Chapter 6A: Moments, Skewness, and Kurtosis  
MATH/STAT 394: Probability I  
Arman Jahangiri  
University of Washington Department of Mathematics  
Summer 2026  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
1
 
Learning Goals  
By the end of this lecture, students should be able to:  
▶ define and interpret the mean, median, and mode;  
▶ understand why mean and variance do not fully determine a distribution;  
▶ define moments, standardized moments, and standardized moments;  
▶ define skewness and kurtosis;  
▶ understand symmetry of distributions;  
▶ define sample moments;  
▶ compute the mean and variance of the sample mean.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
2
Summaries of a Distribution  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
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Why Summaries Matter  
A probability distribution contains a large amount of information.  
Often we want simpler numerical summaries:  
▶ Where is the center?  
▶ How spread out is the distribution?  
▶ Is it symmetric?  
▶ Are extreme values common?  
Main idea  
Moments provide numerical summaries of distributions.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
4
Measures of Standardized Tendency  
Three common summaries of the center are:  
▶ Mean (expected value)  
▶ Median  
▶ Mode  
These summarize different aspects of a distribution.  
Important  
Different notions of “center” can give very different values.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
5
Median  
Definition  
A number m is a median of X if  
1
2
P(X ≤ m) ≥  
and  
1
P(X ≥ m) ≥  
.
2
Intuitively:  
▶ at least half the distribution lies to the left;  
▶ at least half lies to the right.  
Medians are less sensitive to extreme values than means.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
6
Mode  
Definition  
A value c is called a mode of a random variable X if it maximizes:  
p(c), for a discrete random variable with PMF p(x),  
f (c), for a continuous random variable with PDF f (x).  
Equivalently,  
p(c) ≥ p(x), ∀x,  
f (c) ≥ f (x), ∀x.  
Interpretation  
The mode is the most likely value (discrete case) or the point of highest density (continuous case).  
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Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
7
Mean vs Median  
Suppose salaries are:  
40, 42, 45, 46, 48, 50, 1000.  
median = 46,  
Then:  
while  
mean ≈ 181.6.  
Observation  
The mean is heavily affected by extreme values, while the median is robust.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
8
Best Prediction Depends on Loss Function  
Suppose we predict an unknown random variable X by a constant c.  
Two common error measures:  
E(X − c)2  
(mean squared error)  
and  
E|X − c|  
(mean absolute error).  
Key fact  
The mean minimizes mean squared error.  
A median minimizes mean absolute error.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026  
9
Mean and Variance Are Not Enough  
Two distributions can have:  
same mean,  
same variance,  
yet look very different.  
Possible differences:  
▶ asymmetry;  
▶ heavy tails;  
▶ sharp peaks;  
▶ multiple modes.  
Conclusion  
Higher moments help describe these additional features.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
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Moments and Symmetry  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026 11  
Kinds of Population Moments  
Definitions  
The nth population moment:  
µ′n = E(Xn).  
The nth standardized population moment:  
ꢀꢁ  
ꢂ ꢃ  
n
X − µ  
µn = E  
σ
The nth standardized population moment:  
ꢁꢁ  
ꢂ ꢂ  
n
µn  
σn  
X − µ  
µ˜n =  
= E  
,
σ
So, we have the following relations  
▶ Mean is the 1st population moment µ′1 = µ, determines center  
▶ Variance is the 2nd standardized population moment) µ2 = σ2, determines spread  
µ
▶ Skewness is the 3rd standardized population moment 33 , determines asymmetry  
σ
µ
▶ Kurtosis is the 4th standardized population moment 44 , determines tail behavior  
σ
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Chapter 6A: Moments, Skewness, and Kurtosis  
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Symmetry  
Definition  
A random variable X is symmetric about µ if  
X − µ  
has the same distribution as  
µ − X.  
Intuitively:  
▶ the left side mirrors the right side;  
▶ the distribution is balanced around µ.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
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Examples of Symmetry  
Examples of symmetric distributions:  
▶ Normal distribution;  
▶ Uniform distribution on a symmetric interval;  
1
2
▶ Binomial with p =  
.
