Variations occur in nature, be it the tensile strength of a particular
grade of steel, caffeine content in your energy drink or the distance
traveled by your vehicle in a day. Variations are also seen in the observations
recorded during multiple executions of a process, even when all factors
are strictly maintained at their respective levels and all the executions
are run as identically as possible. The natural variations that occur
in a process, even when all conditions are maintained at the same level,
are often termed as noise. When the effect of a particular factor
on a process is studied it becomes extremely important to distinguish
the changes in the process caused by the factor from noise. A number of
statistical methods are available to achieve this. This chapter covers
basic statistical concepts that are useful in understanding the statistical
analysis of data obtained from designed experiments. The initial sections
of this chapter discuss the normal distribution and related concepts.
The assumption of the normal distribution is widely used in the analysis
of designed experiments. The subsequent sections introduce the standard
normal, Chi-Squared, 
and 
distributions that are widely used
in calculations related to hypothesis testing and confidence bounds. The
final sections of this chapter cover hypothesis testing. It is important
to gain a clear understanding of hypothesis testing because this concept
finds direct application in the analysis of designed experiments to determine
whether a particular factor is significant or not. [15]
If you record the distance traveled by your car everyday then these
values would show some variation because it is unlikely that your car
travels the same distance each day. If a variable 
is used to denote these values then
is termed as a random variable
(because of the diverse and unpredicted values 
can have). Random variables are denoted
by uppercase letters while a measured value of the random variable is
denoted by the corresponding lowercase letter. For example, if the distance
traveled by your car on January 1 was 10.7 miles then: 
A commonly used distribution to describe the behavior of random variables
is the normal distribution. When you calculate the mean and standard
deviation for a given data set, you are assuming that the data follows
a normal distribution. A normal distribution (also referred to as the
Gaussian distribution) is a bell shaped curved (see Figure 3.1).
The mean and standard deviation are the two parameters of this
distribution. The mean determines the location of the distribution on
the 
axis and is also called the location
parameter of the normal distribution. The standard deviation determines
the spread of the distribution (how narrow or wide) and is thus called
the scale parameter of the normal distribution. The standard
deviation, or its square called variance, gives an indication
of the variability or spread of data. A large value of the standard deviation
(or variance) implies that a large amount of variability exists in the
data.
|
Figure 3.1: Normal probability density functions for different values of mean and standard deviation. |
Any curve is also referred to as the probability density function or
pdf of the normal distribution as the area under the curve gives
the probability of occurrence of 
for a particular interval. For instance,
if you obtained the mean and standard deviation for the distance data
of your car as 15 miles and 2.5 miles respectively, then the probability
that your car travels a distance between 7 miles and 14 miles is given
by the area under the curve covered between these two values which is
calculated as 34.4% (see Figure 3.2). This means that on 34.4 days out
of every 100 days your car travels, you car can be expected to cover a
distance in the range of 7 to 14 miles.
|
Figure 3.2: Normal probability density function with the shaded area representing the probability of occurrence of data between 7 and 14 miles. |
On a normal probability density function, the area under the curve between
the values of 
and 
is approximately 99.7% of the total
area under the curve. This implies that almost all the time (or 99.7%
of the time) the distance traveled will fall in the range of 7.5 miles
and 22.5 miles 
. Similarly, 
covers approximately 95% of the area
under the curve and 
covers approximately 68% of the area
under the curve.
See Also:
Population Mean, Sample Mean and Variance
