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8/11/2019 Session I212 Standard Deviation
1/18
3/2003 Rev 1 I.2.12slide 1 of 18
Part I Review of Fundamentals
Module 2 Basic Physics and Mathematics
Used in Radiation Protection
Session 12 StatisticsMean, Mode etc
Session I.2.12
IAEA Post Graduate Educational CourseRadiation Protection and Safe Use of Radiation Sources
8/11/2019 Session I212 Standard Deviation
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3/2003 Rev 1 I.2.12slide 2 of 18
In this session we will discuss fundamentalstatistical quantities such as
Mean Mode
Median
Standard deviation
Standard error
Confidence Intervals
Overview
8/11/2019 Session I212 Standard Deviation
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The sum of the values of observations
divided by the number of observations is
called the mean which is designated .
Mean
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Consider the following 6 observations
1.7 3.2 3.2 4.6 1.4 2.8
the mean is calculated as follows
=
= = 2.82
Mean
(1.7 + 3.2 + 3.2 + 4.6 + 1.4 + 2.8)
6
16.9
6
8/11/2019 Session I212 Standard Deviation
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The median is the middle observation when the
observations are arranged in order of their
magnitude (size).
For an even number of observations, the median
is the mean of the two middle observations.
Median
8/11/2019 Session I212 Standard Deviation
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The ordered set of the 6 observations used to
demonstrate the mean is
1.4 1.7 2.8 3.2 3.2 4.6
Median
(2.8 + 3.2)
2
= = 36
2
Because the number of observations is even
(6) the median is calculated as
8/11/2019 Session I212 Standard Deviation
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The mode is the measurement that occurs
most often in a set of observations.
Mode
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For the dataset
1.4 1.7 2.8 3.2 3.2 4.6
the mode is 3.2
Some datasets may have more than one
mode and some have none.
Mode
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The standard deviation () is a measure of how
much a distribution varies from the mean.
Standard Deviation
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For the dataset
1.7 3.2 3.2 4.6 1.4 2.8 (= 2.82)
= =
Standard Deviation
(xi- )2
(n-1)
(1.42.82)2+ (1.72.82)2+ (2.82.82)2+ (3.22.82)2+ (3.22.82)2+ (4.62.82)2
(61)
8/11/2019 Session I212 Standard Deviation
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=
= = 1.16
Standard Deviation
(2.02 + 1.25 + 0.0004 + 0.14 + 0.14 + 3.17)
5
6.725
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The standard deviation is often called the
standard error of the mean, or simply the
standard error.
Standard Deviation
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A confidence interval for a parameter of
interest indicates a measure of assurance
(probability) that the interval includes the
parameter of interest.
Example
We are 95% confident that the mean of a
series of mass measurements is between 8.4
and 10.1 kg.
Confidence Intervals
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Confidence Intervals
95%
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It is possible to define two statistics, t1and t2
such that a parameter being estimated
Pr(t1 t2) = 1 -
where is some fixed probability.
The assertion that lies in this interval will be
true, on average, in proportion to 1 - of the
cases when this is true.
Confidence Intervals
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A confidence interval about the mean of a
normal population assumes:
a two-sided confidence interval about
the population mean is desired
the population variance, 2, is known
the confidence coefficient is 0.95
Confidence Intervals
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For a standardized normal distribution, this
means that 95% of the normal distribution lies
between1.96 and +1.96
Pr { (Y-1.96) < < (Y+1.96) }
where Y =
Confidence Intervals
(xi)n
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Where to Get More Information
Cember, H., Introduction to Health Physics, 3rd
Edition, McGraw-Hill, New York (2000)
Firestone, R.B., Baglin, C.M., Frank-Chu, S.Y., Eds.,Table of Isotopes (8thEdition, 1999 update), Wiley,
New York (1999)
International Atomic Energy Agency, The Safe Useof Radiation Sources, Training Course Series No. 6,
IAEA, Vienna (1995)