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Chapter 3
The Normal Curve
Where have we been?
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
JenniferArthurPatrickMarie
X
7
8357
X = 30 N = 5 = 6.00
X -
+1.00
+2.00-3.00-1.00+1.00
(X- ) = 0.00
(X - )2
1.00
4.009.001.001.00
(X- )2 = SS = 16.00
2 = SS/N = 3.20 = = 1.7920.3
Stem and Leaf Display
Reading time dataReading
Time
2.9
2.9
2.8
2.8
2.7
2.7
2.6
2.6
2.5
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
Transition to Histograms999977777776665555
988666655
3332100
44433332100
9986665555
4433321000
6665
43321110
44442221110
2.50-2.54
2.55-2.59
2.60 –2.64
2.65 –2.69
2.70 –2.74
2.75 –2.79
2.80 –2.84
2.85 –2.89
2.90 –2.94
2.95 –2.99
9998888866
Histogram of reading times
2.50-2.54
2.55-2.59
2.60 –2.64
2.65 –2.69
2.70 –2.74
2.75 –2.79
2.80 –2.84
2.85 –2.89
2.90 –2.94
2.95 –2.99
20181614121086420
Reading Time (seconds)
Frequency
Normal Curve
Principles of Theoretical Curves
Expected frequency = Theoretical relative frequency * 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.
The Normal CurveDescribed mathematically by Gauss in 1851.
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 really represents a histogram whose rectangles have their corners shaved off with calculus.
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.
The normal curve and Z scores
The normal curve is a theoretical 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
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
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.
KEY CONCEPT
The proportion of the curve between any two points on the curve represents the relative frequency of scores between those points.
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.
Normal Curve – Basic Geography
Frequency
Measure
The mean
One standard deviation
|--------------49.87-----------------|------------------49.87------------|
|--------47.72----------|----------47.72--------|
-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00Z scores
|---34.13--|--34.13---|Percentages
3 2 1 0 1 2 3Standard
deviations
The z table
The Z table contains a column of Z scores coordinated with a column of proportions.
The proportion represents the area under the curve between the mean and any other point on the curve. The table represents half the curve
ZScore0.000.010.020.030.04
.1.9602.576
.3.904.004.505.00
Proportionmu to Z
.0000
.0040
.0080
.0120
.0160.
.4750
.4950.
.49995
.49997 .499997
.4999997
Common Z scores – memorize these scores and proportions
Z Proportion Score mu to Z
0.00 .0000
3.00 .4987
2.00 .4772
1.00 .3413
1.960 .4750
2.576 .4950 (* 2 = 99% between Z= –2.576 and Z= + 2.576)
( * 2 = 95% between Z= –1.960 and Z= +1.960)
470
USING THE Z TABLE - Proportion between a score and the mean.
Frequency
score
.
3 2 1 0 1 2 3Standard
deviations
Proportion mu to Z for -0.30
= .1179
Proportion score to mean
=.1179
470
USING THE Z TABLE - Proportion between score
Frequency
score
.
3 2 1 0 1 2 3Standard
deviations
Proportion mu to Z for -0.30
= .1179
Proportion between +Z and -Z
= .1179 + .1179
= .2358
530
470
USING THE Z TABLE – Proportion of the curve above a score.
Frequency
score
Proportion above score.
3 2 1 0 1 2 3Standard
deviations
Proportion mu to Z for .30
= .1179Proportion above score
= .1179 + .5000
= .6179
-1.06
USING THE Z TABLE - Proportion between score and a different point on the other side of the mean.
Frequency
Percent between two scores.
-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00Z scores
+0.37
Proportion mu to Z for -1.06= .3554
Proportion mu to Z for .37= .1443
Area Area Add/Sub Total Per Z1 Z2 mu to Z1 mu to Z2 Z1 to Z2 Area Cent
-1.06 +0.37 .3554 .1443 Add .4997 49.97 %
+1.50
USING THE Z TABLE - Proportion between score and another point on the same side of the mean.
Frequency
Percent between two scores.
-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00Z scores
+1.12
Proportion mu to Z for 1.12= .3686
Proportion mu to Z for 1.50 = .4332
Area Area Add/Sub Total Per Z1 Z2 mu to Z1 mu to Z2 Z1 to Z2 Area Cent
+1.50 +1.12 .4332 .3686 Sub .0646 6.46 %
USING THE Z TABLE – Expected frequency = theoretical
relative frequency * number of participants (EF=TRF*N). Expected frequency between mean and Z = -.30. If N = 300.
.470
Frequency
3 2 1 0 1 2 3Standard
deviations
Proportion mu to Z for -0.30
= .1179
EF= .1179*300 = 35.37
USING THE Z TABLE – Expected frequency = theoretical relative frequency * number of participants (EF=TRF*N). Expected frequency above Z = -.30 if N = 300.EF=.6179 * 300 = 185.37
3 2 1 0 1 2 3Standard
deviations-.30
Frequency
Proportion mu to Z for .30
= .1179Proportion above score
= .1179 + .5000
= .6179
USING THE Z TABLE – Percentage below a score
Frequency
inches
What percent of the population scores at or under a Z score of +1.00
Percentage = 50 % up to mean
3 2 1 0 1 2 3Standard
deviations
+ 34.13% for 1 SD
= 84.13%
USING THE Z TABLE – Percentile Rank is the proportion of
the population you score as well as or better than times 100.
Frequency
inches
What is the percentile rank of someone with a Z score of +1.00
Percentile: .5000 up to mean
3 2 1 0 1 2 3Standard
deviations
+ .3413 =.8413
.8413 * 100 =84.13 =84th percentile
Percentile rank is the proportion of the population you score as well as or better than times 100.
The proportion you score as well as or better than is shown by the part of the curve to the left of your score.
Computing percentile rank
Above the mean, add the proportion of the curve from mu to Z to .5000.
Below the mean, subtract the proportion of the curve from mu to Z from .5000.
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 tg Z = .4821 Proportion at or below Z = .5000 - .4821 =.0179 Percentile = .0179 * 100 = 1.79 = 2nd percentile
A rule about 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(.5000+.3413) = 84.13 = 84th percentile.
Alternatively, someone who scores at Z=+3.00 is at the 100(.5000+.4987)=99.87= 99.87th percentile. Above 99, don’t round to integers.
We never say that someone is at the 0th or 100th percentile.
Calculate percentiles
Z Area Add to .5000 (if Z > 0) Proportion PercentileScore mu to Z Sub from .5000 (if Z < 0) at or below
-2.22 .4868 .5000 - .4868 .0132 1st
-0.68 .2517 .5000 - .2517 .2483 25th
+2.10 .4821 .5000 + .4821 .9821 98th
+0.33 .1293 .5000 + .1293 .6293 63rd
+0.00 .0000 .5000 +- .0000 .5000 50th