Median divides the area under the curve into two equal halves•Medianis aresistant measurebecause it is more robust to extreme valuesor skewness thanthe mean and therefore is a better measure of centre for a very skewed distribution•Meanisnot a resistant measureof centre, because it is seriously influenced by skewness(pulled in the direction of a few extreme observations)•Modeis the only measure of center that can also be used for qualitative data•For skewed distributions, the best measure of center is the median•For symmetric distributions, the best measure of center is the meanPopulation Mean and Sample MeanPopulation mean (μ) =summation of all items in the populationpopulation size12iNyyyyNN+++==Whereμis the Greek letter “mew”Subscript “i” indicates the ithobservationN= population sizeSample mean (y) =summation of all observations in a samplesample size12inyyyyynn+++==Whereyis read “ybar”n= sample size

13medianmodemeanSymmetric distribution(mode)medianmeanRight-skewedLeft-skewed3.8 Measures of Variation (= spread)Range =Max–Min=difference between the highest and lowest observations in a data set•A biased measure of variation (because outliers can give a much wider range than is the realspread of the main data set)Sample Variance and Sample Standard Deviation•Sample standard deviation is the best estimate of population standard deviationSample Variance and Sample Standard DeviationSample variance (s2) = sum of squared deviations from the meanSample size–1()2211niiyysn=−=−Sample standard deviation (s)= positive square root of the sample variance()22sample variance1yyssn−===−

14Example:Calculate the range, standard deviation and variance of the lengths (cm) of a sample of 5gastropods of a certain species.___________________________1.40.80.91.30.7___________________________>>>>>>>>>>Lengths (cm)iyDeviation frommean()iyy−Squareddeviation2()iyy−1.40.80.91.30.7Totals5.1>>>>>>>>>>Degrees of Freedom (df)•df = the number of independent observations•For standard deviation, df = n–1 because the sample mean is used as an estimate of thepopulation mean in calculating it (using defining formula). You first calculate the mean andinclude that as part of the formula. When the sample mean is used and you know (n–1)observations, the nthobservation is fixed and is therefore not independent.•For standard deviation, df is the denominator of the formulaPopulation standard deviation and Population varianceRounding Rules1.Do not perform any rounding until all calculations are complete; otherwise substantial roundingerrors can occur.2.When giving the final answer, keep at least one more decimal place than is given in the raw data.Population standard deviation () (pronounced sigma)2()iyN−=Population variance= (population standard deviation)2=2

153.9 The Five-Number Summary and BoxplotsPercentiles–divide a data set into 100 equal partsDeciles–divide a data set into 10 equal partsQuartiles–divide a data set or distribution into 4 equal parts25%25%25%25%MinQ1Q2Q3MaxFirst quartile (Q1) =

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