BIOL2608 Biometrics Computer lab session II Basic concepts in statistics

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 cm

Mean (= Arithmetic mean) Commonly called average Sum of all measurements in the pop n /sample divided by the pop n /sample size Mean = ( x x x ) / 13 = 12.88cm cm

Median Middle measurement in an ordered dataset Median = the middle (7 th ) of the 13 measurements cm

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 Q 1 = 11.63Q 2 = Median = 13.0 Q 3 = 13.88

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

Measures of dispersion and variability Indicates how the measurements spread around the center of distribution cm Sample A Sample B

Variance and standard deviation Sample ASample B Variance (s 2 )1.17cm cm 2 Standard deviation (s)1.08cm1.63cm cm Sample A Sample B

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

Population or sample?

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

Population or sample?

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

Commonly used symbols PopulationSample Meanμ SizeNn Varianceσ2σ2 s2s2 Standard deviationσs

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

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

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

Normality check 6 criteria: Mean & MedianMean = Median

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

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

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

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

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

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

Box plot

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

P-P Plot / Q-Q Plot

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

K-S one-sample test

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