1. BIOL2608 Biometrics 2011-2012 Computer lab session II Basic concepts in statistics

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. Population or sample? Population – Entire collection of measurements in which one is interested

10. Population or sample?

11. 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

12. Population or sample?

14. ………..…..…………..… ….……... ……..……………………… ……………………………… ……………………………… ……………………………… ……………………………… ……………………………… ……….…………....... Population or sample? Sampling Inference Population (very large size) Sample

15. Commonly used symbols PopulationSample Meanμ SizeNn Varianceσ2σ2 s2s2 Standard deviationσs

16. 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

17. Estimation of variance and standard deviation NOTE: – Variance and standard deviation for a population are calculated using slightly different formulae.

18. Normal distribution A very common bell-shaped statistical distribution of data which allows us to carry out different statistical analysis

20. Normality check 6 criteria: Mean & MedianMean = Median

21. Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape

22. Histogram Bin: Ideal bin size obtained by dividing the range by ideal no. of bin (n = 5logn)

23. Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape Skewness & KurtosisWithin ± 1

24. Skewness Negative skew – longer left tail – data concentrated on the right Positive skew – longer right tail – data concentrated on the left

25. 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

26. Normality check 6 criteria: Mean & MedianMean = Median HistogramLike a bell shape Skewness & KurtosisWithin ± 1 Box plotSymmetric

27. Box plot

28. 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

29. P-P Plot / Q-Q Plot

30. 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

31. K-S one-sample test

32. Related Readings Zar, J. H. (1999). Biostatistical Analysis, 4th edition. New Jersey: Prentice-Hall. – Chapters 2, 3, 4, 6