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Research Methods: 2 M.Sc. Physiotherapy/Podiatry/Pain Descriptive statistics Magic Brain Pills • Research Methods Assessment • mark with • n=15 • Research Methods Assessment • mark without • n=15 • 32 47 76 24 66 43 56 30 43 52 46 28 74 28 65 • 56 29 21 49 56 39 47 34 42 44 42 53 25 41 8 Descriptive/Summary Statistics Measures of centrality Measures of dispersion Summary Statistics Measures of centrality Mode, Median and Mean The Mode • Tally observations in set • Frequency table • Nominal, Ordinal, Interval or Ratio • Uses? Exercise: The Mode • Sample one 5 5 6 9 8 9 7 4 5 8 • Sample two 16 14 12 18 19 14 12 18 • Sample three 81 79 78 75 81 69 75 Sample one Mode = 5 Unimodal Sample two Mode = 14 or 12 or 18 Tri or Multi-Modal Sample three Mode = 75 or 81 Bi-Modal The Median • Put set in rank order • Median lies in middle, half values greater half lower • If n is even, median = mean of the middle two positions • Ordinal, Interval or Ratio • Uses ? Exercise: Median • Sample one 5 5 6 9 8 9 7 4 5 8 • Sample two 16 14 12 18 19 20 14 12 18 • Sample three 81 79 78 75 81 69 75 72 Sample one Median = 6.5 (4 5 5 5 6 7 8 8 9 9) Sample two Median = 16 (12 12 14 14 16 18 18 19 20) Sample three Median = 76.5 (69 72 75 75 78 79 81 81) The Arithmetic Mean • • • • Add all the values Divide by the number of values Mean = Xi/n x Interval or Ratio • Uses ? Exercise: Mean • Sample one 5 5 6 9 8 9 7 4 5 8 • Sample two 16 14 12 18 19 20 14 12 18 • Sample three 81 79 78 75 81 69 75 Sample one Mean = 6.6 Sample two Mean = 15.9 Sample three Mean = 76.8 Summary Statistics Measures of dispersion Range, Inter and Semi Interquartile Range and Standard Deviation The Range • Find Minimum value • Find Maximum value • Subtract Max-Min • Uses? Interquartile Range • • • • • • Put set in rank order Find the median = Q2 Q1 = (n+1)/4th position Q3 = 3(n+1)/4th position Q3-Q1 Interquartile range (Q3-Q1)/2 Semi-Interquartile range Exercise: Interquartile Range • Sample 1: Days to recovery with Rx • Sample 2: Days to recovery without Rx • 12 8 18 22 24 17 15 • 15 12 19 18 25 16 19 14 • Calculate Median and Interquartile range • Calculate Median and Interquartile range Answer Sample 1 12 8 18 22 24 17 15 8 12 15 17 18 22 24 Q2 = 17 (4th), n = 7 Q1= (7+1)/4th position = 2nd = 12 Q3 = 3(7+1)/4th position = 6th = 22 IQR = 10 Answer Sample 2 15 12 19 18 25 16 19 14 12 14 15 16 18 19 19 25 Q2 = 17 (4th + 5th / 2), n = 8 Answer Sample 2 12 14 15 16 18 19 19 25 Q1 = (8+1)/4th position = 2¼th; position 2 = 14 position 3 = 15 Q1 = 14 ¼ Q3 = 3(8+1)/4th position = 6¾th; position 6 = 19 position 7 = 19 Q3 =19 IQR = 4¾ Standard Deviation • • • • • • 2 x ) SD= [(Xi – (n – 1)] (for samples <30) Calculate the mean Subtract each value from the mean Square each answer and sum Divide by n-1 Take square root of that answer Standard Deviation X3 Half x-mean = -ve Half x-mean = +ve X1 X2 X4 Mean So (x - mean) always = 0 So Square then sum and take square root of Sum of Squares/ n-1 Exercise: SD • Sample 1 • Sample 2 • 28.0, 19.2, 25.0, 20.0, 26.6 • 21.7, 19.3, 21.6, 28.7, 10.3 • Calculate Mean, SD, and Range • Calculate Mean, SD, and Range Answer • Sample 1 Mean 23.76, SD 3.95, Range 8.8 • Sample 2 Mean 20.30, SD 6.62, Range 18.4 • Are they different ? • Are they different enough ? Magic Brain Pills • With • Mean 47.7 • SD 17.4 4/15 failed • Without • Mean 39.1 • SD 13.6 6/15 failed