1. Social Science Reasoning Using Statistics
Psychology 138
2019

2. Announcements Exam 2 is Wed. March 6th Quiz 4 Fri March 1nd
Covers z-scores, Normal distribution, and describing correlations
Announcements

3. Outline Transformations: z-scores Normal Distribution
Using Unit Normal Table
Today’s lecture puts lots of stuff together:
Probability
Frequency distribution tables
Histograms
Z-scores
Today
Start the quincnux machine
Outline

4. Flipping a coin example
Number of heads
HHH 3
HHT 2
HTH 2
HTT 1
THH 2
THT 1
TTH 1
TTT
Flipping a coin example

5. Flipping a coin example
Number of heads 3 2
Number of heads 1 2 3 .1 .2 .3 .4
probability X f p 3 1
.125 2
.375 2
.375
.375 1
.125
.125 2 1 1
Flipping a coin example

6. Flipping a coin example
What’s the probability of flipping three heads in a row? .4 .3
probability .2
p = 0.125 .1
.125
.375
.375
.125
Think about the area under the curve as reflecting the proportion/probability of a particular outcome 1 2 3
Number of heads
Flipping a coin example

7. Flipping a coin example
What’s the probability of flipping at least two heads in three tosses? .4 .3
probability .2
p = = 0.50 .1
.125
.375
.375
.125 1 2 3
Number of heads
Flipping a coin example

8. Flipping a coin example
What’s the probability of flipping all heads or all tails in three tosses? .4 .3
probability .2
p = = 0.25 .1
.125
.375
.375
.125 1 2 3
Number of heads
Flipping a coin example

9. Flipping a coin example
Coin flipping results in The Binomial Distribution
As you flip more coins,
N = 20
Flipping a coin example

10. Flipping a coin example
Coin flipping results in The Binomial Distribution
As you flip more coins, the binomial distribution can be approximated by the Normal Distribution
N = 65
Flipping a coin example

11. The Normal Distribution (Sometimes called the “Bell Curve”)
Many commonly occurring distributions in nature are approximately Normal
Originally called the curve of normal errors
Normal Distribution

12. The Normal Distribution (Sometimes called the “Bell Curve”)
Many commonly occurring distributions in nature are approximately Normal
Common approximate empirical distribution for deviations around a continually scaled variable (errors of measurement)
Defined by density function (area under curve) for variable X given μ & σ2
Symmetrical & unimodal;
Mean = median = mode
Converting to standardized z-scores
Mean = 0
±1 σ are inflection points of curve (change of direction)
Originally called the curve of normal errors
Normal Distribution

13. The Normal Distribution (Sometimes called the “Bell Curve”)
Many commonly occurring distributions in nature are approximately Normal
Common approximate empirical distribution for deviations around a continually scaled variable (errors of measurement)
Use calculus to find areas under curve (rather than frequency of a score)
We will use a table rather to find the probabilities rather than do the calculus.
Galileo
Originally called the curve of normal errors
Gauss
Check out the quincnux machine
Normal Distribution

14. The Normal Distribution (Sometimes called the “Bell Curve”)
Important landmarks in the distribution (and the areas under the curve)
%(μ to 1σ) = 34.13
p(μ < X < 1σ) + p(μ > X > -1σ) ≈ .68
%(1σ to 2σ) = 13.59
p(1σ < X < 2σ) + p(-1σ > X > -2σ) = 27%, cumulative = .95
%(2σ to ∞) = 2.28
p(X > 2σ) + p(X < -2σ) ≈ 5%, cumulative = 1.00
We will use the unit normal table rather to find other probabilities
68%
34.13
2.28
Originally called the curve of normal errors
95%
13.59
100%
Normal Distribution

15. Lots of places to get the Unit Normal Table information
But be aware that there are many ways to organize the table, it is important to understand the table that you use
Unit normal table in your reading packet
And online:
copy/TABLES.HTMl - ztable
“Area Under Normal Curve” Excel tool (created by Dr. Joel Schneider)
Bell Curve iPhone app
Do a search on “Normal Table” in Google.
Resources and tools

