1. 1 Statistics for the Behavioral Sciences (5 th ed.) Gravetter & Wallnau Chapter 4 Variability University of Guelph Psychology 3320 — Dr. K. Hennig Winter 2003 Term

2. 2 Chapter in outline 1) Individual Differences in Attachment Quality 2) Factors that Influence Attachment Security 3) Fathers as Attachment Objects 4) Attachment and Later Development

3. 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ****** ******** ******** ******** ******** ******** **** **** ********** ** **** ****** **** ************ ******** ****** ****** **** ***** * * Honours-Yes Honours-No

4. 4 Range Interquartile Range Sum of Squares (Sample) Variance (Sample) Standard Deviation Measures of Variability

5. 5

6. 6 Standard deviation and samples  Goal of inferential stats is to generalize to populations from samples  Representativeness? But, samples tend to be less variable (e.g., tall basketball players) - thus a biased estimate of variance  Need to correct for the bias by making an adjustment to derive a more accurate estimate of the population variability  Variance = mean squared deviation = sum of squared deviations/number of scores

7. 7 Calculating sd and variance: 3 steps (M = 6.8 females) X X-M (Step 1) (X-M) 2 (Step 2) 3-3.814.4 4-2.87.8 92.24.8 Step 3: SS =  (X-M) 2

8. 8 Step 3: SS =  (X-M) 2 - Definition formula (sum of squared deviations) Alternatively: SS = X 2 - (X) 2 /n -computational formula Now correct for the bias with an adjustment, sample variance = s 2 = SS/n - 1 (sample variance) and Calculating variance and sd (contd.)

9. 9 Thus… (text, p. 118) Computational formula X (X) 2 11 636 416 39 864 749 636

10. 10 Degrees of freedom - two points: 1) the sample SS ≤ population SS, always –the difference between the sample mean and the population mean is the sampling error 2) you need to know the mean of the sample to compute the SS; thus one variable is dependent on the rest - df of a sample is n-1 (i.e., the adjustment)  df (defn) - the number of independent scores. Population = 4 SS = 17 Sample of n = 3 scores [8, 3, 4] M = 5 SS = 14

11. 11 Note  Note. an average (mean) = sum/number  thus, variance is the average deviation from the mean –mean squared deviation = sum of squared deviations/ –but to calculate sample variance:

12. 12 Biased and unbiased statistics Table 4.1  63/9 = 7 but 126/9= 14 Population = 4  2 =14 Sample 2 Sample 6 Sample 1 Sample 3 Sample 4 Sample 5 SampleMean s^2 (n) s^2 (n-1) 10.00.00.0 21.52.254.5 34.520.2540.5 41.52.254.5 …99.000.0 total = 36 63126

13. 13 Transformation rules 1) Adding a constant to each score will not change the sd 2) Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant

14. 14 Variance and inferential stats (seeing patterns)  conclusion: the greater the variability the more difficult it is to see a pattern  variance in a sample is classified as error variance (i.e., static noise)  “one suit and lots of bad tailors”

15. 15 Statistics for the Behavioral Sciences (5 th ed. ) Gravetter & Wallnau Chapter 5 z-Scores University of Guelph Psychology 3320 — Dr. K. Hennig Winter 2003 Term

16. 16 Intro to z-scores  Mean & sd as methods of describing entire distribution of scores  We shift to describing individual scores within the distribution - uses the mean and sd (as “location markers”)  “Hang a left (sign is -) at the mean and go down two standard deviations (number) ”  2 nd purpose for z-scores is to standardize an entire distribution

17. 17 z-scores and location in a distribution -2 -1 0 +1 +2 -2 -1 0 +1 +2 Every X has a z-score location In a population:  ---->

18. 18 The z-score formula  A distribution of scores has a  =50 and a standard deviation of  = 8  if X = 58, then z = ___ ?

19. 19 X to z-score transformation: Standardization 80 90 100 110 -2 -1 0 1 2 80 90 100 110 -2 -1 0 1 2  shape stays the same  in a z-score distribution is always 0  the standard deviation is always 1  procedure:  Bob got a 70% in Biology and a 60% in Chemistry - for which should he receive a better grade?

20. 20 Looking ahead to inferential statistics  Is treated sample different from the original population?  Compute z-score of sample; e.g., if X is extreme (z=2.5), then there is a difference Population  = 400  = 20 Sample of n =1 Treatment Treated Sample

21. 21 Statistics for the Behavioral Sciences (5 th ed. ) Gravetter & Wallnau Chapter 6 Probability University of Guelph Psychology 3320 — Dr. K. Hennig Winter 2003 Term

22. 22 Example  Jar = population of 3 checker, 1 red dotted, 3 yellow dotted, 3 tiled marbles  if you know the population you know the probability of picking a n =1 tiled sample –3/10 (almost a 30% chance)  but we don’t know the population (reality)  inferential statistics works backwards

23. 23 Sample Population

24. 24 Introduction to probability  probability of A = number of outcomes A/ total number of possible outcomes  p(spade) = 13/52 = ¼ (or 25%)  p (red Ace) = ?  random sample: –each individual in the population has an equal chance (no selection bias) –if sample > 1, then there must be constant probability for each and every selection  e.g., p(jack) if first draw was not a jack?  sampling with replacement

25. 25 * 0 1 2 3 4 ******** ****** ****** *

26. 26 “God loves a normal curve” 34.13% 13.59% 2.28% = 68 74 80 = 6 What is the probability of picking a 6’ 8” (80”) tall person from the population? or p(X>80) = 80-68/6 = +2.0 p(z>2,0) = ?

27. 27 Unit normal table (Fig. 6.6) (A)z(B)(C)(D).01.504.496.004.02.508.492.008 B C D

28. 28 Finding scores corresponding to specific proportions or ps X z-score proportions or ps unit normal table

29. 29 Binomial distribution  probability of A (heads) = p(A)  probability of B (tails) = p(B)  p + q = 1.00 1 st toss 2 nd toss 000 011 101 112 0 1 2 p=.50.25 -With more tosses -> normal & mean increases (M=3 with 6 tosses)

30. 30 The normal approximation to the binomial distribution  With increases in n the distribution approaches a normal curve  Given 10 tosses the expectation is to obtain around 5 heads; unlikely to get values far from 5  Samples with n>10 (the criteria)  Mean:  = pn (e.g., p (heads given 2 tosses) = ½(2)=1  standard deviation:  = npq

31. 31 Example 6.4a (text)  A PSYC dept. is ¾ female. If a random sample of 48 students is selected, what is p(14 males)? (i.e., 12 males)  pn=¼(48)=12=  qn=3/4(48)=36=  p(X= 14) = are under curve 13.5-14.5

32. 32 Example 6.14a (cond.) 12 14 X values.50.83 z-scores

33. 33 Looking ahead to inferential statistics