KNR 445 Statistics t-tests Slide 1 Variability Measures of dispersion or spread 1

KNR 445 CT/Spread & Z-scores Slide 2 Variability defined  Measures of Central Tendency provide a summary level of performance  Recognizes that performance (scores) vary across individual cases  Variability quantifies the spread of performance (how scores vary)  parameter or statistic 1

KNR 445 CT/Spread & Z-scores Slide 3 To describe a distribution  Measure of Central Tendency  Mean, Mode, Median  Variability  how scores cluster  multiple measures  Range, Interquartile range  Mean of Absolute Deviations, Variance, Standard Deviation 1

KNR 445 CT/Spread & Z-scores Slide 4 The Range  # of hours spent watching TV p/wk  2, 5, 7, 7, 8, 8, 10, 12, 12, 15, 17, 20  Range = (Max - Min) Score  = 18  Very susceptible to outliers  Dependent on sample size 1

KNR 445 CT/Spread & Z-scores Slide 5 Semi-Interquartile range  What is a quartile??  Rank values from largest to smallest  Divide sample into 4 parts  Q1, Q2, Q3 => Quartile Points (25 th, 50 th & 75 th percentiles)  Interquartile Range = Q 3 - Q 1  SIQR = IQR / 2  Related to the Median  For ordinal data, or skewed interval/ratio

KNR 445 CT/Spread & Z-scores Slide 6 Standard Deviation  Most commonly accepted measure of spread  Take the mean, then add up the deviations of all numbers from the mean  E.g. take 3 values as a “distribution”  3,4,5  Mean is 4  First: 3-4 = -1, 4-4 = 0, 5-4 = 1.  Then square these deviations, and add them up.  Then divide by the number of values in the original distribution (3)  Then take the square root of this.  Your answer?  The final number is an estimate of the typical (standard) difference (deviation) between a score and the mean  Why square deviations and square root them again…?

KNR 445 CT/Spread & Z-scores Slide 7 Key points about SD  SD small  data clustered round mean  SD large  data scattered from the mean  Affected by extreme scores (as per mean)  Consistent (more stable) across samples from the same population  just like the mean - so it works well with inferential stats (where repeated samples are taken) 1 3 2

KNR 445 CT/Spread & Z-scores Slide 8 Reporting descriptive statistics in a paper  1. “Descriptive statistics for vertical ground reaction force (VGRF) are presented in Table 3, and graphically in Figure 4.”  2. “The mean (± SD) VGRF for the experimental group was 13.8 (±1.4) N/kg, while that of the control group was 11.4 (± 1.2) N/kg.” 1 2

KNR 445 CT/Spread & Z-scores Slide 9 SD and the normal curve X = 70 SD = 10 34% About 68% of scores fall within 1 SD of mean 1 2

KNR 445 CT/Spread & Z-scores Slide 10 About 68% of scores fall between 60 and 70 The standard deviation and the normal curve X = 70 SD = 10 34% 1

KNR 445 CT/Spread & Z-scores Slide About 95% of scores fall within 2 SD of mean X = 70 SD = 10 The standard deviation and the normal curve 1

KNR 445 CT/Spread & Z-scores Slide About 95% of scores fall between 50 and X = 70 SD = 10 The standard deviation and the normal curve 1

KNR 445 CT/Spread & Z-scores Slide 13 The standard deviation and the normal curve 70 About 99.7% of scores fall within 3 S.D. of the mean X = 70 SD =

KNR 445 CT/Spread & Z-scores Slide 14 The standard deviation and the normal curve 70 About 99.7% of scores fall between 40 and X = 70 SD =

KNR 445 CT/Spread & Z-scores Slide 15 What about = 70, SD = 5?  What approximate percentage of scores fall between 65 & 75?  What range includes about 99.7% of all scores? 1 2 3

KNR 445 CT/Spread & Z-scores Slide 16 Descriptive statistics for a normal population  n  Mean  SD  Allows you to formulate the limits (range) including a certain percentage (Y%) of all scores. Allows rough comparison of different sets of scores. 1

KNR 445 CT/Spread & Z-scores Slide 17 Interpreting The Normal Table  Area under Normal Curve  Specific SD values (z) including certain percentages of the scores  Values of Special Interest  1.96 SD = 47.5% of scores (95%)  2.58 SD = 49.5% of scores (99%)    Info on using tables:  1