Understanding Mean Squared Deviation and Its Usage in Statistical Analysis

[zeeshahmad]

Could someone explain the meaning of "Mean Squared Deviation"?

Also, in <x1+x2+..xn>
what is the meaning of the pointy brackets <..> ?

 

[I like Serena]

Welcome to PF, zeeshahmad!


Hmm, didn't I see you somewhere else?

The term "mean squared deviation" is a bit ambiguous and can mean 2 things.
I'll try to explain.


Suppose you have a sample of n measurements x₁, x₂, ..., x_n.

Then the sum of squared deviations, often abbreviated as SS is:
$$SS = \sum (x_i - \bar x)^2$$
where $\bar x$ is the mean.

This set of measurements come with a "degrees of freedom", abbreviated DF.
For a "normal" repeated measurement, we have:
$$DF = n - 1$$

In statistics, when the term "mean squared deviation" is used, it usually means:
$$MS = {SS \over DF}$$
This is exactly the variance (or squared standard deviation) of the sample.


However, taken literally, "mean squared deviation" means just the average of the squared deviations, which is:
$$SS \over n$$



I can't tell you what <x1+x2+..xn> means.
Do you have a context for that?

 

[zeeshahmad]

Nice posting to you again
Actually I have got the lecture notes, in which it tells the meaning, but I don't understand it:

"Consider a distribution with average value μ and standard deviation σ from which a sample measurements are taken, i.e.

\mu = \left\langle x \right\rangle
\sigma^2 = \left\langle x^2 \right\rangle - {\left\langle x \right\rangle}^2

"where the brackets <..> mean an average with respect to the whole distribution."

 

[I like Serena]

Nice posting to you again
Actually I have got the lecture notes, in which it tells the meaning, but I don't understand it:

"Consider a distribution with average value μ and standard deviation σ from which a sample measurements are taken, i.e.

\mu = \left\langle x \right\rangle
\sigma^2 = \left\langle x^2 \right\rangle - {\left\langle x \right\rangle}^2

"where the brackets <..> mean an average with respect to the whole distribution."

Ah, I see what you mean.
<...> as you show it, is also called the "expected value".
The expected valueof a variable X is also written as EX or E(X).

If the variable x can take only specific values $x_i$ with an associated chance of $p_i$, then in general, the expectation of a function f(x) is:
$$\langle f(x) \rangle = \sum f(x_i)p_i$$
Or if x is a continuous variable, it is:
$$\langle f(x) \rangle = \int f(x)p(x)dx$$
where p(x) is the so called probability density function.So <x1+x2+...xn> would be the expected value of the sum.
This is equal to <x1>+<x2>+...+<xn>.

 

[zeeshahmad]

Thankyou for the detailed explanation