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Statistics for the Behavioral Sciences (5th ed.) Gravetter

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Statistics for the Behavioral Sciences (5th ed.) Gravetter & Wallnau Chapter 4 Variability University of Guelph Psychology 3320 Dr. K. Hennig – PowerPoint PPT presentation

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Title: Statistics for the Behavioral Sciences (5th ed.) Gravetter


1
Statistics for the Behavioral Sciences (5th ed.)
Gravetter Wallnau
  • Chapter 4 Variability

University of Guelph Psychology 3320 Dr. K.
Hennig Winter 2003 Term
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
Honours-No
Honours-Yes
4
Measures of Variability
Range Interquartile Range Sum of Squares
(Sample) Variance (Sample) Standard Deviation
5
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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
Calculating sd and variance 3 steps (M 6.8
females)
X X-M (Step 1) (X-M)2 (Step 2)
3 -3.8 14.4
4 -2.8 7.8
9 2.2 4.8
Step 3 SS ? (X-M)2
8
Calculating variance and sd (contd.)
Step 3 SS ? (X-M)2 - Definition formula (sum
of squared deviations) Alternatively SS ?X2 -
(?X)2/n -computational formula Now correct for
the bias with an adjustment, sample variance s2
SS/n - 1 (sample variance) and
9
Thus (text, p. 118) Computational formula
X (X)2
1 1
6 36
4 16
3 9
8 64
7 49
6 36
10
Degrees of freedom - two points
  • Population
  • 4 SS 17

Sample of n 3 scores 8, 3, 4 M 5 SS 14
  • the sample SS population SS, always
  • the difference between the sample mean and the
    population mean is the sampling error
  • 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.

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
Biased and unbiased statistics Table 4.1
Sample 1
  • Population
  • 4 ?214

Sample 2
Sample 3
Sample Mean s2 (n) s2 (n-1)
1 0.0 0.0 0.0
2 1.5 2.25 4.5
3 4.5 20.25 40.5
4 1.5 2.25 4.5
9 9.0 0 0.0
total 36 63 126
Sample 4
Sample 5
Sample 6
  • 63/9 7 but 126/9 14

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
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
Statistics for the Behavioral Sciences (5th ed.)
Gravetter Wallnau
  • Chapter 5 z-Scores

University of Guelph Psychology 3320 Dr. K.
Hennig Winter 2003 Term
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)
  • 2nd purpose for z-scores is to standardize an
    entire distribution

17
z-scores and location in a distribution
  • Every X has a z-score location
  • In a population
  • -2 -1 0 1 2

? ----gt?
18
The z-score formula
  • A distribution of scores has a ? 50 and a
    standard deviation of ? 8
  • if X 58, then z ___ ?

19
X to z-score transformation Standardization
  • 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
Looking ahead to inferential statistics
Population ? 400 ? 20
Treatment
Sample of n 1
Treated Sample
  • Is treated sample different from the original
    population?
  • Compute z-score of sample e.g., if X is extreme
    (z2.5), then there is a difference

21
Statistics for the Behavioral Sciences (5th ed.)
Gravetter Wallnau
  • Chapter 6 Probability

University of Guelph Psychology 3320 Dr. K.
Hennig Winter 2003 Term
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 dont know the population (reality)
  • inferential statistics works backwards

23
Population
Sample
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 gt 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
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26
God loves a normal curve
2.28
What is the probability of picking a 6 8 (80)
tall person from the population? or p(Xgt80)
80-68/6 2.0 p(zgt2,0) ?
? 68 74 80 ? 6
13.59
34.13
27
Unit normal table (Fig. 6.6)
(A) z (B) (C) (D)
.01 .504 .496 .004
.02 .508 .492 .008
B
C
D
28
Finding scores corresponding to specific
proportions or ps
z-score
X
unit normal table
proportions or ps
29
Binomial distribution
  • probability of A (heads) p(A)
  • probability of B (tails) p(B)
  • p q 1.00

1st toss 2nd toss
0 0 0
0 1 1
1 0 1
1 1 2
p .50 .25
0 1 2
-With more tosses -gt normal mean increases
(M3 with 6 tosses)
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 ngt10 (the criteria)
  • Mean ? pn (e.g., p (heads given 2 tosses)
    ½(2)1
  • standard deviation ? ?npq

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)
  • pn¼(48)12?
  • qn3/4(48)36?
  • p(X 14) are under curve 13.5-14.5

32
Example 6.14a (cond.)
12 14 X values .50 .83 z-scores
33
Looking ahead to inferential statistics
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