# Segment 7. Central Tendency and Moments

#### 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.)

{{#widget:Iframe |url=http://www.youtube.com/v/ZWOmsKWQ7Fw&hd=1 |width=800 |height=625 |border=0 }}

The direct YouTube link is http://youtu.be/ZWOmsKWQ7Fw

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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)}**
, with a maximum at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0}**
, whose third moment **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_3}**
exists, but whose fourth moment **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)}**
is a known probability distribution. But what if you know **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_8}**
, the eighth semi-invariant?