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Bicen Maths
Statistics
Normal Distribution
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The normal distribution is for
continuous
variables
Normal Distribution:
Y
∼
N
(
μ
,
σ
2
)
Y\ \sim\ N\left(\mu,\ \sigma^2\right)
Y
∼
N
(
μ
,
σ
2
)
Where are the points of inflection on the normal distribution?
μ
±
σ
\mu\pm\sigma
μ
±
σ
What percentage of data is within 2
σ
\sigma
σ
of
μ
\mu
μ
?
95
%
How many standard deviations away from the mean contains 99.7% of the population?
3
To find the probabilities and the inverse normal, you need to use your
calculator
Standard Normal Distribution:
Z
∼
N
(
0
,
1
2
)
Z\ \sim\ N\left(0,\ 1^2\right)
Z
∼
N
(
0
,
1
2
)
What letter do we use for the Normal Distribution?
Y
What letter do we use for the Standard Normal Distribution?
Z
If you are missing
μ
,
σ
\mu,\ \sigma
μ
,
σ
or both:
Use
coding
, or
simultaneous equations
for both
What does n have to be to approximate binomial distributions as normal distributions?
large
What does p have to be to approximate binomial distributions as normal?
approximately
0.5
To approximate a binomial as normal:
μ
\mu
μ
=
n
p
np
n
p
σ
\sigma
σ
=
n
p
(
1
−
p
)
\sqrt{np\left(1-\ p\right)}
n
p
(
1
−
p
)
Continuity Correction: if approximating binomial as normal, change discrete values to continuous.
P
(
X
>
5
)
P\left(X\ >\ 5\right)
P
(
X
>
5
)
=
P
(
Y
>
5.5
)
P\left(Y\ >\ 5.5\right)
P
(
Y
>
5.5
)
P
(
3
<
X
≤
11
)
P\left(3\ <\ X\ \le11\right)
P
(
3
<
X
≤
11
)
=
P
(
3.5
<
Y
<
11.5
)
P\left(3.5\ <\ Y\ <\ 11.5\right)
P
(
3.5
<
Y
<
11.5
)