Page 228 - DMGT524_TOTAL_QUALITY_MANAGEMENT

Unit 14: Statistical Process Control

Both control charts and sampling inspection make use of statistics and probability. Before Notes
discussing statistical process control in detail, a brief description of the statistical and probability
concepts is necessary.

14.6.1 Variations and Their Representation

Variation is the law of nature. Examples of variations are marks of students in a class, dimensions
of similar products made in mass production and so on. In statistical process control efforts are
made to reduce these variations. The first requirement however is to represent the variation.
There are two categories of methods to represent variation. One of them is to draw a frequency
distribution from the actual data by counting frequencies of each value. Histograms, frequency
polygon and frequency bar charts fall in this category. The other measure is to calculate the
average and dispersion of various values of the data and then from these two statistics represent
the variation. The measures of central tendency and dispersion are briefly explained below.

14.6.2 Measures of Central Tendency: Mean, Median and Mode of a Set
of Data

A collection of specific values, or “scores”, x , x , ..., x of a random variable X is called a sample.
1 2 n
If {x , x , ..., x } is a sample, then the sample mean of the collection is
1 2 n
X + X + X + K + X
X = 1 2 3 n
n
where n is the sample size i.e. the number of scores.

The sample median m is the middle score (in the case of an odd-size sample), or average of the two
middle scores (in the case of an even-size sample), when the scores in a sample are arranged in
ascending order.

A sample mode is a score that appears most often in the collection. (There may be more than one
mode in a sample.)
If the sample x , x , ..., x we are using consists of all the values of X from an entire population (for
1 2 n
instance, the marks of the students in a subject, we refer to the mean, median, and mode above
as the population mean, median, and mode.
We write the population mean as instead of X.

14.6.3 Measures of Dispersion

Sample Variance and Sample Standard Deviation

Given a set of numbers x , x , ..., x the sample variance is
1 2 n
(X − X) 2
2
S = ∑ i
−
n1
2
2
+
(X − X) + (X − X) + ... (X − X) 2
= 1 2 n
−
n1
The sample standard deviation is the square root, s, of the sample variance.

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