Proposition  
If a distribution is symmetric and the mean exists, then  
mean = median.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026 14  
Symmetry Implies Odd Standardized Moments Are Zero  
Proposition  
Let X be a random variable symmetric about µ. Then, for every integer k ≥ 0,  
h
i
E (X − µ)2k+1 = 0,  
provided the moment exists.  
Proof.  
ꢄ
ꢅ
ꢄ
ꢅ
d
Let Y = X − µ. By symmetry, Y = −Y .. Therefore, E Y 2k+1 = E (−Y )2k+1 . But since 2k + 1 is odd,  
ꢄ
ꢅ
ꢄ
ꢅ
ꢄ
ꢅ
ꢄ
ꢅ
(−Y )2k+1 = −Y 2k+1. Hence, E Y 2k+1 = −E Y 2k+1 . So, 2E Y 2k+1 = 0, and therefore E Y 2k+1 = 0. Since  
ꢄ
ꢅ
Y = X − µ, E (X − µ)2k+1 = 0.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026 15  
Skewness  
Definition  
The skewness of X is  
 
!
ꢁ
ꢂ
3
X − µ  
Skew(X) = E  
.
σ
Interpretation:  
▶ positive skewness: long right tail;  
▶ negative skewness: long left tail;  
▶ zero skewness: often indicates symmetry.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
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Examples of Skewness  
▶ Exponential distributions are right-skewed.  
▶ Income distributions are often right-skewed.  
▶ Normal distributions have skewness 0.  
Important  
Zero skewness does not necessarily imply symmetry.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
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Kurtosis  
Definition  
The kurtosis of X is  
 
!
ꢁ
ꢂ
4
X − µ  
Kurt(X) = E  
.
σ
▶ Kurtosis measures tail heaviness and extremeness.  
▶ Kurt(N(µ, σ2))=3. If Kurt(X) >< 3 then fX has heavier/lighter tails compared to normal.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
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Common Discrete Distributions  
Distribution  
Mean  
Median  
Mode  
Variance  
Skewness  
Kurtosis  
0, p < 1/2  
1/2, p = 1/2  
0, p < 1/2  
0, 1, p = 1/2  
1 − 2p  
p(1 − p)  
1 − 6p(1 − p)  
p(1 − p)  
Bernoulli(p)  
p
p(1 − p)  
p
+3  
+3  
1,  
p > 1/2  
1,  
p > 1/2  
1 − 2p  
np(1 − p)  
1 − 6p(1 − p)  
np(1 − p)  
Binomial(n, p)  
np  
≈ np  
⌊(n + 1)p⌋  
np(1 − p)  
p
2
2
n + 1  
n + 1  
n
− 1  
12  
9
5
n
+ 1  
Discrete  
Uniform(1, . . . , n)  
none  
0
·
2
2
2
n
− 1  
&
'
2
1
−1  
log (1 − p)  
1 − p  
2 − p  
1 − p  
p
Geometric(p)  
1
6 +  
1
+ 3  
√
2
p
p
1 − p  
2
1
1
0.02  
λ
Poisson(λ)  
λ
≈ ⌊λ+  
−
⌋
⌊λ⌋ − 1, ⌊λ⌋  
λ
√
+ 3  
3
λ
λ
Click here to see a bigger table, which you can bring a copy on the exams too. It is on the instructor’s course website.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026 19  
Common Continuous Distributions  
Distribution  
Mean  
Median  
Mode  
Variance  
Skewness  
Kurtosis  
2
a + b  
a + b  
(b − a)  
12  
9
5
Uniform(a, b)  
none  
µ
0
0
2
2
2
2
Normal(µ, σ  
)
µ
µ
σ
3
9
1
log 2  
1
Exponential(λ)  
0
2
2
√
2
λ
λ
α
λ
α − 1  
λ
α
6
Gamma(α, λ)  
(no closed form)  
(α > 1)  
+ 3  
α
2
λ
λ
α
√
α
α − 1  
α + β − 2  
αβ 2(β − α) α + β + 1  
(α + β) (α + β + 1) (α + β + 2) αβ  
Beta(α, β)  
(no closed form)  
*
√
2
α + β  
h
Click here to see a bigger table, which you can bring a copy on the exams too. It is on the instructor’s course website.  
