Sheep - AgRIS

The Mean
Everyone has calculated averages, or means. Nevertheless, to help the student to become
familiar with the use of symbols and formulas, the mean may be stated as follows:
Many statisticians refer to the mean for a group of individual observations as bar X or X.
The symbol X refers to each individual item or observation, and the Greek symbol z means to
add all items in the group. The letter n refers to the number of X items in the group of data to be
summed.
The Range
The range is a very rough measure of the variation within a population. It is determined by
finding the lowest and the highest values within a series or group of figures.
The chief disadvantages of the range as a measure of variation are that it is subject to
chance fluctuations and that it becomes larger as the size of the sample increases.
The Variance
No doubt we can use the average of the deviations from the mean as a measure of
variation, but when we add all of the deviations and pay strict attention to the sign of each
deviation, the sum is equal to zero. Therefore, the average of the deviations from the means
would be zero also. This difficulty could be eliminated, however, by disregarding signs method
is by the deviations and averaging them. Squaring the deviations and averaging them also
makes it possible to perform other statistical measurements in a correct mathematical snanner.
Once the sum of the squarred deviations from the mean is determined, may be calculated
easily by dividing by the number one less than the total number of observations in the sample.
Instead of dividing by the number of individuals in a sample n we divide by the total minus one
(n-l), at Ic-ast for samples that include less than 25 observations.
Short-cuts have been devised by statisticians in which the variance may be determined by
using a calculator. By using the following equation, the variance can be calculated:
373
X
n
X 2 – X 2 /n
n –1
Where Xis the sum of all items, X 2 is the sum of all items squared, and n is the number
of observations in the sample.
By using the calculator, we could, of course, obtain the answer more quickly.
Another important point to mention here is that the numerator of the fomula for the
variance,
X 2 /n
n
actually is the sum of the squared deviations from the mean and is often written as X 2 .
The sum of littel x squared divided by the number of observations minus one is the variance.
One of the most useful properties of the variance is that it can be separated by a special
analysis into its various component parts. Special adaptations of the analysis of variance can be
used to determine the percentage of the variation in a population that is due to inheritance and
that due to environment.
The Standard Deviation
The standard deviation is a much more accurate measure of variation in a population
than is the range, and can be used very effectively, together with the mean, to describe
a population. Statisticians use various symbols to denote the standard deviation, but the
one used here is S.D. The standard deviation is the square root of the variance.