#### Transcript 04 Dispersion

Dispersion Outline What is Dispersion? I Ordinal Variables 1.Range 2.Interquartile Range 3.Semi-Interquartile Range II Ratio/Interval Variables 1.Variance 2.Standard Deviation Anthony J Greene 1 Significant Differences? μ1= 40 μ2=60 Anthony J Greene 2 Significant Differences? μ1= 40 μ2=60 Anthony J Greene 3 Dispersion is the Measure of Spread Anthony J Greene 4 Measures of Dispersion Ordinal Interval/Ratio Range Variance Interquartile Range Standard Deviation Semi-Interquartile Range (as well as range, I.R. and S.I.R.) Nominal Variables have no dispersion Anthony J Greene 5 Range Anthony J Greene 6 Range • The range of a data set is the difference between its maximum and minimum observations: Range = Max – Min. – Use Lower Real Limits: The Min is not merely the lowest score its any score that could be rounded up to the lowest score. – Use Upper Real Limits: Likewise the Max is any score that could be rounded down to the lowest score. – For integer values this generally amounts to adding 0.5 to the highest to get the max, and subtracting 0.5 from the lowest score to get the min. Anthony J Greene 7 Quartiles • Let n denote the number of observations. Arrange the data in increasing order. • The first quartile is at position (n + 1)/4. • The second quartile is the median, which is at position (n + 1)/2. • The third quartile is at position 3(n + 1)/4. • If a position is not a whole number, linear interpolation is used to find the fraction representing the quartile. Anthony J Greene 8 Interquartile Range • The interquartile range, denoted IQR, is the difference between the first and third quartiles; that is, IQR = Q3 – Q1 • Roughly speaking, the IQR gives the range of the middle 50% of the observations. Anthony J Greene 9 The Interquartile Range Anthony J Greene 10 Five Number Summary • The five-number summary of a data set consists of the minimum, maximum, and quartiles written in increasing order: Min, Q1, Q2, Q3, Max. Anthony J Greene 11 Quartiles Anthony J Greene 12 Box & Whiskers Plots Anthony J Greene 13 Box & Whiskers Plots Anthony J Greene 14 Box & Whiskers Plots Anthony J Greene 15 Standard Deviation 68% 95% Anthony J Greene 16 Standard Deviation 68% 95% Anthony J Greene 17 Standard Deviation 68% Anthony J Greene 95% 18 Standard Deviation of a Discrete Random Variable The population standard deviation of a discrete random variable X is denoted by and is defined by x N 2 Or the computational formula x N 2 2 The variance, V, is the square of the standard deviation V=2 Anthony J Greene 19 Variance is the Average Squared Deviation 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 -1 +1 +2 -6 +4 -15 -17 x 2 +6 +9 x 3 -20 +11 +14 x 2 +15 -22 -23 -27 +16 +18 +20 μ = 33 Average Deviation is Zero Average Squared Deviation: V = Σ(x-μ)2/N Anthony J Greene 20 Samples and Populations Anthony J Greene 21 Population and Sample Variability Anthony J Greene 22 Sample Standard Deviation • For a variable x, the standard deviation of the observations for a sample is called a sample standard deviation. It is denoted by sx or, when no confusion will arise, simply by s. We have x M s or n 1 2 x x 2 2 n 1 n • where n is the sample size: n-1 is referred to as the degrees of freedom Anthony J Greene 23 Deviation from the Sample Mean M Anthony J Greene 24 Deviation From the Sample Mean M Anthony J Greene 25 Sample Variance and Standard Deviation Using Conceptual Formula M x M 2 s n 1 24 6 4 Anthony J Greene M 26 Computational Columns Using Conceptual Formula M M x M s n 1 2 353 s 10.85 4 -1 Anthony J Greene 27 Computational Columns Using Computational Formula 2 x 2 x s s n 1 3582 32, 394 4 1 n 4 32,394 32,041 s 4 1 353 s 10.85 3 Anthony J Greene 28 APA Format For Mean and St.Dev Anthony J Greene 29 Sample Standard Deviation • Almost all of the observations in any data set lie within three standard deviations to either side of the mean • 95% of the observations lie within two standard deviations to either side of the mean • 68% of the observations lie within one standard deviation to either side of the mean Anthony J Greene 30 Sample Standard Deviation 68% 95% Anthony J Greene 31 Summary of Descriptives Central Tendency 1. Mode Dispersion 1. -- 2. Median 2. Interquartile range or Semi-interquartile range 3. Variance or Standard deviation* 3. Mean Anthony J Greene 32 Again, The Basic Idea of Experiments 1. Are there differences between means? 2. Is that difference large enough so that it is not likely to be due to chance factors? Answer: It depends on how far apart the means are and how much dispersion you have in your variables Anthony J Greene 33 Effect Size Compared to Random Variation The variability within samples is small and it is easy to see the 5-point mean difference between the two samples. Anthony J Greene 34 Effect Size Compared to Random Variation The 5point mean difference between samples is obscured by the large variability within samples. Anthony J Greene 35 Significant Differences? μ1= 40 μ2=60 Anthony J Greene 36 Significant Differences? μ1= 40 μ2=60 Anthony J Greene 37