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1

Chapter 1-3

Introduction to Statistics

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1 Definitions - Statistics

The science that looks into the study of generic phenomena, normally complex and framed in a variable universe by means of the employment of reduction models of information and validation analysis of the results in representative terms.

The science and practice of developing human knowledgethrough the use of empirical data. It is based on statistical theory which is a branch of applied mathematics. Within statistical theory, randomness and uncertainty are modeled by probability theory.

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1 Descriptive Statistics

In order to be meaningful, data must be

presented and summarized so as to transmit the

information they represent.

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1 Methods of Summarizing Data

• Graphical

– Histograms

• Numerical

– Descriptive Statistics

• Means

• Standard Deviations

• Excel Applications

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1 Histogram

• A histogram is a way of summarizing data that are measured on an interval scale (either discrete or continuous). It is often used in exploratory data analysis to illustrate the major features of the distribution of the data in a convenient form.

• It divides up the range of possible values in a data set into classes or groups.

• For each group, a rectangle is constructed with a base length equal to the range of values in that specific group, and an area proportional to the number of observations falling into that group. This means that the rectangles might be drawn of non-uniform height.

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1 Histograms Give an Indication of

• Central Location or Balance Point

– Compare with customer target

• Dispersion or Spread

– Compared with customer upper and lower

performance requirements

• Shape

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1

Histogram

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8

Hours

# o

f O

bse

rva

tio

ns

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© 2009 Institute of Industrial Engineers1-3-9

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1 Descriptive Statistics

Measures of central

tendency show the

locations of the data. Measures of dispersion

indicate how widely

distributed the data are.

Measures of shape indicate how

well the data fits the standard

normal and how skewed the data

is.

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1 Statistical Symbols

• Greek letters are used for population statistics.

• English letters are used for sample statistics.

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1 Arithmetic Mean

Sum of the individual

values in the sample

divided by the number of

values in the sample.

XX

n=∑

. symbol theusesmean population The µ

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1 Median

The median is another measure of central

tendency. It is the middle value of a sample with

an odd number of values and the average of the

two middle values in a sample with an even

number of values.

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1 Excel®

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1 Measures of Dispersion

The range is the difference between the largest

and the smallest value within a sample. The

symbol is R.

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1 Measures of Dispersion

The standard deviation is a mathematically

derived statistic that shows the average

variability, or dispersion, of the individual sample

values about the sample mean.

A population standard deviation uses the symbol,

σ, while a sample standard deviation uses s.

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1 Sample Standard Deviation

1

)( 2

−

−=∑

n

xxS

x

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© 2009 Institute of Industrial Engineers1-3-18

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1 Measures of Shape

• Skewness is a measure of symmetry, or more precisely,

the lack of symmetry. A distribution, or data set, is

symmetric if it looks the same to the left and right of the

center point.

• Kurtosis is a measure of whether the data are peaked or

flat relative to a normal distribution. That is, data sets

with high kurtosis tend to have a distinct peak near the

mean, decline rather rapidly, and have heavy tails. Data

sets with low kurtosis tend to have a flat top near the

mean rather than a sharp peak. A uniform distribution

would be the extreme case.

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1 Skewness

3

x

n

1

3

i

1)s-(n

)x-(x

Skewness∑

=

Negative values for the skewness indicate data that

are skewed left and positive values for the

skewness indicate data that are skewed right.

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1 Kurtosis

4

x

n

1

4

i

1)s-(n

)x-(x

Kurtosis∑

=

The kurtosis for a standard normal distribution is

three. For this reason, excess kurtosis is defined as

so that the standard normal distribution has a

kurtosis of zero. Positive kurtosis indicates a

"peaked" distribution and negative kurtosis indicates

a "flat" distribution.

31)s-(n

)x-(x

4

x

n

1

4

i

−

∑

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1