1. Chapter 3
The Normal Curve

2. Where have we been?

3. Example: Scores on a Psychology quiz
To calculate SS, the variance, and the standard deviation: find the deviations from , square and sum them (SS), divide by N (2) and take a square root().
Example: Scores on a Psychology quiz
Student
John
Jennifer
Arthur
Patrick
Marie X 7 8 3 5
X - 
+1.00
+2.00
-3.00
-1.00
(X - )2
1.00
4.00
9.00
(X- )2 = SS =
X = 30
N = 5
 = 6.00
(X- ) = 0.00
2 = SS/N = 3.20
 = = 1.79

4. The variance and standard deviation are numbers that describe how far, on the average, scores are from their mean, mu.
But we often want additional detail about how scores will fall around their mean.
We may also wish to theorize about how scores should fall around their mean.

5. Describing and theorizing about how scores fall around their mean.
Frequency distributions
Stem and leaf displays
Bar graphs and histograms
Theoretical frequency distributions

6. Frequency distributions
Cumulative
Relative
Frequency
.165
.387
.610
.773
.883
.945
.975
.983
.993
.994
.999
1.000
# of acdnts 1 2 3 4 5 6 7 8 9 10 11
Absolute
Frequency
117
157
158
115 78 44 21 7 6 1 3
708
Cumulative
Frequency
117
274
432
547
625
669
690
697
703
704
707
708
Cumulative frequencies show number of scores at or below each point.
Calculate by adding all scores below each point.
Cumulative relative frequencies show the proportion of scores at or below each point.
Calculate by dividing cumulative frequencies by N at each point.

7. Stem and Leaf Display Reading time data i = .05 #i = 10 Reading Time
2.9
2.8
2.7
2.6
2.5
Leaves
5,5,6,6,6,6,8,8,9
0,0,1,2,3,3,3
5,5,5,5,5,6,6,6,7,7,7,7,7,7,7,8,9,9,9,9
0,0,1,2,3,3,3,3,4,4,4
5,5,5,5,6,6,6,8,9,9
0,0,0,1,2,3,3,3,4,4
5,6,6,6
0,1,1,1,2,3,3,4
6,6,8,8,8,8,8,9,9,9
0,1,1,1,2,2,2,4,4,4,4
i = .05
#i = 10

8. Transition to Histograms9 7 6 5 4 2 1 4 3 2 1 9 8 6 4 3 2 1 9 8 6 5 9 8 6 5 4 3 2 1 3 2 1 6 5

9. Histogram of reading times20 18 16 14 12 10 8 6 4 2 F r e q u n c y
Reading Time (seconds)

10. Normal Curve

11. Principles of theoretical frequency distributions
Expected frequency = Theoretical relative frequency X N
Expected frequencies are your best estimates because they are closer, on the average, than any other estimate when we square the error.
Law of Large Numbers - The more observations that we have, the closer the relative frequencies should come to the theoretical distribution.

12. Using the theoretical frequency distribution known as the normal curve

13. The Normal Curve
Described mathematically by Gauss in So it is also called the “Gaussian”distribution. It looks something like a bell, so it is also called a “bell shaped” curve.
The normal curve is a figural representation of a theoretical frequency distribution.
The frequency distribution represented by the normal curve is symmetrical.
The mean (mu) falls exactly in the middle.
68.26% of scores fall within 1 standard deviation of the mean.
95.44% of scores fall within 2 standard deviations of the mean.
99.74% of scores fall within 3 standard deviations of mu.

14. IMPORTANT CONCEPT: Since the curve is symmetrical around the mean, whatever happens on one side of the curve is exactly mirrored on the other side.

15. The normal curve and Z scores
The normal curve is the theoretical relative frequency distribution that underlies most variables that are of interest to psychologists.
A Z score expresses the number of standard deviations that a score is above or below the mean in a normal distribution.
Any point on a normal curve can be referred to with a Z score

16. Concepts behind Z scores
AGAIN, Z scores represent standard deviations above and below the mean.
Positive Z scores are scores higher than the mean. Negative Z scores are scores lower than the mean.
If you know the mean and standard deviation of a population, then you can always convert a raw score to a Z score.
If you know a Z score, the Z table will show you the proportion of the population between the mean and that Z score.

