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VII. MEASURE - STATISTICS

BOK

BASIC STATISTICS/BASIC TERMS

V.D.1

Measure - Statistics
Measure - Statistics is described in the following topic areas:

.
.
.

Basic statistics
Probability
Process capability

Basic Statistics
Basic Statistics is presented in the following topic areas:

.
.
.

Basic terms
Central limit theorem

.

Graphical methods
Statistical conclusions

Descriptivestatistics

Basic Terms
Gontinuous
Distribution:

A distribution containing infinite (variable) data points that
may be displayed on a continuous measurement scale.
Examples: normal, uniform, exponential, and Weibull
distributions.

Discrete
Distribution:

A distribution resulting from countabte (attribute) data that has

a finite number of possible values. Examples: binomial,
Poisson, and hypergeometric distributions.

Parameter:

The true numeric population value, often unknown, estimated
by a statistic.

Population:

All possible observations of similar items from which
is drawn.

Statistic:

A numerical data value taken from a sample that may be used
to make an inference about a population.

a

sample

(Omdahl,2009)12

Other terms are defined in the following content.

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VII.

TUEASURE . STATISTICS

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VII. MEASURE . STATISTICS

BOK
V.D.2

BASIC STATISTICS/CENTRAL LIMIT THEOREM

Central Limit Theorem
lf a random

variable, x, has mean p, and finite varianceol, as n increases,x

approaches a normal distribution with mean p and varianceo|. Wtrere, ofl =
and n is the number of observations on which each mean is based.

*

Normal Distribution
of Sample Means

Figure 7.1 Distributions of tndividuals Versus Means

The Central Limit Theorem States:
The sample meansX,will be more normally distributed around trr than
individual readings \. The distribution of sample means approaches normal
regardless of the shape of the parent population. This is why X - n control
charts work!

.

The spread in sample means X, is less than

\

with the standard deviation of
X,equal to the standard deviation of the population (individuals) divided by
the square root of the sample size. s* is referred to as the standard error of
the mean:

"*

=

ft

Which is estimated by

s, =

sx

.6

Example 7.1: Assume the following are weight variation results: X= 20 grams and
o = 0.124 grams. Estimate ox for a sample size of 4:

solution:

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--* -

sx o'Eo
t,.r'v& grams
=
=
- 0.062
Jn
J4

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.3

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QUALIW COUNCIL OF INDIANA

VII. MEASURE . STATISTIGS

BOK
V.D.2

-

BASIC STATISTICS'CENTRAL LIMIT THEOREM

Central Limit Theorem (Continued)
The significance of the centrat limittheorem on controt charts is thatthe distribution
of sample means approaches a normal distribution. Refer to Figure 7.2 below:

Population Distribution

Sampling Distribution of

Population Distri bution

X

Sampling Distribr-rtion of

X

Population Distribution

Sampling Distribution of

Population Distribution

X

Sampting Distribution of X

Figure 7.2 lllustration of Central Tendency
(Lapin, 1982)e

ln Figure 7.2, a variety of population distributions approach normality for the
sampling distribution of Xas n increases. For most distii-butions, but not ill, a near
normal sampling distribution is attained with a sample size of 4 or 5.

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VII. MEASURE . STATISTICS

BOK
V.D.3

BASIC STATISTICS'DESCRIPTIVE STATISTICS

Measures of Central Tendency (Continuedl
The Mode
The mode is the most frequently occurring number in a data set.

Example 7.3: (9 Numbers). Find the mode of the following data set:

5 3 7I

I5

4 5

8

Answer: 5
Note: lt is possible for groups of data to have more than one mode.
Advantages of using the mode:

.
.
.
.

No calculations or sorting are necessary
lt is not influenced by extreme values
lt is an actual value
lt can be detected visually in distribution plots

Disadvantaqe of usinq the mode:

.

The data may not have a mode, or may have more than one mode

The Median (Midpoint)
The median is the middle value when the data is arranged in ascending or
descending order. For an even set of data, the median is the average of the middle

two values.

Examples 7.4: Find the median of the following data set:

(l0Numbers) 2 2 2 3 4 6 7 7

8

(9Numbers) 2 2 3 4 S I 8 8

9

Answer: 5 for both examples

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VII. MEASURE . STATISTICS

BOK
V.D.3

BASIC STATISTICS/DESCRIPTIVE STATISTICS

Measures of Central Tendency (Gontinued)
Advantages of usinq the median:

.
.
.

