Transcript An Pragmatic Introduction to the Gaussian Curve

A Pragmatic Introduction to the
Gaussian Curve
John Behrens
Arizona State University
[email protected]
Version of 9/98
As we have seen, data occur in many shapes
including...
As we have seen, data occur in many shapes
including...
• Positively Skewed
As we have seen, data occur in many shapes
including...
• Positively Skewed
• Negatively Skewed
WRITING
As we have seen, data occur in many shapes
including...
• Positively Skewed
• Negatively Skewed
• Bell-shaped
Curves with a single mode, and symmetric
sides are often called . . .
• Bell-shaped (remember the Liberty Bell?)
Curves with a single mode, and symmetric
sides are often called . . .
• Bell-shaped (remember the Liberty Bell?)
• or Gaussian (after the mathematician who
identified the exact shape)
Curves with a single mode, and symmetric
sides are often called . . .
• Bell-shaped (remember the Liberty Bell?)
• or Gaussian (after the mathematician who
identified the exact shape)
• or “Normal” (a misnomer to get away from a
dispute about authorship!).
Karl Pearson gave the name “Normal” to
this shape:
“Many years ago I called the Laplace-Gaussian
curve the NORMAL curve, which name,
while it avoids an international question of
priority, has the disadvantage of leading
people to believe that all other distributions
of frequency are in one sense or another
"abnormal." That belief is, of course, not
justifiable.”
Karl Pearson, 1920, p 25
Karl regretted it, and we will honor him by
using the other terms.
• Normalcy is a social, not a statistical concept.
• In our culture, abnormal is valued in
intelligence, but not in “moral” behavior.
• Remember Adolph Quetelet and La Homme
Moyen.
The Gaussian shape is not a general
appearance, but a very specific shape.
With a very specific formula:
(x) =
1
2
e-(x-)
2
/ 22
What makes the shape Gaussian, is the
relative height of the curve at the different
locations
0
0.1
0.2
0.3
0.4
• Whether the curve is tall
-4
-3
-2
-1
0
1
2
3
4
What makes the shape Gaussian, is the
relative height of the curve at the different
locations
0.3
0 0.1 0.2 0.3 0.4
0.4
• Whether the curve is tall
• Or flat
-3
0
0.1
0.2
-4
-4
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
What makes the shape Gaussian, is the
relative height of the curve at the different
locations
0.3
0 0.1 0.2 0.3 0.4
0.4
• Whether the curve is tall
• Or flat
-3
-2
4
• Each of these shapes are
Gaussian, because of the
relative height at each
point of the horizontal
scale.
0
0.1
0.2
-4
-4
-3
-2
-1
0
1
2
3
-1
0
1
2
3
4
We have already talked about the peak of the
distribution, which occurs at the mean.
Each side of the curve has inflection points
where the curve makes shifts in direction.
Each side of the curve has inflection points
where the curve makes shifts in direction.
Inflection points occur at very specific places.
Mean
The first inflection point to the right of the
mean occurs one standard deviation above the
mean.
1
SD
Mean Mean
+
1 SD
The second inflection point to the right of the
mean occurs two standard deviations above
the mean.
Mean Mean Mean
+
+
1 SD 2 SD
Inflection points below the mean occur at one
and two standard deviations below the mean.
Mean Mean Mean Mean Mean
+
+
1 SD 2 SD
2 SD 1 SD
Because all points are in reference to the
mean, we will indicate the differences with
the mean implied.
-2
SD
-1 Mean
SD
+1
SD
+2
SD
One of the most helpful aspects of the normal
curve is that there are specific areas under
each part of the curve.
-2
SD
-1 Mean
SD
+1
SD
+2
SD
As we noted before, 50% of the data falls on
each side of the mean.
50%
50%
-2
SD
-1 Mean
SD
+1
SD
+2
SD
Of this 50%, 34% falls between the mean and
one standard deviation above and below the
mean.
50%
50%
34% 34%
-2
SD
-1 Mean
SD
+1
SD
+2
SD
The area between one and two standard
deviations from the mean holds 14% of the
distribution.
50%
50%
34% 34%
14%
-2
SD
-1 Mean
SD
14%
+1
SD
+2
SD
Since the total area on each side must sum to
50%, we know there is 2% of the distribution
beyond two standard deviations in each
direction.
50%
50%
34% 34%
14%
14%
2%
2%
-2
SD
-1 Mean
SD
+1
SD
+2
SD
Turn your attention to the tails for a moment.
There are two things to notice.
+2
SD
+3
SD
+4
SD
+5
SD
+6
SD
First, while most of the data is in the first few
standard deviations, the tails go on forever.
+2
SD
+3
SD
+4
SD
+5
SD
+6
SD
Second, notice that the 2% in the tails covers
all the tails including the area of all
subsequent standard deviations. When we
work with all these areas, we will look their
areas up in a table.
+2
SD
+3
SD
+4
SD
+5
SD
+6
SD