1
Key concepts from week 2
Idealized/Smoothed density curves
A smoothed
density curve
is a mathematical model of a distribution.
It plots percent density distribution.
The total area under the curve is 100%.
The area under the curve for a range of values is the percentage of all
counts for that range.
Density curves come in any
imaginable shape.
Some are well-known
mathematically and others aren’t.
Normal distributions
e = 2.71828… The base of the natural logarithm
π
= pi = 3.14159…
Normal—or Gaussian—distributions are a family of symmetrical, bell-
shaped density curves defined by a mean
µ
(
mu
) and a standard
deviation
σ
(
sigma
):
N
(
µ
,
σ
).
2
2
1
2
1
)
(
⎟
⎠
⎞
⎜
⎝
⎛
−
−
=
σ
µ
π
x
e
x
f
x
x
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
A family of density curves
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Here the means are different
(
µ
= 10, 15, and 20) while the
standard deviations are the same
(
σ
= 3).
Here the means are the same (
µ
=
15) while the standard deviations
are different (
σ
= 2, 4, and 6).
mean
μ
= 64.5
standard deviation
σ
= 2.5
N
(
μ
,
σ
) =
N
(64.5, 2.5)
All Normal curves
N
(
µ
,
σ
) share the same
properties
Reminder
:
μ (mu) is the mean of the idealized curve, while
is the mean of a sample.
σ
(sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample.
About 68% of all observations or
counts are
within 1 standard
deviation (
σ
)
of the mean (
µ
).
About 95% of all observations
are
within 2
σ
of the mean
µ
.
Almost all (99.7%) observations
are
within 3
σ
of the mean.
Inflection point
x

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