Provides an idea of where most data is located
Little calculation required
Insensitivity to extreme values

Disadvantages of using the median:

.
.
.
.

The data must be sorted and arranged
Extreme values may be important
Two medians cannot be averaged to obtain a combined distribution median
The median will havc,ritore variation (between samples) than the average

For a Normal Distribution

MEAN=MEDIAN=MODE

For a Skewed Distribution

Figure 7.3 A Comparison of Central Tendency in Normal and Skewed Distributions

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VII. MEASURE - STATISTIGS

BOK
V.D.3

BASIC STATISTICS/DESC RIPTIVE STATISTICS

Measures of Dispersion
Other than central tendency, the other important parameter to describe a set of data
is spread ordispersion. Three main measures of dispersion will be reviewed: range,
variance, and standard deviation.

Range (R)
The range of a set of data is the difference between the largest and smallest values.

Example 7.5: (9 Numbers). Find the range of the following data set:

53798545

8

Answer:9-3=6
Variance (o2, s2)
The variance, 02 or s', is equal to the sum of the squared deviations from the mean,
divided by the sample size. The formula for variance is:

Population, 02 =

I(x

-

p)'

sample,

"'

=

I$--r*)'

The variance is equal to the standard deviation squared.

Standard Deviation (o, s)
The standard deviation is the square root of the variance.

Populatior, o

=

I(x -r,)'

I(*-x)'

Sample, s =

N is used fora population and n - 1 fora sample to remove potential bias in relatively
small samples (less than 30).

Coefficient of Variation (COV)
The coefficient of variation equals the standard deviation divided by the mean and
is expressed as a percentage.

cov = 91rooz") or cov =

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VII. MEASURE - STATISTICS

BOK
V.D.3

BASIC STATISTICSIDESCRIPTIVE STATISTICS

The Classic Method of Calculating Standard Deviation
Calculate the standard deviation of the following data set using the formula:

mple 7.6: Determine s from the following data:

(,-x)

(,'x)'

+16
+30
+14

256

SAMPLE

x

x

1

2

162
176

146
146

3

160

146

4

142
125

146

-4

146

-21

41

159

746

+13

169

145

-1

I

167

146
146

+21

441

114
120

146
146

-32

1024
676

119
180

146

-26
-27
+34

5
6
7
8

I

10
11

12

{3

146

154
125
142

14

{5

fX

16

146

+8

146

-21

.l46

900
196

729
1156
64

Ml

-4

16

= 2190

Calculate the average: x

=

E*

2190

n

15

E(x-X)'=.trt
= 146

. Compute the deviation (.-.)
. Square each deviation (* -

i)'

. Sum the squares of the deviation" >(r - r)'
. Calculate standard deviation:

,=1

=,m=J46s

=21,6

Summary:

X=

146
n= 15
s = 21.6
R= 66

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s is the standard deviation of the sample (21.6) which
is used as an estimate for the population from which
the sample was taken.

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QUALIry COUNCIL OF INDIANA

VII. MEASURE . STATISTICS

BOK
V,D.3

BASIC STATISTICS/DESCRIPTIVE STATISTICS

Shortcut Formula for Standard Deviation
$=

n(r*')-(>*)'
n(n - 1)

This formula will yield the same results as shown on the previous page. lt is called
a "shortcut" because it is convenient to use with some computers and calculators
when working with messy data.

Determine

x andsUsingaCalculator

Formerly the authors attempted to instruct students on how to determine X and
standard deviation on a Sharp calculator. However, many varieties of Texas
lnstrument, Casio, Hewlett Packard, and Sharp calculators can accomplish this task.
The functions on all of these calculators are subiect to change. lt should be
recognized that most technical people determine the mean and dispersion using a
calculator. The following general procedures apply:

1. Turn on the calculator. Put it in statistical mode.

.

2.

Enter all observation values following the model instructions.

3.

Determine the sample mean 1X

1.

4. Determine the population standard deviation, o, or the sample standard
deviation, s.

Alternative Methods to Determine standard Deviation
Standard deviation can be determined using probability paper. However, since the
advent of computer programs, this is rarely done. Standard deviation can also be
estimated from control charts using R. This technique is discussed tater in this
Section and relates to the determination of process capability.

The control chart method of estimating standard deviation makes the big

assumption thatthe process being charted is in controland many processes are not.
Using a calculator or software program to determine s from individual data is often
more accurate.

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