PSY201: Chapter 5: The Normal Curve and Standard Scores
– Normal curve
+ a very important distribution in behavior sciences
+ three principal reasons why...
- 1. many of the variables measured in behavioral science research have distributions that quite closely
approximate the normal curve (ie: height, weight, intelligence and achievement are few examples)
- 2. many of the inference tests used in analyzing experiments have sampling distributions that become normally
distributed with increasing sample size. (ie: sign test & Mann-Whitney U test)
- 3. many inference tests require sampling distributions that are normally distributed. The z test, Student's t test,
and the F test are examples of inference tests that depend on this point → much of importance of normal curve
occurs in conjunction with inferential statistics.
The Normal Curve:
– normal curve is a theoretical distribution of population scores.
+ a theoretical curve and is only approximated by real data
+ bell-shaped curve that is described by equation:
– curve has two inflection points, one on each side of the mean
+ inflection points are located where the curvature changes direction
+ ie: inflection points are located where curve changes from being convex downward to being convex upward
- if the bell-shaped curve is a normal curve, inflection points are at 1 standard deviation from the mean (
- as the curve approaches the horizontal axis, it is slowly changing its Y value.
- the curve never quite reaches the axis
- it approaches the horizontal axis and gets closer and closer to it, but it never quite touches it.
- curve is asymptotic to the horizontal axis
– infection points under the curve, horizontal... in the diagram on page 97
Area Contained Under the Normal Curve:
– in distributions that are normally shaped, there is a special relationship between the mean and the standard
deviation with regard to the area contained under the curve
– when a set of scores is normally distributed, 34.13% of the area under the curve is contained between the mean
(u) and a score that is equal to u + 1o ; 13.59% of the area is contained between a score equal to + 1 and a
score of u+ 2 o; 2.15%of the area is contained between scores of u+ 2o and u + 3o ; and 0.13% of the area
exists beyond u+ 3o . This accounts for 50% of the area
+ since curve is symmetrical, same percentages hold for scores below the mean
+ since frequency is plotted on vertical axis, these percentages represent the percentage of scores contained
within the area – ie:
+ have a population of 10,000 IQ scores
+ distribution normally shaped with u = 100 and o = 16
+ since scores are normally distributed, 34.13% of scores are contained between scores of 100 and 116 ( u+ 1o
= 100 + 16 = 116), 13.59% between 116 and 132 ( u+ 2o = 100+32 = 132), 2.15% between 132 and 148, and
0.13% above 148
+ similarly, 34.13% of scores fall between 84 and 100, 13.59% between 68 and 84, 2.15% between 52 and 68,
and 0.13% below 52.
– to calculate the number of scores in each area, multiply the relevant percentage by the total number of scores →
there are 34.13% x 10,000 = 3413 scores between 100 and 116, 13.59% x 10.000 = 1359 scores between 116
and 132, and 215 scores between 132 and 148; 13 scores are greater than 148.
+ for other half of distribution, there are 3413 scores between 84 and 100, 1359 scores between 68 and 84, and
215 scores between 52 and 68; there are 13 scores below 52.
+ these frequencies would be true only if distribution is exactly normally distributed
+ in actual practice, the frequencies would vary slightly depending on how close the distribution is to this
Standard Scores (z Scores):
– IQ of 132...
+ a score is meaningless unless you have a reference group to compare against
+ without one, can't tell whether the score is high, average, or low
– score is one of the 10,000 scores of distributions → gives IQ of 132 some meaning
+ ie: can determine the percentage of scores in distribution that are lower than 132 → determining the percentile
rank of score of 132 (percentile rank of a score is defined as the percentage of scores that is below the score in
– 132 is 2 standard deviations above the mean
+ in normal curve, there are 34.13 + 13.59 = 47.72% of the scores between the mean and a score that is 2
standard deviations above the mean
+ to fine percentile rank of 132, need to add this percentage the 50.00% that lie below the mean → 97.72%
(47.72 + 50.00) of the scores fall below your IQ score of 132.
+ should be happy to be intelligent
– to solve problem, had to determine how many standard deviations the raw score of 132 was above or below the
+ transformed the raw score into a standard score, also called a z score
– a z score is a transformed score that designated how many standard deviation units the corresponding raw score
is above or below the mean – process which by the raw score is altered – score transformation
+ z transformation results in a distribution having a mean of 0 and a standard deviation of 1
+ reason z scores are called standard deviation is they are expressed relative to a distribution mean of 0 and a
standard deviation of 1
– in conjunction with a normal curve, z scores allow to determine the number or percentages of scores that fall
above or below any score in the distribution
+ z scor