Definitions

Moment	Raw(R) Moment (Moment about the origin/zero)	Central(C)/Mean(M) Moment (Moment about the mean 𝜇)
𝑀𝑘	𝑘^𝑡ℎ Raw Moment 𝜇_𝑘’ = 𝐄[𝑋^𝑘]	𝑘^𝑡ℎ Central Moment\| 𝜇_𝑘 = 𝐄[(𝑋 - 𝜇)^𝑘]
𝑀0	Zeroth Raw Moment 𝜇₀’ = 𝐄[𝑋⁰] = 1	Zeroth Central Moment 𝜇₀ = 𝐄[(𝑋 - 𝜇)⁰] = 1
𝑀1	First Raw Moment (Mean) 𝜇₁’ = 𝜇 = 𝐄[𝑋¹] = 𝐄[𝑋]	First Central Moment 𝜇₁ = 𝐄[(𝑋 - 𝜇)¹] = 0
𝑀2	Second Raw Moment 𝜇₂’ = 𝐄[𝑋²]	Second Central Moment (Variance) = 𝐄[𝑋²] - 𝜇² 𝜇₂ = 𝐄[(𝑋 - 𝜇)²] 𝜇₂= 𝐄[𝑋² - 2𝜇𝑋 + 𝜇²] 𝜇₂ = 𝐄[𝑋²] - 𝐄[2𝜇𝑋] + 𝐄[𝜇²] 𝜇₂ = 𝐄[𝑋²] - 2𝜇𝐄[𝑋] + 𝜇² 𝜇₂ = 𝐄[𝑋²] - 2𝜇𝜇 + 𝜇² 𝜇₂ = 𝐄[𝑋²] - 𝜇²
𝑀3	Third Raw Moment 𝜇₃’ = 𝐄[𝑋³]	Third Central Moment (Skewness) 𝜇₃ = 𝐄[(𝑋 - 𝜇)³]
𝑀4	Fourth Raw Moment 𝜇₄’ = 𝐄[𝑋⁴]	Fourth Central Moment (Kurtosis) 𝜇₄ = 𝐄[(𝑋 - 𝜇)⁴]

Significance of moments (raw, central, normalized) and cumulants (raw, normalized), in connection with named properties of distributions

Moment ordinal	Moment			Cumulant
Moment ordinal	Raw	Central	Standardized	Raw	Normalized
1	Mean	0	0	Mean	—
2	–	Variance	1	Variance	1
3	–	–	Skewness	–	Skewness
4	–	–	(Non-excess or historical) kurtosis	–	Excess kurtosis
5	–	–	Hyperskewness	–	–
6	–	–	Hypertailedness	–	–
7+	–	–	–	–	–