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BIOL2608 Biometrics 2011-2012 Computer lab session II Basic concepts in statistics
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Measures of central tendency Also known as measure of location Indicates the location of the pop n /sample along the measurement scale Useful for describing and comparing pop n 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 cm
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Mean (= Arithmetic mean) Commonly called average Sum of all measurements in the pop n /sample divided by the pop n /sample size Mean = (10.5 + 11.5 x 2 + 12 + 12.5 + 13 x 3 + 13.5 x 2 + 14 + 14.5 + 15) / 13 = 12.88cm 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 cm
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Median Middle measurement in an ordered dataset 10.5 11.5 11.5 12.0 12.5 13.0 13.0 13.0 13.5 13.5 14.0 14.5 15.0 Median = the middle (7 th ) of the 13 measurements 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 cm
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Quartile Describes an ordered dataset in four equal fractions – 1/4 of the data smaller than 1 st quartile (Q 1 ) – 1/4 lies between Q 1 and Q 2 – 1/4 lies between Q 2 and Q 3 – 1/4 bigger than the Q 3 10.5 11.5 11.5 12.0 12.5 13.0 13.0 13.0 13.5 13.5 14.0 14.5 15.0 Q 1 = 11.63Q 2 = Median = 13.0 Q 3 = 13.88
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Percentile Describes an ordered dataset in 100 equal fractions – 25 th percentile = 1 st quartile – 50 th percentile = 2 nd quartile = median – 75 th pecentile = 3 rd quartile
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Measures of dispersion and variability Indicates how the measurements spread around the center of distribution 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 cm Sample A Sample B
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Variance and standard deviation Sample ASample B Variance (s 2 )1.17cm 2 2.67cm 2 Standard deviation (s)1.08cm1.63cm 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 cm Sample A Sample B
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Population or sample? Population – Entire collection of measurements in which one is interested
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Population or sample?
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Population – Entire collection of measurements in which one is interested – Often large and hard to obtain all measurements Sample – Subset of all measurements in the population
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Population or sample?
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………..…..…………..… ….……... ……..……………………… ……………………………… ……………………………… ……………………………… ……………………………… ……………………………… ……….…………....... Population or sample? Sampling Inference Population (very large size) Sample
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Commonly used symbols PopulationSample Meanμ SizeNn Varianceσ2σ2 s2s2 Standard deviationσs
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Estimation of mean Confidence Interval – Allows us to express the precision of the estimate of population mean (μ) from sample mean ( ) – When we say at 95% confidence level μ = ± y, it means that we are 95% confident that μ lies between - y and + y
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Estimation of variance and standard deviation NOTE: – Variance and standard deviation for a population are calculated using slightly different formulae.
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Normal distribution A very common bell-shaped statistical distribution of data which allows us to carry out different statistical analysis
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Normality check 6 criteria: Mean & MedianMean = Median
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Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape
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Histogram Bin: Ideal bin size obtained by dividing the range by ideal no. of bin (n = 5logn)
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Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape Skewness & KurtosisWithin ± 1
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Skewness Negative skew – longer left tail – data concentrated on the right Positive skew – longer right tail – data concentrated on the left
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Kurtosis Measure of “peakedness” and “tailedness” Positive kurtosis (leptokurtic) – More acute peak around mean – Longer, fatter tails Negative kurtosis (platykurtic) – Lower, wider peak around mean – Shorter, thinner tails
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Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape Skewness & KurtosisWithin ± 1 Box plotSymmetric
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Box plot
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Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape Skewness & KurtosisWithin ± 1 Box plotSymmetric P-P plot / Q-Q plotDots follow the incline straight line
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P-P Plot / Q-Q Plot
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Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape Skewness & KurtosisWithin ± 1 Box plotSymmetric P-P plot / Q-Q plotDots follow the incline straight line Goodness of fit testK-S one-sample test; p > 0.05
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K-S one-sample test
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Related Readings Zar, J. H. (1999). Biostatistical Analysis, 4th edition. New Jersey: Prentice-Hall. – Chapters 2, 3, 4, 6
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