Download Normal Distribution

Normal Distribution
Sampling and Probability
Properties of a Normal Distribution
• Mean = median = mode
• There are the same number of scores
below and above the mean.
• 50% of the scores are on either side of the
mean.
• The area under the normal curve totals
100%..
Not all distributions are bell-shaped
curves
•
•
•
•
Some distributions lean to the right
Some distributions lean to the left
Some distributions are tall or peaked
Some distributions are flat on top
Skewed Distribution (positive skew)
300
200
100
Std. Dev = 17.81
Mean = 45.6
N = 1514.00
0
20.0
30.0
25.0
40.0
35.0
50.0
45.0
A ge of Respondent
60.0
55.0
70.0
65.0
80.0
75.0
90.0
85.0
Things to remember about skewed
distributions
• Most distributions are not bell-shaped.
• Distributions can be skewed to the right or
the left.
• The direction of the skew pertains to the
direction of the tail (smaller end of the
distribution).
• Tails on the right are positively skewed.
• Tails on the left are negatively skewed.
Other things to know about
skewness
• Skewness is a measure of how evenly
scores in a distribution are distributed
around the mean.
• The bigger the skew the larger is the
degree to which most scores lie on one
side of the mean versus the other side.
• Skew scores produced in SPSS are either
negative or positive, indicating the
direction of the skew.
Negative Skew
600
500
400
300
200
100
Std. Dev = 2.84
M ean = 13.9
N = 1845.00
0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
HIGHEST YEAR OF SCHOOL COMPLETED
20.0
Information about previous graph
Mean
2.845
Standard Deviation
13.93
Skew
-.084
Kurtosis
.252
Knowing whether a distribution is
skewed:
• Tells you if you have a normal distribution. (If the
skew is close to zero, you may have a normal
distribution)
• Tells you whether you should use mean or
median as a measure of central tendency. (The
greater the skew, the more likely that the median
is the better measure of central tendency)
• Tells you whether you can use inferential
statistics.
Another measure of the shape of the distribution is
kurtosis. This measure tells you:
• Whether the standard deviation (degree to
which scores vary from the mean) are
large or small)
• Curves that are very narrow have small
standard deviations.
• Curves that are very wide have large
standard deviations.
• A kurtosis measure that is near zero (less
than 1) may indicate a normal distribution.
Other Important Characteristics of
a Normal Distribution
• The shape of the curve changes
depending on the mean and standard
deviation of the distribution.
• The area under the curve is 100%
• A mathematical theory, the Central Limit
Theorem, allows us to determine what
scores in the distribution are between 1, 2,
and 3 standard deviations from the mean.
Central Limit Theorem:
• 68.25% of the scores are within one
standard deviation of the mean.
• 94.44% of the scores are within two
standard deviations of the mean.
• 99.74%(or most of the scores) are within 3
standard deviations of the mean.
Central Limit Theorem also means
that:
• 34.13% of the scores are within one
standard deviation above or below the
mean.
• 47.12% of the scores are within two
standard deviations above or below the
mean.
• 49.87% of the scores are within three
standard deviations above or below the
mean.
One standard deviation:
Two standard deviations from the
mean:
Three standard deviations from the
mean:
This theory allows us to:
• Determine the percentage of scores that fall within any
two scores in a normal distribution.
• Determine what scores fall within one, two, and three
standard deviations from the mean.
• Determine how a score is related to other scores in the
distribution (What percentage of scores are above or
below this number).
• Compare scores in different normal distributions that
have different means and standard deviations.
• Estimate the probability with which a number occurs.
Determining what scores fall within
1, 2, and 3 SD from the Mean
Mean = 25
SD = 4
1 SD
Above
Below
= 25 + 4 = 29
= 25 – 4 = 21
2 SD
= 25 + (2 * 4) =
25 + 8 = 33
3 SD
= 25 + (3 * 4) =
25 + 12 = 37
= 25 – (2 * 4) =
25 – 8 = 17
25 – (3 * 4) =
25 – 12 = 13
To do some of these things we need to convert a
specific raw score to a z score. A z score is:
• A measure of where the raw score falls in
a normal curve.
• It allows us to determine what percentage
of scores are above or below the raw
score.
The formal for a z score is:
(Raw score – Mean)
Standard Deviation
If the raw score is larger than the mean, the z
score will be positive. If the raw score is smaller
than the mean, the z score will be negative. This
means that when we try to compare the raw
score to the distribution, a positive score will be
above the mean and a negative score will be
below the mean!
For example, (Assessment of
Client Economic Hardship)
Raw Score = 23
Mean = 25
SD = 1.5
23-25
1.5
-2
1.5
= - 1.33
Percent of Area Under Curve = 40.82%
But where does this client fall in
the distribution compared to
other clients. What percent
received higher scores?
What percent received lower
scores?
To convert z scores to percentages;
• Use the chart (Table 8.1) on p. 132 of your
textbook (Z charts can be found in any
statistics book).
• Area under curve for a z score of -1.33 is
40.82
Location of Z scores
To find out how many clients had lower or
higher economic hardship scores:
Lower Scores – Area of curve below the
mean = 50%
50% – 40.82% = 9.18%
Consequently 9.18% had lower scores.
Higher scores = 40.82% + 50% (above the
mean) = 90.82% had higher scores.
Positive Z score example:
Raw Score = 15
Mean = 12
Standard Deviation = 5
= 15 – 12 = 3 = .60 = 22.57
5
5
50 – 22.57 = 27.43% had higher scores on the
economic hardship scale
50 + 22.57 = 72.57 had lower scores on the scale
How do scores from different
distributions compare:
Distribution 1
Raw Score = 10 Mean = 5 SD = 2
= 10 – 5 = 5 = 2.5 = 49.38 = .62% above
2
2
99.38% below
Distribution 2
Raw Score= 9 Mean = 4 SD = 2.2
= 9 - 4 = 5 = 2.27 = 48.84 = 1.16% above
2.2 2.2
98.84% below