i
2
6
(α−β) (α+β+1)−αβ(α+β+2)  
αβ(α+β+2)(α+β+3)  
* Kurt(Beta(α, β)) =  
+ 3  
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Chapter 6A: Moments, Skewness, and Kurtosis  
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Sample Moments  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026 21  
Sample Moments  
Suppose  
X1, . . . , Xn  
are i.i.d. random variables.  
Definition  
The kth sample moment is  
n
X
1
Mk =  
Xjk .  
n
j=1  
▶ Sample moments estimate population moments.  
P
n
1
n
▶ The first sample moment, M1 =  
j=1 Xj , is an important quantity in Statistics, also known as the Sample  
′
¯
Mean, denoted by X. It estimates the first population moment, i.e., µ1 = E[X], the population mean.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
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i.i.d. Random Variables (Random Samples)  
Definition: i.i.d. Random Variables  
We say that the random variables X1, . . . , Xn are independent and identically distributed (i.i.d.) with distribution F,  
and write  
i.i.d.  
X1, . . . , Xn ∼ F,  
if:  
▶ X1, . . . , Xn are independent;  
▶ each Xj has distribution F.  
i.i.d.  
Terminology: If X1, . . . , Xn ∼ F, then we also call a random sample of size n from the population distribution F.  
Quantities associated with F are called population quantities, while quantities computed from the random sample are called  
sample quantities.  
Example  
If  
i.i.d.  
X1, . . . , Xn ∼ N(µ, σ2),  
then for every j = 1, . . . , n,  
E(Xj ) = µ,  
Var(Xj ) = σ2.  
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Chapter 6A: Moments, Skewness, and Kurtosis  
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Example of a Random Sample  
Suppose the population consists of the GPAs of all University of Washington undergraduate students.  
Let F denote the population distribution of undergraduate GPAs.  
Now suppose we randomly select 12 undergraduate students and record their GPAs:  
i.i.d.  
X1, . . . , X12 ∼ F.  
Here:  
▶ each Xj represents one student’s GPA;  
▶ all Xj ’s have distribution F;  
▶ the observations are assumed to be independent.  
A possible sample is:  
3.5, 3.7, 4.0, 2.9, 3.1, 3.8, 3.4, 2.7, 3.9, 3.2, 3.6, 3.3.  
Important  
In practice, we usually do not know the population distribution F or its parameters, such as the population mean µ  
and variance σ2.  
Instead, we observe a random sample  
X1, . . . , Xn  
from the population and use the random sample to estimate unknown population quantities.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
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Sample Mean and Sample Variance  
Let  
i.i.d.  
X1, . . . , Xn ∼ F, E(Xj ) = µ,  
Var(Xj ) = σ2.  
Definitions  
The sample mean and sample variance are  
n
n
X
X
1
¯
1
2
¯
Xn  
=
Xj ,  
Sn2  
=
(Xj − Xn) .  
n
n − 1  
j=1  
j=1  
Mean and Variance  
σ2  
.
¯
¯
E(Xn) = µ,  
Var(Xn) =  
n
ꢀ
ꢁ
1
n − 3  
E(Sn2) = σ2,  
Var(Sn2) =  
µ4 −  
σ4  
,
n
n − 1  
where  
ꢂ
ꢃ
µ4 = E (Xj − µ)4  
.
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026 25  
Summary  
▶ Mean, median, and mode describe center differently.  
▶ Higher moments describe asymmetry and tail behavior.  
▶ Skewness measures asymmetry.  
▶ Kurtosis measures tail heaviness.  
▶ Sample moments estimate population moments.  
▶ The variance of the sample mean decreases like  
1
.
n
Next lecture  
Moment generating functions.  
Arman Jahangiri  
Chapter 6A: Moments, Skewness, and Kurtosis  
Summer 2026 26