16. Unit Normal Table (linked online in labs)X f p cp 3 1
.125
1.0 2
.375
.875
.500
|z|
.00
.01 :
1.0
2.0
3.0
0.5000
0.1587
0.0228
0.0013
0.4960
0.1562
0.0222
Number of heads 1 2 3 .1 .2 .3 .4
probability
.125
.875
We will use the unit normal table rather to find other probabilities
Unit Normal Table (linked online in labs)

17. Unit Normal Table (in textbook)
z p
Proportions beyond z-scores
Same p-values for + and - z-scores
p-values = 0.50 to
|z|
In body
In tail
0.00 :
1.00
2.0
3.0
0.5000
0.8413
0.9772
0.9987
0.1587
0.0228
0.0013
tail
body
“body” is defined as the larger part of the distribution
Note. : indicates skipped rows
Unit Normal Table (in textbook)

18. Unit Normal Table (in textbook)
z p
Proportions beyond z-scores
Same p-values for + and - z-scores
p-values = 0.50 to
|z|
In body
In tail
0.00 :
1.00
2.0
3.0
0.5000
0.8413
0.9772
0.9987
0.1587
0.0228
0.0013
= .1587
For z = 1.00
, % beyond, p(z > 1.00) =
1.01, 15.62% beyond, p(z > 1.00) =
Note. : indicates skipped rows
Unit Normal Table (in textbook)

19. Unit Normal Table (in textbook)
Proportions beyond z-scores
Same p-values for + and - z-scores
p-values = 0.50 to
|z|
In body
In tail
0.00 :
1.00
2.0
3.0
0.5000
0.8413
0.9772
0.9987
0.1587
0.0228
0.0013
For z = -1.00
, 15.87% beyond, p(z < -1.00) =
100% % = 84.13% =
Note. : indicates skipped rows
Unit Normal Table (in textbook)

20. Unit Normal Table (in textbook)
Proportions beyond z-scores
Same p-values for + and - z-scores
p-values = 0.50 to
|z|
In body
In tail
0.00 :
1.00
2.0
3.0
0.5000
0.8413
0.9772
0.9987
0.1587
0.0228
0.0013
For z = 2.00, 2.28% beyond, p(z > 2.00) =
2.01, 2.22% beyond, p(z > 2.01) =
As z increases, p decreases
Note. : indicates skipped rows
Unit Normal Table (in textbook)

21. Unit Normal Table (in textbook)
Proportions beyond z-scores
Same p-values for + and - z-scores
p-values = 0.50 to
|z|
In body
In tail
0.00 :
1.00
2.0
3.0
0.5000
0.8413
0.9772
0.9987
0.1587
0.0228
0.0013
For z = -2.00, 2.28% beyond, p(z < -2.00) =
Note. : indicates skipped rows
Unit Normal Table (in textbook)

22. Unit Normal Table cumulative version (in some other books)z
.00
.01
-3.4 :
-1.0
1.0
3.4
0.0003
0.1587
0.5000
0.8413
0.9997
0.1562
0.5040
0.8438
Proportions left of z-scores: cumulative
Requires table twice as long
p-values to
50%
= 84.13% to the left
Cumulative % of population starting with the lowest value
For z = +1,
Unit Normal Table cumulative version (in some other books)

23. SAT examples Population parameters of SAT:
μ = 500, σ = 100, normally distributed
Example 1
Suppose you got 630 on SAT. What % who take SAT get your score or better?
From table:
z(1.3) =.0968
9.68% above this score
Solve for z-value of 630.
Find proportion of normal distribution above that value.
Hint: I strongly suggest that you sketch the problem to check your answer against
SAT examples

24. SAT examples Population parameters of SAT:
μ = 500, σ = 100, normally distributed
Example 2
Suppose you got 630 on SAT. What % who take SAT get your score or worse?
From table:
z(1.3) =.0968
100% % = 90.32% below score (percentile)
Solve for z-value of 630.
Find proportion of normal distribution below that value.
SAT examples

25. Wrap up In lab Questions? Using the normal distribution
Check the quincnux machine
Brandon Foltz: Understanding z-scores (~22 mins)
StatisticsFun (~4 mins)
Chris Thomas. How to use a z-table (~7 mins)
Dr. Grande. Z-scores in SPSS (~7 mins)
Wrap up