17. The Z table and the curve
The Z table shows the normal curve in tabular form as a cumulative relative frequency distribution.
That is, the Z table lists the proportion of a normal curve between the mean and points further and further from the mean.
The Z table shows only the cumulative proportion in one half of the curve. The highest proportion possible on the Z table is therefore .5000

18. Why does the Z table show cumulative relative frequencies only for half the curve?
The cumulative relative frequencies for half the curve are all one needs for all relevant calculations.
Remember, the curve is symmetrical.
So the proportion of the curve between the mean and a specific Z score is the same whether the Z score is above the mean (and therefore positive) or below the mean (and therefore negative).
Separately showing both sides of the curve in the Z table would therefore be redundant and (unnecessarily) make the table twice as long.

19. IMPORTANT CONCEPT: The proportion of the curve
between any two points on the
curve represents the theoretical
relative frequency (TRF) of
scores between those points.

20. With a little arithmetic, using the Z table, we can determine:
The proportion of the curve above or below any Z score.
Which equals the proportion of the scores we can expect to find above or below any Z score.
The proportion of the curve between any two Z scores.
Which equals the proportion of the scores we can expect to find between any two Z scores.

21. Normal Curve – Basic GeographyF r e q u n c y
The mean
One standard
deviation
Standard
deviations
Measure
Z scores
Percentages
| | |
| | |
| | |

22. The z table See pages 54 and 55 Z Score Proportion mu to Z
0.00
0.01
0.02
0.03
0.04 .
1.960
2.576
3.90
4.00
4.50
5.00
Proportion
mu to Z
.0000
.0040
.0080
.0120
.0160 .
.4750
.4950
.49995
.49997
The Z table contains pairs of columns: columns of Z scores coordinated with columns of proportions from mu to Z.
The columns of proportions show the proportion of the scores that can be expected to lie between
the mean and any other point on the curve.
The Z table shows the cumulative relative frequencies for half the curve.
See pages 54 and 55

23. Another important concept: Most scores are close to the mean
Another important concept: Most scores are close to the mean! So if you have two equal sized intervals, the one closer to mu contains a higher proportion of scores
What proportion of scores falls in the interval between Zs of -.50 to +.50 (an interval of one standard deviation right around the mean)?
= (almost 40%)
Note: this is the one-standard-deviation-wide interval with the highest proportion anywhere on the curve.
Proportion = = .2586

24. Intervals further from mu
What proportion of scores falls in the interval between Zs of 0.00 to (an interval of one standard deviation starting at the mean, but not right around it)?
This one can be read directly from the table
(It is a little over a third)
What proportion of scores falls in the interval between Zs of +.50 to (an interval of one standard deviation a little further from the mean)?
= .2417
This time we are down to less than a quarter of the population.

25. Common Z scores – memorize these scores and proportions
Z Proportion
Score mu to Z
( x 2 = 95% between Z= –1.960 and Z= )
(x 2 = 99% between Z= –2.576 and Z= )

26. USING THE Z TABLE - Proportion of the scores between a specific Z score and the mean.q u n c y
Proportion mu to Z for -0.30
= .1179
Proportion score to mean
=.1179
Standard
deviations
score
470 .

27. USING THE Z TABLE - Proportion of the scores in a population between two Z scores that are identical in size, but have opposite signs. F r e q u n c y
Proportion mu to Z for -0.30
= .1179
Proportion between +Z and -Z =
= .2358
Standard
deviations
score
470
530 .

28. The critical values of the normal curve
Critical values of a distribution show which symmetrical interval around mu contains 95% and 99% of the curve.
In the Z table, the critical values are starred and shown to three decimal places
95% (a proportion of .9500) is found between Z scores of –1.960 and
99% (a proportion of .9900) is found between Z scores of –2.576 and

29. USING THE Z TABLE – Proportion of the scores in a population above a specific Z score.q u n c y
Proportion mu to Z for .30
= .1179
Proportion above score =
= .6179
Standard
deviations
score
470

30. USING THE Z TABLE - Proportion of scores between a two different Z scores on opposite sides of the mean. (ADD THE TWO PROPORTIONS!).
Proportion mu to Z for = .3554
Proportion mu to Z for .37
= .1443 F r e q u n c y
-1.06
+0.37
Z scores
Percent between two scores.
Area Area Add/Sub Total Per
Z Z2 mu to Z1 mu to Z2 Z1 to Z2 Area Cent
Add %

31. USING THE Z TABLE - Proportion of scores between two Z scores on the same side of the mean. (Subtract the smaller proportion from the larger one.)
Proportion mu to Z for 1.12 = .3686
Proportion mu to Z for 1.50
= .4332 F r e q u n c y
+1.12
+1.50
Z scores
Percent between two scores.
Area Area Add/Sub Total Per
Z Z2 mu to Z1 mu to Z2 Z1 to Z2 Area Cent
Sub %

32. Obtaining expected frequencies (EF) from the normal curve.
Basic rule: To find an expected frequency, multiply the proportion of scores expected in the part of the curve by the total N. Expected frequency = theoretical relative frequency x N.

