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Statistics for the Behavioral Sciences (5th ed.)

Gravetter Wallnau

- Chapter 4 Variability

University of Guelph Psychology 3320 Dr. K.

Hennig Winter 2003 Term

Chapter in outline

- Individual Differences in Attachment Quality
- Factors that Influence Attachment Security
- Fathers as Attachment Objects
- Attachment and Later Development

Honours-No

Honours-Yes

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

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

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

Thus (text, p. 118) Computational formula

X (X)2

1 1

6 36

4 16

3 9

8 64

7 49

6 36

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.

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

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

Transformation rules

- Adding a constant to each score will not change

the sd - Multiplying each score by a constant causes the

standard deviation to be multiplied by the same

constant

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

Statistics for the Behavioral Sciences (5th ed.)

Gravetter Wallnau

- Chapter 5 z-Scores

University of Guelph Psychology 3320 Dr. K.

Hennig Winter 2003 Term

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

z-scores and location in a distribution

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

- -2 -1 0 1 2

? ----gt?

The z-score formula

- A distribution of scores has a ? 50 and a

standard deviation of ? 8 - if X 58, then z ___ ?

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?

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

Statistics for the Behavioral Sciences (5th ed.)

Gravetter Wallnau

- Chapter 6 Probability

University of Guelph Psychology 3320 Dr. K.

Hennig Winter 2003 Term

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

Population

Sample

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

Unit normal table (Fig. 6.6)

(A) z (B) (C) (D)

.01 .504 .496 .004

.02 .508 .492 .008

B

C

D

Finding scores corresponding to specific

proportions or ps

z-score

X

unit normal table

proportions or ps

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)

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

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

Example 6.14a (cond.)

12 14 X values .50 .83 z-scores

Looking ahead to inferential statistics