Transcript The Normal Distribution

The Normal Distribution
Unimodal
Symmetrical
N(  ,  
Abscissa – the different values of x
Ordinate – density, probability, or frequency of each value of x
1
f ( x) 
(e)
 2
( x )2
2
2
  3.1416
e  2.7183


and
Thus, we can derive the probability of any x[f(x)] if we know
By integrating the function ranging from x1 to x2, derive the probability of
selecting an x between these two points.
Using the Z Table
Z-score = standardized score
Mean to score (A) – (either to the left of mean or to the right), area or probability
Larger portion (B) – area to the furthest tail from the score
Smaller portion (C) – area to the closest tail from the score
Larger + Smaller portion = 1 or 100% of the area
[A ]
--- B ---- ---- ] [ ---C ----
Standard Normal Distribution
To solve the problem of there being an infinite number of normal distributions (differing in terms
of means and standard deviations) we can transform the scores.
 0
 2 1
 1
With the rules of linear transformation, we change any normal distribution into a standard normal
distribution. Then we only need one table to determine the probability of obtaining a given score,
a greater score, or a smaller score.
Transform
N(   2 )
into
N(0,1)
We know:
If ynew equals yold plus or minus a constant, then the new mean equals the old mean plus or
minus the constant.
If ynew equals yold divided by a constant, then the new standard deviation is equal to the
old standard deviation divided by the constant.
Thus to create a new mean of zero…..
y
old
 75
yold  15
S
2
old
 62.5
Sold  7.9
y
new
0
ynew  0
S
2
new
 62.5
Snew  7.9
Yold
Ynew
5
-15
-10
10
-15
-5
15
-15
0
20
-15
5
25
-15
10
To create a variance and a standard deviation of one…
2
S new
2
62.5
S old

1
 2
62.5
S old
S new
S old
7.9

1

7.9
S old
5
10
15
20
25
-10
-5
0
5
10
-0.6
0
0.6
1.3
-1.3
Now we can use the z-table to to find the probability of…
1.
2.
3.
A score less than Xold.
A score greater than Xold
Or any interval between two Xold points
A. Xold ---- to ---- Xnew or z-score ----- to ---- z-table.
B. z-table ----- to ---- z-score ------ to ----- Xold.
Examples Using the z-table
y: 5,10,15,20,25
 y  75
y  15
S?
1.
What is the chance of a score being less than or equal to 5?
2.
What is the chance of a score being greater than or equal to 5?
3.
What is the chance of a score falling between 10 and 20?
4.
What is the chance of a score falling between –10 and 18?
5.
Within what two raw scores (be symmetrical) will 95% of the scores fall?
Setting Probable Limits: 95%
Remember:
Z
(y   )

Go to the z-table to find the z-scores that cover the range of 95%.
Mean to z will be 0.475. Why?
z  196
.
(y   )
 196
. 
s
( y  15)
 196
. 
7.9
*We only have estimates.
y  15  196
. (7.9)
y  15  196
. (7.9)
= 30.484 and
-0.484