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3/2003 Rev 1 I.2.12 – slide 1 of 18 Part I Review of Fundamentals Module 2Basic Physics and Mathematics Used in Radiation Protection Session 12Statistics – Mean, Mode etc Session I.2.12 IAEA Post Graduate Educational Course Radiation Protection and Safety of Radiation Sources

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3/2003 Rev 1 I.2.12 – slide 2 of 18 In this session we will discuss fundamental statistical quantities such as Mean Mode Median Standard deviation Standard error Confidence Intervals Overview

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3/2003 Rev 1 I.2.12 – slide 3 of 18 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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3/2003 Rev 1 I.2.12 – slide 4 of 18 Consider the following 6 observations 1.73.23.24.61.42.8 the mean is calculated as follows = = = 2.82 Mean (1.7 + 3.2 + 3.2 + 4.6 + 1.4 + 2.8) 6 16.96

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3/2003 Rev 1 I.2.12 – slide 5 of 18 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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3/2003 Rev 1 I.2.12 – slide 6 of 18 The ordered set of the 6 observations used to demonstrate the mean is 1.41.72.83.23.24.6 Median (2.8 + 3.2) 2 = = 3 62 Because the number of observations is even (6) the median is calculated as

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3/2003 Rev 1 I.2.12 – slide 7 of 18 The mode is the measurement that occurs most often in a set of observations. Mode

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3/2003 Rev 1 I.2.12 – slide 8 of 18 For the dataset 1.41.72.83.23.24.6 the mode is 3.2 Some datasets may have more than one mode and some have none. Mode

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3/2003 Rev 1 I.2.12 – slide 9 of 18 The standard deviation ( ) is a measure of how much a distribution varies from the mean. Standard Deviation

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3/2003 Rev 1 I.2.12 – slide 10 of 18 For the dataset 1.73.23.24.61.42.8 ( = 2.82) = = Standard Deviation (x i - ) 2 (n-1)½ (1.4 – 2.82) 2 + (1.7 – 2.82) 2 + (2.8 – 2.82) 2 + (3.2 – 2.82) 2 + (3.2 – 2.82) 2 + (4.6 – 2.82) 2 (6 – 1) ½

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3/2003 Rev 1 I.2.12 – slide 11 of 18 = = = 1.16 Standard Deviation (2.02 + 1.25 + 0.0004 + 0.14 + 0.14 + 3.17) 5½ 6.72 5½

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3/2003 Rev 1 I.2.12 – slide 12 of 18 The standard deviation is often called the standard error of the mean, or simply the standard error. Standard Deviation

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3/2003 Rev 1 I.2.12 – slide 13 of 18 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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3/2003 Rev 1 I.2.12 – slide 14 of 18 Confidence Intervals 95%

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3/2003 Rev 1 I.2.12 – slide 15 of 18 It is possible to define two statistics, t 1 and t 2, such that a parameter being estimated Pr(t 1 t 2 ) = 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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3/2003 Rev 1 I.2.12 – slide 16 of 18 A confidence interval about the mean of a normal population assumes that: 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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3/2003 Rev 1 I.2.12 – slide 17 of 18 For a standardized normal distribution, this means that 95% of the normal distribution lies between –1.96 and +1.96 Pr { (Y-1.96) < < (Y+1.96) } where Y = Confidence Intervals (xi)(xi)(xi)(xi)n

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3/2003 Rev 1 I.2.12 – slide 18 of 18 Where to Get More Information Cember, H., Johnson, T. E., Introduction to Health Physics, 4th Edition, McGraw-Hill, New York (2008) Martin, A., Harbison, S. A., Beach, K., Cole, P., An Introduction to Radiation Protection, 6 th Edition, Hodder Arnold, London (2012) Jelley, N. A., Fundamentals of Nuclear Physics, Cambridge University Press, Cambridge (1990) Firestone, R.B., Baglin, C.M., Frank-Chu, S.Y., Eds., Table of Isotopes (8 th Edition, 1999 update), Wiley, New York (1999)

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