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Psyc_315_-_Winter_2010_-_Class_5

# Psyc_315_-_Winter_2010_-_Class_5 - Recap of Last Class...

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1 Recap of Last Class Measures of Central Tendency – The Mean – The Mode – The Median – Variance • Questions? Chapter 5 Standard deviation and Z-scores 3 Problem A spread measure should have the same units as the original data. E.g., weight of students - variance would be lbs squared. Also, difficult to interpret the actual spread of the scores (e.g., variance of 1000 does not give insight into actual variation of weights). The solution is to take the square root – The Standard Deviation.

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4 Measures of Variability: The Standard Deviation The standard deviation (SD) is the square root of the variance (averaged squared deviation from the mean). The standard deviation tells us approximately how far the scores vary from the mean on average. The value of the SD is that it permits a measure of relative standing within a distribution of scores. The bigger the standard deviation, the greater the spread 2 S D S D = 5 Measures of Variability: The Standard Deviation Definitional Formula: Step 1 : Find X. Step 2 : Subtract the mean from each score. Step 3 : Square each deviation (X - M). Step 4 : Sum the squared deviations (X - M) 2 to obtain SS. Step 5 : Enter SS and N in formula below to obtain SD. 2 2 ( ) X M SS SD SD N N - = = = 6 Measures of Variability: The Standard Deviation Computational Formula: Get the same answer with either formula. The major difference is how SS is calculated. Computing SS using the computational formula: – Step 1 : Square each score – Step 2 : Calculate Σ X 2 (sum the squared scores) – Step 3 : Calculate Σ X (sum of scores) and square it ( Σ X) 2 – Step 4 : Calculate SS using the formula. – Step 5: Enter SS and N in SD formula. ( ) 2 2 X SS X N = - SS SD N =
7 Example Using Definitional Formula X = [75,70,80,40,60,85,75,90,70,55] M= X = 700 = 70 N 10 SS = (X - M) 2 = 2000 SD 2 = SS = 2000 = 200 N 10 SD = SD 2 = 200 = 14.14 Scores vary, on average, by about 14.14 in each direction from the mean (55.86 – 84.14). X X-M (X - M) 2 75 5 25 70 0 0 80 10 100 40 -30 900 60 -10 100 85 15 225 75 5 25 90 20 400 70 0 0 55 -15 225 8 Example Using Computational Formula X = [75,70,80,40,60,85,75,90,70,55] SS = X 2 – ( X) 2 = 51000 – (700) 2 = 2000 N 10 SD 2 = SS = 2000 = 200 N 10 SD= SD 2 = 200 = 14.14 X X 2 75 5625 70 4900 80 6400 40 1600 60 3600 85 7225 75 5625 90 8100 70 4900 55 3025 9 Your Turn Calculate the Mean (M), Variance (SD 2 ) and Standard Deviation (SD) of these 2 sets of numbers: a) X= [1,2,3,5,7,9] b) X= [22, 33, 44, 88] M= X N SS = (X - M) 2 SD 2 = SS N SD = SD 2

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10 11 Answers X (X-M) (X-M) 2 1 -3.5 12.25 2 -2.5 6.25 3 -1.5 2.25 5 0.5 0.25 7 2.5 6.25 9 4.5 20.25 2 2 4.5 47.5 47.5 7.92 6 7.92 2.81 M SS SS SD N SD SD = = = = = = = = X (X-M) (X-M) 2 22 -24.75 612.56 33 -13.75 189.06 44 -2.75 7.56 88 41.25 1701.56 2 2 46.75 2510.75 2510.75 627.69 4 627.69 25.05 M SS SS SD N SD SD = = = = = = = = 12 An important note about the standard deviation In practice, the estimated standard deviation is used because statistical analyses are based on samples drawn from the population In theory, you never test the entire population A minor change to the formula: instead of dividing by N, divide by N - 1 N: when you are dealing with entire population N – 1: results based on a sample
13 The Normal Distribution

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Psyc_315_-_Winter_2010_-_Class_5 - Recap of Last Class...

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