Download Normal Distribution Curve

Normal Distributions
OBJECTIVE
Find the area and
probability of a
standard normal
distribution.
RELEVANCE
Can find probabilities
and values of populations
whose data can be
represented with a
normal distribution.
Definition……


Normal Distribution Curve – a symmetric
distribution where the data values are evenly
distributed about the mean.
2 other names for the normal distribution:
a. Guassian
b. Bell Curve
Skewness……

Three Possible Distributions where the tail
indicates skewness:
1. Normal
2. Left (Negative)
3. Right (Positive)
Normal……
Normal – mean, median, and mode are all
the same.
Mean, median, mode
Left Skew (Negative)……
Negative Skew – from left to right: mean,
median, mode.
mean
median
mode
Right Skew (Positive)……
mode
median
mean
Positive Skew- from right to left: mean, median, mode
Standard Normal
Distributions…..
Properties of a Standard Normal
Distribution……
1.
2.
3.
4.
5.
Total area under the curve = 1 (or 100%).
Mounded and symmetric; Never touches the
x-axis.
Mean = 0; St. Deviation = 1
The mean divides the area in half → 0.50 on
each side.
 68%
Percentages under the curve:  12stst..dev
dev  95%
 3 st . dev  99.7%
Does this look familiar?
68%
95%
99.7%
THE EMPIRICAL RULE!
Z-score


All values can be
transformed from a
normal distribution to
a standard normal
distribution by using
the z-score.
It represents how many
standard deviations “x”
is away from the mean.
z
x

Finding area under the curve……
1.
2.
3.
Draw the distribution curve.
Shade the area in which you are interested.
Use the table to find the areas. You might
have to add or subtract or both, depending on
your interest.
****The table we are using gives areas from
z = 0 to the other z-score.
Example……Find the area under the
curve between z = 0 and z = 1.52.

Look up z = 1.52

The area between z = 0
and z = 1.52 is .4357.
You try…Find the area between z = 0
and z = 2.34.

Answer:
.4904
Find the area between z = 0 and
z = -1.75.

Answer:
.4599
Find the area to the right of z = 1.11

Answer:
Half is 0.5.
Look up z = 1.11 to get
an area of 0.3665.
Final Answer:
0.5 – 0.3665 = 0.1335
.3665
x
0 .5
Now you try……

Find the area to the left of z = -1.93.

Answer: 0.5 – 0.4732 = 0.0268
Find the area between z = 2.00 and
z = 2.47.



Look up z = 2.47 to get
0.4932.
Look up z = 2 to get
0.4772.
Subtract the two areas:
0.4932 – 0.4772 =
0.0160
You try……

Find the area between z = -2.48 and z = -0.83.

Answer: 0.4934 – 0.2967 = 0.1967
Find the area between z = -1.37 and
z = 1.68.



The area for z = 1.68 is
0.4535.
The area for z = =1.37
is 0.4147.
Add the two areas
together for the final
answer:
0.4535 + 0.4147 = 0.8682
Find the area to the left of z = 1.99.



The area for z = 1.99 is
0.4767.
The area for the entire
left half of the
distribution is 0.5000.
Add the two areas
together for the final
answer: 0.4767 +
0.5000 = 0.9767
Find the area to the right of z = -1.16.



The area for z = -1.16
is 0.3770.
The area for the entire
right half of the
distribution is 0.5000.
Add the two areas
together to get your
final answer: 0.3770 +
0.5000 = 0.8770
Find the area to the right of z = 2.43
and to the left of z = -3.01.



The area for z = 2.43 is 0.4925.
The area for z = -3.01 is 0.4987.
Add the two areas together and then subtract
from 1:
0.4925 + 0.4987 = 0.9912
1 – 0.9912 = 0.0088
Probability and the Normal
Curve
Finding Probability Under the
Curve……

Use the same procedure as finding the area
under the curve; however, there is a different
notation.
Notations……

Area Notation:
a. Between z=0
and z=2.32
b. To the left of
z=1.65
c. To the right of
z=1.91
d. Between z = -1.2
and z=2.3

Probability Notation:
a. P(0  z  2.32)
b.
P( z  1.65)
c.
P( z  1.91)
d.
P(1.2  z  2.3)
Find
P(0  z  2.32)

The area for z = 2.32 is
0.4898.

The probability is
0.4898.
Find P( z  1.65)



The probability for
z = 1.65 is 0.4505.
The probability for the
entire left half of the
distribution is 0.5000.
The final probability is
0.4505 + 0.5000 =
0.9505.
Find P( z  1.91)



The probability for z =
1.91 is 0.4719.
The probability for the
entire right half of the
distribution is 0.5000.
Subtract the two
answers to get your
final answer: 0.5000 –
0.4719 = 0.0281.
Find P(1.2  z  2.3)



The probability for z =
-1.2 is 0.3849.
The probability for z =
2.3 is 0.4893.
Add the two
probabilities together
to get your final
answer: 0.3849 +
0.4893 = 0.8742.