11.2. Relationship between Raw and Central Moments

The first central moment:

𝜇 = 𝐸[𝑋−𝜇] = 0

The first central moment is a measure that reflects the average of the deviations of random variable X from its expected value (mean). In mathematical terms, when we calculate E[X - μ], we find that it equals zero. This is because the positive and negative deviations from the mean cancel each other out. Thus, the first central moment is always zero.

Think of the first central moment like a seesaw perfectly balanced in the middle. When a child sits on one side, there’s usually another child on the opposite side to balance them out. Just as the seesaw balances at its center (the mean), the first central moment reflects that the average distance from the center (the deviations from the mean) is zero.

The second central moment (Variance):

𝜇 = 𝜇′ − (𝜇′ )²

2 2 1

The second central moment, commonly referred to as variance, quantifies how far the values of a random variable deviate from their mean. The formula shows that the variance can be derived from the second raw moment (𝜇′) minus the square of the first raw moment (the mean squared). Variance tells us about the spread or dispersion of the distribution; higher variance means wider spread.

Imagine you are measuring the heights of students in a class. If the variance is low, it means most students are of similar heights, clustering closely around the mean height. If the variance is high, students have a wider range of heights, with some much shorter or taller than the average. Just as variance helps identify how spread out the heights are, it does the same for any set of data.

The third central moment:

𝜇 = 𝜇′ − 3𝜇′ 𝜇′ + 2(𝜇′ )³

3 3 1 2 1

The third central moment provides insight into the skewness of the distribution, indicating whether the values tend to lean to one side of the mean. This formula expresses the third central moment in terms of raw moments. A positive third central moment means that the distribution is skewed to the right (more values on the left), while a negative value indicates a left skew (more values on the right).

Consider a basketball game where most scores are between 70-90 points, but a few games might score only 20 points; this creates a left skew. Imagine a friend who regularly scores around 80 points but occasionally scores terribly (20 points). The overall stats would skew left due to those few low scores, just as the third central moment reflects how distributions can become unbalanced.

The fourth central moment:

𝜇 = 𝜇′ − 4𝜇′ 𝜇′ + 6𝜇′ (𝜇′ )² − 3(𝜇′ )⁴

4 4 3 1 2 1 1

The fourth central moment is related to the kurtosis of the distribution and provides information about the heaviness of the tails. The formula indicates that it takes into account the contributions from the first, second, and third moments as well as the fourth raw moment. High kurtosis means more data in the tails compared to a normal distribution (peaked), while low kurtosis indicates lighter tails (flat).

Picture an ice cream shop where most customers order vanilla, but on hot days, many order unusual flavors. This creates a heavy tail in your sales data on those days—a fourth central moment indicating high kurtosis. If everyone only orders vanilla all the time, your sales become flat, which correlates with low kurtosis. The fourth central moment helps gauge how unusual the customers’ choices can be.

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