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Statistical Inference > The Normal Distribution
Statistical Inference: Normal Distribution
The Normal Distribution and Z-Scores
What is a "normal" distribution?
The distribution of equally-likely events in the long run through accidents of nature.
In an infinite amount of time, a random process could ultimately generate structured results: e.g., a group of monkeys seated at typewriters could peck out all the great works of literature.
This would be an extremely rare event, but it is conceivable.
The normal curve is a mathematical formula that assigns probabilities to the occurrence of rare events.
Statistically speaking, it is a probability distribution for a continuous random variable:
The ordinate represents the probability density for the occurrence of a value.
The baseline represents the values.
The exact shape of the curve is given by a complicated formula that you do NOT need to know.
The area under the curve is interpreted as representing all occurrences of the variable, X.
We can consider the area as representing 100 PERCENT of the occurrences; in PROPORTIONS this is expressed as 1.0.
We can then interpret AREAS under the curve as representing certain PROPORTIONS of occurrences or "probabilities".
We cannot assign a probability to any point, but we can attach probabilities to INTERVALS on the baseline associated with AREAS under the curve: e.g., the mean has 50% of the cases standing to each side.

Special properties of the normal distribution:
Their shape is such that it
Embraces 68.26% of the cases within 1 s.d. around the mean.
Embraces 95.46% of the cases within 2 s.d. around the mean.
Embraces 99.74% of the cases within 3 s.d. around the mean.
More roughly speaking, 68%, 95%, and 99% of the cases are embraced within 1, 2, and 3 standard deviations from the mean in a normal distribution.
Determining whether a distribution is "normal"
The "Eyeball" test
Is the distribution unimodal?
Is the distribution symmetrical?
More exacting mathematical tests: measured according to "moments" or "deviations" from the mean
FIRST MOMENT: = 0 <----you know this already
SECOND MOMENT: = variance <--- you had this
THIRD MOMENT: = skewness
Formula calculated within SPSS
Positive values greater than 0 mean right-skew
FOURTH MOMENT: - 3 = kurtosis
Positive values means more peaked (LEPTOKURDIC) than the normal curve
Negative values means flatter (PLATYKURDIC)
If skewness and kurtosis values tend toward 0, then the distribution approximates a normal distribution.
Suppose the distribution is not normal?
No matter how the original observations are distributed, the mean plus or minus two standard deviations will include at least 75% of the observations.
No matter how the original observations are distributed, the mean plus or minus three standard deviations witll include 89% or more. (Freeman, p. 62)

Using the Table of Areas under the Normal Curve: The z-score
One determines the probability of occurrence of a random event in a normal distribution by consulting a tables of areas under a normal curve (e.g., Table D.2, pp.702-705 in Kirk).
Tables of the normal curve have a mean of 0 and a standard deviation of 1.
To use the table., you must convert your data to have a mean of 0 and standard deviation of 1.
This is done by transforming your raw values into z-scores, according to this formula:
z-score =

Example: percent black in Washington, D.C. in 1980 (Note: not 1990)

D.C's Raw score = 70.3



Mean for all states = 10.3




(70.3 - 10.3) = 60

60 / 12.5 = 4.8


standard deviation = 12.5



z-score for D.C. = 4.8

Example: D.C.'s percent vote for Reagan in 1984

D.C's Raw score = 13



Mean for all states = 60




(13 - 60) = 47

47 / 8.8 = -5.3


standard deviation = 8.8



z-score for D.C. = -5.3

Comparison with Florida: percent vote for Reagan in 1984

Florida's percent black is 13.8



z-score = (13.8 - 10.3) / 12.5 = .28

Florida's percent for Reagan was 65



z-score = (65 - 60) / 8.8 = .57

Computing z-scores for raw data Transforming raw standardizes the data, which makes it easier to compare values in different distributions.
Limitations of raw data values
Raw scores of individual cases do not disclose how they vary from the central tendency of the distribution
One needs to know also the mean of the distribution and its variability to determine if any given score is "far" from the mean
Properties of the z-score transformation
It is a LINEAR transformation: does not alter relative positions of observations in the distribution nor change the shape of the original distribution.
The transformed observations have positive AND negative DECIMAL values expressed in STANDARD DEVIATION UNITS.
The SIGN of the z-score tells whether the observation is above or below the mean.
The VALUE of the z-score tells how far above or below it is.
When transformed into z-scores, all distributions are standardized
The mean of the transformed distribution is equal to 0.
The standard deviation of the distribution is equal to 1.
The variance of the distribution is equal to 1. (Old Chinese proverb.)
When subjected to a z-score transformation, any set of raw scores that conform to a normal distribution, will conform exactly to the table of areas under the normal curve.
That is, the likelihood of observing z-scores of certain magnitudes can be read directly from Table D.2.
Using z-scores to read a table of areas under the normal curve