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 GuelphPsychology 3320 Dr. K.
HennigWinter 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

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Honours-No
Honours-Yes
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Measures of Variability
Range Interquartile Range Sum of Squares
(Sample) Variance (Sample) Standard Deviation
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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

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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
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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
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Thus (text, p. 118)Computational formula
X (X)2
1 1
6 36
4 16
3 9
8 64
7 49
6 36
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Degrees of freedom - two points

• Population
• 4SS 17

Sample of n 3 scores8, 3, 4M 5SS 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.

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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

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Biased and unbiased statisticsTable 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

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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

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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

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

• Chapter 5 z-Scores

University of GuelphPsychology 3320 Dr. K.
HennigWinter 2003 Term
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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

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z-scores and location in a distribution

• Every X has a z-score location
• In a population

• -2 -1 0 1 2

? ----gt?
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The z-score formula

• A distribution of scores has a ? 50 and a
  standard deviation of ? 8
• if X 58, then z ___ ?

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X to z-score transformationStandardization

• 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?

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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

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

• Chapter 6 Probability

University of GuelphPsychology 3320 Dr. K.
HennigWinter 2003 Term
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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

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Population
Sample
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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

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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
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Unit normal table (Fig. 6.6)
(A) z (B) (C) (D)
.01 .504 .496 .004
.02 .508 .492 .008
B
C
D
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Finding scores corresponding to specific
proportions or ps
z-score
X
unit normaltable
proportionsor ps
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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)
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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

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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

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