221-chapter-4

# 221-chapter-4 - Statistics Introduction 1 All measurements...

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Statistics Introduction 1.) All measurements contain random error results always have some uncertainty 2.) Uncertainty are used to determine if two or more experimental results are equivalent or different Statistics is used to accomplish this task Masuzaki, H., et. al Science (2001), 294(5549), 2166 Is the mutant (transgenic) mouse significantly fatter than the normal (wild-type) mouse? Statistical Methods Provide Unbiased Means to Answer Such Questions.

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Statistics Gaussian Curve 1.) For a series of experimental results with only random error: (i) A large number of experiments done under identical conditions will yield a distribution of results. ( ii) Distribution of results is described by a Gaussian or Normal Error Curve Number of Occurrences Value High population about correct value low population far from correct value
Statistics Gaussian Curve 2.) Any set of data (and corresponding Gaussian curve) can be characterized by two parameters: (i) Mean or Average Value ( ) where: n = number of data points x i = value of data point number i = value 1 + value 2 + value 3 value n ( ii) Standard Deviation ( s ) x n x x n 1 i i = = = n 1 i i x ( 29 ( 29 1 n x x s n 1 i 2 i - - = = Smaller the standard deviation is, more precise the measurement is.

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Statistics Gaussian Curve 3.) Other Terms Used to Describe a Data Set (i) Variance : Related to the standard deviation Used to describe how “wide” or precise a distribution of results is variance = ( s ) 2 where: s = standard deviation ( ii) Range : difference in the highest and lowest values in a set of data Example: From the 4 light bulb measurements High Value = 855 hours Low Value = 783 hours Range = High Value – Low Value = 855 – 783 = 72 hours
Gaussian Curve 3.) Other Terms Used to Describe a Data Set (iii) Median : The value in a set of data which has an equal number of data values above it and below it For odd number of data points, the median is actually the middle value For even number of data points, the median is the value halfway between the two middle values Example: Data Set: 1.19, 1.23, 1.25, 1.45 ,1.51 mean( ) = 1.33 Data Set: 1.19, 1.23, 1.25, 1.45 mean( ) = 1.28 median = 1.24 Statistics Median value x x Median value

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Statistics Gaussian Curve ( iii) Example: For the following bowling scores 116.0, 97.9, 114.2, 106.8 and 108.3, find the mean, median, range and standard deviation.
Statistics Gaussian Curve 4.) Relating Terms Back to the Gaussian Curve (i) Formula for a Gaussian curve where e = base of natural logarithm (2.71828…) μ (mean) σ ≈ s (standard deviation) 2 2 2 ) x ( e 2 1 y σ μ π - - = x mean ± standard deviation Entire area under curve is normalized to one

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Statistics Standard Deviation and Probability
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221-chapter-4 - Statistics Introduction 1 All measurements...

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