session i212 standard deviation

8/11/2019 Session I212 Standard Deviation

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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


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In this session we will discuss fundamentalstatistical quantities such as

Mean Mode

Median

Standard deviation

Standard error

Confidence Intervals

Overview


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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


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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


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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


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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)


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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.



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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.



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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



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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 =


(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)

session i212 standard deviation

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