Central moments

The preceding pages introduced variance and standard deviation. Variance summarizes squared deviations from the mean. A psycholinguistic response distribution may also be asymmetric, with occasional values much farther above the mean than below it. Changing the exponent on the centered deviation gives a family of summaries sensitive to different aspects of that shape.

The kth central moment of X is

\mu_k \equiv\mathbb{E}\!\left[(X-\mathbb{E}[X])^k\right],

when this expectation exists. The word central indicates that each value is centered by subtracting the mean before the power is applied.

Working through a constructed response distribution

Suppose a response variable J has this PMF.

response j	p_J(j)
1	.25
2	.50
5	.25

Its mean is

\mathbb{E}[J] =1(.25)+2(.50)+5(.25) =2.50.

The centered values are -1.5, -.5, and 2.5.

The first central moment

For k=1,

\begin{aligned} \mu_1 &=(-1.5)(.25)+(-.5)(.50)+(2.5)(.25)\\ &=0. \end{aligned}

The first central moment is zero because signed deviations from the mean cancel in expectation.

The second central moment

For k=2,

\begin{aligned} \mu_2 &=(-1.5)^2(.25)+(-.5)^2(.50)+(2.5)^2(.25)\\ &=2.25. \end{aligned}

The second central moment is the variance. Its square root is the standard deviation, \sqrt{2.25}=1.5.

The third central moment

For k=3, the sign of each centered value remains:

\begin{aligned} \mu_3 &=(-1.5)^3(.25)+(-.5)^3(.50)+(2.5)^3(.25)\\ &=3. \end{aligned}

The positive result reflects the larger positive departure at 5. A standardized third central moment divides by \sigma_J^3 so that the units cancel. This unitless summary is often called skewness.

Higher orders

An even power makes departures on both sides positive, while an odd power preserves direction. Increasing the power gives progressively more weight to values far from the mean.

The fourth central moment is

\mu_4 =\mathbb{E}\!\left[(X-\mathbb{E}[X])^4\right].

Dividing \mu_4 by the fourth power of the standard deviation gives a unitless summary often called kurtosis. This quantity is sensitive to probability placed far from the mean, but one value does not reveal where that probability lies.

Computing several moments in base R

Code

response <- c(1, 2, 5)
response_mass <- c(.25, .50, .25)
response_mean <- sum(response * response_mass)

central_moment <- function(order) {
  sum((response - response_mean)^order * response_mass)
}

sapply(1:4, central_moment)

The first four central moments are 0, 2.25, 3, and 11.0625.

A moment does not determine a distribution

One central moment does not provide a complete account of distribution shape. A positive third central moment indicates an imbalance in cubed deviations, but it does not identify the observation process that produced the imbalance. Different distributions can share the same first several moments.

Moments provide compact summaries. Plots and later model checks are needed to locate which values or regions create a difference.

Check your understanding

Why is the first central moment zero whenever the mean exists?
Which previously introduced summary equals the second central moment?
Why do odd central moments retain information about direction while even moments do not?
Explain why two distributions with the same first three central moments need not have the same shape.

This completes the shared representation and summary sequence. The next page begins the distribution family pages using only objects already introduced here.

--- title: "Central moments" --- The preceding pages introduced [variance](variance-and-covariance.qmd) and [standard deviation](standard-deviation.qmd). Variance summarizes squared deviations from the mean. A psycholinguistic response distribution may also be asymmetric, with occasional values much farther above the mean than below it. Changing the exponent on the centered deviation gives a family of summaries sensitive to different aspects of that shape. The [**$k$th central moment**](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/04%3A_Expected_Value/4.01%3A_Definitions_and_Basic_Properties) of $X$ is $$ \mu_k \equiv\mathbb{E}\!\left[(X-\mathbb{E}[X])^k\right], $$ when this expectation exists. The word *central* indicates that each value is centered by subtracting the mean before the power is applied. ## Working through a constructed response distribution Suppose a response variable $J$ has this PMF. | response $j$ | $p_J(j)$ | |---:|---:| | 1 | $.25$ | | 2 | $.50$ | | 5 | $.25$ | Its mean is $$ \mathbb{E}[J] =1(.25)+2(.50)+5(.25) =2.50. $$ The centered values are $-1.5$, $-.5$, and $2.5$. ## The first central moment For $k=1$, $$ \begin{aligned} \mu_1 &=(-1.5)(.25)+(-.5)(.50)+(2.5)(.25)\\ &=0. \end{aligned} $$ The first central moment is zero because signed deviations from the mean cancel in expectation. ## The second central moment For $k=2$, $$ \begin{aligned} \mu_2 &=(-1.5)^2(.25)+(-.5)^2(.50)+(2.5)^2(.25)\\ &=2.25. \end{aligned} $$ The second central moment is the variance. Its square root is the standard deviation, $\sqrt{2.25}=1.5$. ## The third central moment For $k=3$, the sign of each centered value remains: $$ \begin{aligned} \mu_3 &=(-1.5)^3(.25)+(-.5)^3(.50)+(2.5)^3(.25)\\ &=3. \end{aligned} $$ The positive result reflects the larger positive departure at 5. A standardized third central moment divides by $\sigma_J^3$ so that the units cancel. This unitless summary is often called skewness. ## Higher orders An even power makes departures on both sides positive, while an odd power preserves direction. Increasing the power gives progressively more weight to values far from the mean. The fourth central moment is $$ \mu_4 =\mathbb{E}\!\left[(X-\mathbb{E}[X])^4\right]. $$ Dividing $\mu_4$ by the fourth power of the standard deviation gives a unitless summary often called kurtosis. This quantity is sensitive to probability placed far from the mean, but one value does not reveal where that probability lies. ## Computing several moments in base R ```{r} #| eval: false response <- c(1, 2, 5) response_mass <- c(.25, .50, .25) response_mean <- sum(response * response_mass) central_moment <- function(order) { sum((response - response_mean)^order * response_mass) } sapply(1:4, central_moment) ``` The first four central moments are 0, 2.25, 3, and 11.0625. ## A moment does not determine a distribution One central moment does not provide a complete account of distribution shape. A positive third central moment indicates an imbalance in cubed deviations, but it does not identify the observation process that produced the imbalance. Different distributions can share the same first several moments. Moments provide compact summaries. Plots and later model checks are needed to locate which values or regions create a difference. ## Check your understanding 1. Why is the first central moment zero whenever the mean exists? 2. Which previously introduced summary equals the second central moment? 3. Why do odd central moments retain information about direction while even moments do not? 4. Explain why two distributions with the same first three central moments need not have the same shape. This completes the shared representation and summary sequence. The [next page](bernoulli-and-categorical-distributions.qmd) begins the distribution family pages using only objects already introduced here.