Page 229 - DMGT524_TOTAL_QUALITY_MANAGEMENT

Total Quality Management

Notes Population Variance and Population Standard Deviation

The population variance and standard deviation have slightly different formulas from those of
the corresponding statistics for samples. Given a set of numbers x , x , ..., x the population
1 2 n
variance, σ, is found from the expression
(X − X) 2
2
σ = ∑ i
n
2 2 2
+
(X − X) + (X − X) + ... (X − X)
= 1 2 n
n
The population standard deviation, s, is the square root of the population variance.
14.6.4 Random Variables

A random variable is an abstraction of the concept of chance into the theoretical domains of
mathematics, forming the foundations of probability theory and mathematical statistics. Intuitively,
a random variable describes a system that can exist in several states, with each state having a
certain probability. For example, a coin used for tossing can be described as a random variable
with two states, ‘head’ and ‘tail’, with each state having probability one half.
The theory and language of random variables were formalized over the last few centuries
alongside ideas of probability. Full familiarity with all the properties of random variables
requires a strong background in the more recently developed concepts of measure theory, but
random variables can be understood intuitively at various levels of mathematical fluency; set
theory and calculus are fundamentals.

There are two types of random variables – discrete and continuous.
A random variable has either an associated probability distribution (discrete random variable)
or probability density function (continuous random variable).

The outcome of an experiment need not be a number, for example, the outcome when a coin is
tossed can be ‘heads’ or ‘tails’. However, we often want to represent outcomes as numbers.
A random variable is a function that associates a unique numerical value with every outcome of
an experiment. The value of the random variable will vary from trial to trial as the experiment
is repeated.

Examples:

1. A coin is tossed ten times. The random variable X is the number of tails that are noted.
X can only take the values 0, 1, ..., 10, so X is a discrete random variable.
2. A light bulb is burned until it burns out. The random variable Y is its lifetime in hours.
Y can take any positive real value, so Y is a continuous random variable.

Discrete Random Variables

A discrete random variable is one which may take on only a countable number of distinct values
such as 0, 1, 2, 3, 4, .... . Discrete random variables are usually (but not necessarily) counts. If a
random variable can take only a finite number of distinct values, then it must be discrete.
Examples of discrete random variables include the number of children in a family, the Friday
night attendance at a cinema, the number of patients in a doctor’s surgery, the number of
defective light bulbs in a box of ten.

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