Variability Lecture Notes - Week 6 PDF

VARIABILITY Overview  Variability  What is it & why is it important?  Measures of variability (i.e., variation) Range Variance Standard deviation  Calculations Computation vs. definition formulas for Sums of Squares (SS)...

VARIABILITY Overview  Variability  What is it & why is it important?  Measures of variability (i.e., variation) Range Variance Standard deviation  Calculations Computation vs. definition formulas for Sums of Squares (SS) Samples versus populations Variability  Why do we care about variability?  Where would you like to vacation?  Gulfside BungalowsDay = 72 M = 70 Night =  Kalahari Condos 68 Day = M = 70 110 Night = 30 Variability  In the past month, how many times have you had a serious disagreement or conflict with your significant other? Sample 1: (n=3) Sample 2: (n=3) 4 1 5 5 6 9 Mean = 5 Mean = 5 Median = 5 Median = 5 Mode = none Mode = none Measures of variability  Degree to which scores are spread out in a distribution  Gives us an idea about how well an individual or sample might represent population Measures of Variability  Range: one measure of variability  (Crude Range) Largest score minus smallest score  1, 3, 5, 7, 9  Range = 9 – 1 = 8  Only takes into consideration extreme scores The Range 1 person 1 person 60 70 80 90 100 60 70 80 90 100 Mean = 80; Crude Range = 40 Measures of Variability (con’t)  2 common measures  Variance  Standard deviation  Degree to which a distribution varies from mean Deviations from mean X M (X-M) 4 5 -1 Deviation 5 5 0 Scores 6 5 1 Σ(X- M)=0 Subtract Mean from the Score= Standard Deviation We will always get 0 when we sum the deviation scores. We have to make the deviations from the mean stop cancelling each other out. Variability X M (X-M) (X-M)2 4 5 -1 +1 Squared 5 5 0 0 deviation s 6 5 +1 +1 Σ(X-M)2 = Squaring each 2 Sum of deviation & then squared summing gives a deviation value greater than 0 s Sum of Squared Deviations from Mean  Σ(X-M)2 = Sum of squares or SS  Definitional formula  Computational formula 2 (x) 2 SS x  N  Sum of squares (1, 2, 6) Definitional Computational 2 ( x ) 2 x  SS =Σ(X-M)2 N X X-M (X-M)2 X X2 1 -2 4 1 1 2 -1 1 2 4 6 3 9 6 36 Σ(X- ΣX= ΣX 2 = M)=0 M= 9 41 Sum of squares = 41- 3 Sum of squares = Σ(X-M) (92)/3 2 =14 41-(81/3)=41-27=14 Variance  Sample Variance – average squared deviations  s2 or SD2 or Means square (MS) SS s2  N1 Steps for computing variance:  Compute sum of squares (SS)  Use definitional or computational formulas 2 ( x ) X  M  x 2  2 N  Then divide by N - 1 Practice  Compute Sample variance X  M  2 2 SS Definitional s   Formula N1 N1 2 (  x ) x 2  Computational 2 SS N s   Formula N1 N1  1, 2, 4, 4, 10 Practice: Answer: SS Definitional Formula Practice: Answer: SS Computational Formula Practice: Answer: Variance SS 48.8 s2  s2  12.2 N1 5 1 Population vs. sample variance Sample symbol for s2 variance s2 = SS =  ( X  X ) 2 N-1 N1 Population symbol for σ2 variance σ2= SS =  ( X  ) 2 N N Why N -1?  Usually cannot collect data from entire population  Infer population from sample  Problem: Spread of scores in sample tends to be smaller  Dividing by N-1 gives us a more unbiased estimate Standard deviation  Avg (or typical) deviation of scores from mean  Remember 1, 2, 4, 4, 10 M= 4.2  s2 = 12.2  Square root of variance  Square root of 12.2 is 3.49, which makes more sense to us because it takes us back to the original units  Standard deviation Sample Population Variance: 2 s   ( X  X ) 2 Variance: σ2=  ( X   ) 2 N1 N (take square root) (take square root) Standard Standard Deviation: s =  (X  M ) 2 Deviation: σ =  ( X   ) 2 N1 N Review: Measures of variability  First, compute the sum of squares Definitional formula: Computational formula: 2 (x) SS X  M  2 2 SS x  N Review: Measures of variability  Second, determine whether your data are for a population or a sample. Then compute the variance term using the sum of squares. If population: If sample: σ2 = Sum of Squares s2 = Sum of Squares N N -1  Third, find the standard deviation by taking the square root of the variance term Transformations of scale  Adding/subtracting a constant to each score  Example (population): 1 3 5 7  What is the standard deviation? SD = 2.24  Add 2 points to each score  What is the new standard deviation? SD = 2.24  Will not change the standard deviation Transformations of scale  Multiplying/dividing each score  Example (population): 1 3 5 7  What is the standard deviation? Already know SD = 2.24  Multiply each score by 2  What is the new standard deviation? SD = 4.47  Causes SD to be multiplied/divided by the same constant  2 x 2.24 = 4.48 z-Scores  Transform each X-value into a signed (+ or -) number:  The sign tells us whether the score is above (+) or below (-) the Mean.  The number tells us the distance between the score and the mean (in terms of the number of standard deviations). Example: IQ Scores: Mean = 100 SD = 15 The score (X) is 130 = Z=+2.00 The score is ABOVE the mean by a distance of 2 standard deviations. Z= (X-M)/SD Reporting Results in APA format  Report standard deviations when reporting the means  Use statistical abbreviations when the info is in parentheses  “Students in the high-stress condition talked less (M = 22.07, SD = 27.14) than those in the low-stress condition (M = 45.20, SD = 24.97).”  Summarizing Research 28  Suppose 15 women participate in an experiment designed to examine the effects of sleep deprivation on math performance.  Participants are randomly assigned to one of three groups. Based on their assigned condition, they will be woken after 2 hours of sleep, 4 hours of sleep, or 8 hours of sleep.  Researchers assess the number of errors participants make on a math test.  Summarizing Research 29 2 hrs/sleep 4 hrs/sleep 8 hrs/sleep 10 8 0 12 8 6 12 10 5 14 8 4 12 6 0  What is the mean number of errors for each condition?  What is the sample (to be used to estimate population) standard deviation for each condition?  Report the findings in APA format.  Summarizing Research 30  APA Format  Women with eight hours of sleep made fewer errors (M = 3.00, SD = 2.83) than women with four hours of sleep (M = 8.00, SD = 1.41), who made fewer errors than women with two hours of sleep (M = 12.00, SD = 1.41).  Women with eight hours of sleep made fewer errors than women with four hours of sleep, who made fewer errors than women with two hours of sleep (M = 3.00, SD = 2.83; M = 8.00, SD = 1.41; M = 12.00, SD = 1.41; respectively).  You would need to compute an ANOVA to determine if the mean for each condition are significantly different from one another (later lectures).