33. Expected frequencies are another least squared, unbiased prediction.
Expected frequencies usually must be wrong, as they are routinely written to two decimal places, while it is impossible to actually find 65 hundreths of a score anywhere.
So, expected frequencies are another set of least squared, unbiased predictions.
Such predictions can be expected to be wrong, but close.

34. EF=TRF x N
In the examples that I’ve solved that follow, let’s assume we have a population of size 300 (N=300)
To find the expected frequency, compute the proportion of the curve between two specific Z scores, just as we have been doing.
Then multiply that proportion (also called the theoretical relative frequency or TRF) by N.

35. Expected frequency = theoretical relative frequency x number of participants (EF=TRF*N). TRF from mean to Z = -.30 = If N = 300: EF= .1179*300 =
EF= .1179x300 = 35.37
470 F r e q u n c y
Standard
deviations
Proportion mu to Z for -0.30
= .1179 .

36. Expected frequency above a score.
This is like asking the EF between your Z score and a Z score of Half the curve lies between mu and a Z score of
If Z is below mu, TRF is between two Z scores on opposite sides of the mean. To get TRF, add half of the curve (.5000) to the area from mu to Z. To get EF, then multiply TRF by N.
If Z is above mu, TRF is between two Z scores on the same side of the mean. To get TRF, subtract the area from mu to Z from half of the curve (.5000). To get EF, then multiply TRF by N

37. TRF above Z of -. 30 is. 1179 +. 5000 =. 6179. If N = 300: EF=
TRF above Z of is = If N = 300: EF=.6179 x 300 =
Standard
deviations
-.30 F r e q u n c y
Proportion mu to Z for .30
= .1179
Proportion above score =
= .6179

38. EF below a score.
This is the opposite of expected frequencies above a score. It is like asking the EF between your Z score and a Z score of Half the curve lies between mu and a Z score of
If Z is above mu, TRF is between two Z scores on opposite sides of the mean. To get TRF, add half of the curve (.5000) to the area from mu to Z. To get EF, then multiply TRF by N.
If Z is below mu, TRF is between two Z scores on the same side of the mean. To get TRF, subtract the area from mu to Z from half of the curve (.5000). To get EF, then multiply TRF by Z

39. If N = 300, what is the EF of scores below a Z of 1.00.
Expected frequency below a score: If Z is above mu, to get TRF, add half of the curve (.5000) to the area from mu to Z. TRF below Z = is =

40. F r e q u n c y inches If N = 300: EF=.8413 x 300 = 252.39.
Proportion = up to mean
for 1 SD =
Standard
deviations
inches

41. Percentile Rank

42. Percentile rank is the proportion of the population you score as well as or better than times The proportion you score as well as or better than is shown by the part of the curve below (to the left of) your score.

43. Computing percentile rank
Above the mean, add the proportion of the curve from mu to Z to
Below the mean, subtract the proportion of the curve from mu to Z from
In either case, then multiply by 100 and round to the nearest integer (if 1st to 99th).
For example, a Z score of –2.10
Proportion mu to Z = .4821
Proportion at or below Z = =.0179
Percentile = x 100 = 1.79 = 2nd percentile

44. Percentile Rank is the percent of the population you score as well as or better = TRF below your Z score times 100. What is the percentile rank of someone with a Z score of +1.00 F r e q u n c y
Percentile: up to mean
=.8413
.8413 x 100 =84.13 =84th percentile
Standard
deviations
inches

45. A rule about rounding percentile rank
Between the 1st and 99th percentiles, you round off to the nearest integer.
Below the first percentile and above the 99th, use as many decimal places as necessary to express percentile rank.
For example, someone who scores at Z=+1.00 is at the 100( ) = = 84th percentile.
Alternatively, someone who scores at Z=+3.00 is at the 100( )=99.87= 99.87th percentile. Above 99 and below 1, don’t round to integers.
We never say that someone is at the 0th or 100th percentile.

46. Calculate percentiles
Z Area Add to (if Z > 0) Proportion Percentile
Score mu to Z Sub from (if Z < 0) at or below st th th rd th

47. Below the 1st percentile and above the 99th: Don’t round!
What percentile are you at if your Z score is +3.04?
Area mu to Z =
Since Z is above the mean, add proportion mu to Z to .5000
Percentile = ( )*100 = 99.88
Above 99th percentile, DON”T ROUND!
The answer is the 99.88th percentile