Quantiles
Definition
The τth quantile of a random variable is defined as:
q(τ ) := inf{ x →
min
: F ( x ) ↑ τ },
0<τ<1
• If F is strictly increasing, then:
q(τ ) := the unique x such that F ( x ) = τ
In this case,
τ = F (q(τ )) =↓ q(τ ) = F ↔1 (τ )
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xy
x*
>
-
27
Visualizing the Relationship Between F ( x ) and q(τ )
Ley*x - d X
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Graphical Representations of τth quantile
F (q(τ )) := P( X ↗ q(τ )) =
)︄ q(τ )
∞
f ( x ) dx = τ
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A More Complicated Case
Example. F is flat in [ x2 , x3 ) and has two jumps at x1 and x3 .
Flat spots in F ( x ) corresponds to jumps in q(τ ) and vice versa.
Jumps (
x4
q(τ )
x3
1
τ3
x2
τ↘
τ2
F(x)
τ1
x1
τ1
x2
x3
x4
x1
τ2
τ3 1
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9121t 10 11
• q(τ ) is also the 100 ≃ τth percentile of the distribution.
,
• Special quantiles:
• The median is the 0.5th quantile of the distribution. ***
• The quartiles are the 0.25, 0.5, and 0.75 quantiles.
• The quintiles are the 0.2, 0.4, 0.6, and 0.8 quantiles.I
• The deciles are the 0.1, 0.2, . . . , 0.9 quantiles.
• Quantiles are useful summaries of the spread of the
distribution.
inequality
eg wage
.
,
• q(τ ) is used to determine critical values for hypothesis
testing.
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7 : 911-E)
:
1
.
96 for 4 05
= 0
.
Finding Critical Values Through Quantiles gorm 10 995,
.
mean a
• Example. Let X ⇐ F with symmetry & strict increasing.
• Find the value c > 0 such that:
=
1 96
.
P(↔c ↗ X ↗ c) = 1 ↔ α =↓ c = q(1 ↔ α/2)
• When F = Φ, zα/2 := q(1 ↔ α/2) = Φ↔1 (1 ↔ α/2).
cdE of N10 1)
0 4d7 of N101/
:
,
q()
↑
*
-q11-
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