Variance and Standard Deviation calculation

Very important statistical guide

1. Variance and Standard Deviation
The variance of a sample is always represented by S
2
and the standard deviation will be
represented by the square root of S
2
, i.e., S = √S
2
. While the standard deviation is an absolute
measure of dispersion, it is however, measured in units—does it depend on the units of
measurement? The coefficient of variation on the other hand is a relative measure of dispersion
based on the standard deviation and is defined as,
The coefficient of variation being a ratio, it is a dimensionless quantity. Thus, for comparing the
variability of say, two distributions, we compute their CVs. The distribution with the smaller CV
would be more homogeneous than the other with a higher CV. We consider in the next section
two cases of obtaining sample variances for both ungrouped and grouped data.
Case I: Variance of Ungrouped Data
Given a set of values x1, x2, · · · , xn, the variance is defined as
Example
To find the variance of the age of boys in School I above, we note that ¯x = 18, hence

2. Steps to Follow when Calculating S
2
for Case I
i.Obtain the mean.
ii.From each observation, deduct the mean to obtain the deviations x - x¯.
iii.Square each deviation to obtain (x - x¯)
2
.
iv.Obtain the sum
v.Divide this sum by n - 1.
Calculating the variance of grouped data
The data below gives the weight in kilograms of 100 students at a given University.
To find the variance of the above grouped data, we form the table below.

3. From the above table, we have
RESEARCH EXPERIMENTAL DESIGNS
a.Completely Randomized Designs (CRD)
This layout works best in tightly controlled situations and very uniform conditions. For this
reason, the completely randomized design is not commonly used in field experiments. It can be
applied in a greenhouse or growth chamber, on a fairly homogeneous field, or in situations when
you are unsure of the variability in your field. The completely randomized design functions well
in very homogeneous and strictly regulated environments. A farmer wants to study the effects of
four different fertilizers (A, B, C, D) on maize productivity. Three replicates of each treatment
are assigned randomly to 12 plots, as illustrated in Figure 1.

4. b.Randomized Complete Block Designs (RCBD)
The randomized complete block design is used to evaluate three or more
treatments. Similar to the paired comparison, plot orientation and blocking aid in
addressing the issue of field variability. Put another way, the ideal solution is to
utilize randomized complete block design when the field is not homogeneous, for
example, in terms of slope, fertility level, soil texture, moisture gradient, etc. In this
case, every block comprises a complete set of treatments, and inside each block,
the treatments are assigned at random. Four to six replications of a complete block
are sufficient for most on-farm research projects, as illustrated in Figure 2.
Figure 1: An illustration of completely randomized design
Plots 1-12
Rep 1
Rep 2
Rep 3

5. c.Incomplete Block designs
Incomplete block design (IBD) gets its name from the incomplete set of treatments in each block.
With this approach, a large number of entries may be evaluated in a number of incomplete and
more uniform blocks since plots are divided into blocks that are not large enough to
accommodate all the treatments. These designs were introduced with the intention of reducing
variability when there are many treatments, more so than with RCBD and Latin square designs.
When the number of replications of all pairs of treatments in a design is same, then the class of
IBD is called Balanced Incomplete Block (BIB) designs and when there are unequal number of
replications for different pair of treatments, then the designs are called as Partially Balanced
Incomplete Block (PBIB) designs, as shown in Figure 3.
Figure 2: An illustration of randomized complete block design with three treatments