of probability f ( x ) is given by:

..

This distribution depends on only two terms: the mean X. and the standard deviation o

418 The curve representing the function is shown below (Figure A2.1). This Gaussian or bellshaped curve is characteristic of a normal distribution of the random variable x. As the curve is

Figure A2.1: The Gaussian curve. continuous, the mean X and the standard deviation CT are defined for an infinite number o f x values. As it is impossible to make an infinite number of measurements, we distinguish between the population and a sample.

Population This includes all possible values of x. Theoretically a population has an infinite number of individuals, but in practice this is a great number of individuals.

Sample This comprises a few individuals (several values of x) drawn randomly from the population. The larger the sample, the more representative of the population will be. The problem for the experimenter is to estimate the mean and standard deviation of the population from a sample. Statistics theory shows that:

..The sample mean is the best estimate of the population mean. The best estimate of the population standard deviation is given, from the sample, by:

419

where:

of individuals in the sample, and ..xi-isXtheis number the deviation of measurement from the sample mean X. N

x1

The denominator N-1 is important when N is small, as is the case in many experiments.

Variance The variance is the square of the standard deviation u for a population; it can therefore be written 02,The variance of a sample is the square of the estimation s of the standard deviation The standard deviation is more important to the experimenter than the variance because the standard deviation is measured with the same units as the value itself But the standard deviation is calculated from the variance, according to the variance theorem.

2.

VARIANCE THEOREM

The random variables xl, x2....xn and the constants ao, at ....a, are assumed to satisfy the relationship:

+ al x, +

y=

xz + ...+a,,xn

If we also assume that the random variables xl, xz ..,xnare independent, the variance of y , Vb), is (Variance Theorem): V b ) = V(

or

+ V(a, XI) + V(% ~ 2 +) .. , + V(% xn)

~ ( y=)O+a: v ( x l ) + a: where:

v(x,)+... + a:

~(x,)

.V( .V(

V( x,) is the variance of xI x2) is the variance of x2 x,) is the variance of x,

The above formula can be used to deduce the standard deviation in the two following cases: One random variable

The fkctiony is simply: Applying the variance theorem:

y=ax

420

v(y)= a2 v ( x ) This can also be written:

=a

CT.~

( T ~

The error of the variable x is the error of y multiplied by the coefficient a.

Error of the mean If xis the mean of n values of xi 1 x=-[x, n

+...+X,]

+x2

The variance theorem allows us to write: 1 v, = -[v(

n2

XI)

+ v ( x 2 )+ ... + v(Xn)]

and if all the variances of xi are equal: 2 V(XJ = v ( x 2 ) = ... = v ( x , ) = 0,

V,

= -[1no:]

n2

2 6F

2

Ox

=-

n

The error of the mean is the error of one measurement divided by the square root of n. This is the formula that we use most frequently. It allows us to calculate the error of an effect (or interaction) from the known error of the responses.