1-3 Variability lecture_ Week 6.pptx
Transcript
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 N1 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 N1 N1 2 ( x ) x 2 Computational 2 SS N s Formula N1 N1 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 N1 5 1 Population vs. sample variance Sample symbol for s2 variance s2 = SS = ( X X ) 2 N-1 N1 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 N1 N (take square root) (take square root) Standard Standard Deviation: s = (X M ) 2 Deviation: σ = ( X ) 2 N1 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).
