Numerical Statistics Measures of Variability Packet #22

Measures in Variability Why is it important? Used to better describe data If a researcher has two samples/populations with the same mean, how are they different? Can be determined with measures of variability using the range for example These measures quantify the extent of dispersion in one’s data Basically, looking at how far apart certain numbers are and why. 9/14/2019 2:06:10 PM

Measures in Variability Range The difference between the largest and smallest values [+1]. The range measures the spread of the two extreme values. Interquartile Range Difference between the 75th and 25th percentile The range of the middle 50% of scores 9/14/2019 2:06:10 PM

Range II Range = Highest – Lowest Range = Highest – Lowest plus 1 Ordered Array 98 80 94 92 78 90 75 86 72 84 70 83 68 82 62 58 36

Interquartile Range II Interquartile range is the score representing the 75th percentile minus the score at the 25th percentile 84 – 70 = 14 Raw Score Percentile Rank 98 100 80 50 94 95 92 90 78 40 85 75 35 86 72 30 84 70 25 83 68 20 82 65 62 15 58 10 36 5

Interquartile Range III If there is no specific score that falls at the 75th or 25th percentile, one must estimate it through extrapolation.

Interquartile Range IV 75th percentile Determine where in the distribution the 75th percentile would be Determine how many percentages fall between the known percentages 80 – 60 = 20 percentiles/ticks Subtract the lower score from the higher score and divide the answer from by the answer of the previous step (8 – 7) / 20 = 1/20 or 0.05 Percentile Score 100 10 90 9 80 8 60 7 50 6 40 5 2

Interquartile Range V 75th percentile continued Determine how many “ticks” one will need to either go up, from the lower percentile, or down, from the upper percentile 80 is the closet percentile to 75 Therefore, one must move down 5 ticks At each tick, one must remove, if moving down, or add, if moving up, the answer from step #3 % Score 80 8.00 79 8 – 0.05 = 7.95 78 7.95 – 0.05 = 7.90 77 7.85 76 7.80 75 7.75

Interquartile Range VI 25th percentile Utilize the same steps used when solving for the 75th percentile Percentile Score 100 10 90 9 80 8 60 7 50 6 40 5 2

Interquartile Range VII 25th percentile Falls between the 10th and 40th percentile Therefore, the score MUST be between 2 and 5 40 – 10 = 30 percentiles or ticks Each “tick” is worth (5 – 2)/30 = 3/30 = 1/10 = 0.1 Percentile Score 4 30 3.9 29 3.8 28 3.7 27 3.6 26 3.5 25 3.4 24 3.3 23 3.2 22 3.1 21 3 20

Interquartile Range VIII 25th percentile Go up or go down Does not matter since the distance, to the 25 percentile, from 40 or 10 is 15 This time, the researcher will go UP from the 10th percentile Percentile Score 4 30 3.9 29 3.8 28 3.7 27 3.6 26 3.5 25 3.4 24 3.3 23 3.2 22 3.1 21 3 20

Interquartile Range IX % Score 10 2.0 11 2.1 12 2.2 13 2.3 14 2.4 15 2.5 16 2.6 17 2.7 % Score 18 2.8 19 2.9 20 3.0 21 3.1 22 3.2 23 3.3 24 3.4 25 3.5

Measures in Variability II Deviation These tell you how far away the “raw score” (the value that you are looking at) is from the mean x - x X - mean (sample data) x -  X - mu (population data) 9/14/2019 2:06:10 PM

Measures in Variability III Standard Deviation (Variation) This is the average of all deviations, for all “raw scores,” from your mean s = standard deviation (sample)  (pronounced sigma)= standard deviation (population) Equation for Standard Variation (Deviation) √ (( (x – mean )2) / (N – 1)) √ (SS / (N – 1)) 9/14/2019 2:06:10 PM

Measures in Variability IV Variance The degree of spread within the distribution The larger the spread, the larger the variance This is simply the square of the standard deviation NOT THE DEVIATION, BUT THE STANDARD DEVIATION s2 (s squared) denotes the variance of a sample 2 denotes the variance of a population Equation for Variance ( (x – mean )2) / (N – 1) SS / (N – 1) The square root sign is removed since the standard deviation is squared. Will be written on board to be clearer **The sum of the squared deviations of n measurements from their mean divided by (n-1) 9/14/2019 2:06:10 PM

Variance Chart x (x – x) (x – x)2 2 3.2 -1.2 1.44 3 -0.2 0.04 4 0.8 0.64 5 1.8 3.24 16 6.8

Standard Deviation & Variance √ (( (x – mean )2) / (N – 1)) 6.8 / 4 = 1.7 Variance s2 2.89