Variability Measures of dispersion or spread 1
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
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
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 • 20 - 2 = 18 • Very susceptible to outliers • Dependent on sample size 1
Semi-Interquartile range • What is a quartile?? • Rank values from largest to smallest • Divide sample into 4 parts • Q1 , Q2 , Q3 => Quartile Points (25th, 50th & 75th percentiles) • Interquartile Range = Q 3 - Q 1 • SIQR = IQR / 2 • Related to the Median • For ordinal data, or skewed interval/ratio 1 2 3 4
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…? 1 2 5 3 6 4 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 2 3
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
SD and the normal curve 1 About 68% of scores fall within 1 SD of mean X = 70 SD = 10 34% 34% 60 70 80 2
The standard deviation and the normal curve About 68% of scores fall between 60 and 70 X = 70 SD = 10 34% 34% 60 70 80 1
The standard deviation and the normal curve About 95% of scores fall within 2 SD of mean X = 70 SD = 10 1 50 60 70 80 90
The standard deviation and the normal curve About 95% of scores fall between 50 and 90 X = 70 SD = 10 1 50 60 70 80 90
The standard deviation and the normal curve About 99.7% of scores fall within 3 S.D. of the mean X = 70 SD = 10 1 40 50 60 70 80 90 100
The standard deviation and the normal curve About 99.7% of scores fall between 40 and 100 X = 70 SD = 10 1 40 50 60 70 80 90 100
What about = 70, SD = 5? 1 • What approximate percentage of scores fall between 65 & 75? • What range includes about 99.7% of all scores? 2 3
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
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%) • http://psych.colorado.edu/~mcclella/java/normal/tableNormal.html • http://davidmlane.com/hyperstat/z_table.html • Info on using tables: • http://www.statsoft.com/textbook/sttable.html 1
