# Segment 7. Central Tendency and Moments

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## Contents

#### Watch this segment

(Don't worry, what you see statically below is not the beginning of the segment. Press the play button to start at the beginning.)

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Links to the slides: PDF file or PowerPoint file

### Problems

#### To Calculate

1. Prove the result of slide 3 the "mechanical way" by setting the derivative of something equal to zero, and solving.

2. Give an example of a function $\displaystyle p(x)$ , with a maximum at $\displaystyle x=0$ , whose third moment $\displaystyle M_3$ exists, but whose fourth moment $\displaystyle M_4$ doesn't exist.

3. List some good and bad things about using the median instead of the mean for summarizing a distribution's central value.

#### To Think About

1. This segment assumed that $\displaystyle p(x)$ is a known probability distribution. But what if you know $\displaystyle p(x)$ only experimentally. That is, you can draw random values of x from the distribution. How would you estimate its moments?

2. High moments (e.g., 4 or higher) are algebraically pretty, but they are rarely useful because they are very hard to measure accurately in experimental data. Why is this true?

3. Even knowing that it is useless, how would you find the formula for $\displaystyle I_8$ , the eighth semi-invariant?