# Statistic deviation.

1. Jun 24, 2009

### icystrike

I went for a lecture and the lecturer said that the square of the difference between the x sub i and the mean is the take precaution of the negative value. This has been bugging me , i was wondering why dont they just take absolute because there is a difference between :
$$\sqrt{\frac{\sum(x-\mu)^2}{f}}$$ and$$\frac{\sum \left|(x-\mu)\right|}{f}$$

2. Jun 24, 2009

Yes, there is a difference, as

$$\sqrt{\sum(x-\mu)^2} \ne \sum |x - \mu |$$

There is actually quite a history about whether a measure based on

$$\sqrt{\frac{\sum (x-\mu)^2 }{f}}$$

or

$$\sqrt{\frac{\sum |x-\mu|}{f}}$$

should be used. Basically, the measure based on the sum of squared deviations won out because, statistically, when it is assumed that the data are drawn from a normal distribution (equivalently, when it is assumed the random noise is Gaussian).

3. Jun 25, 2009

### icystrike

Random noise, i got to check this out !

4. Jun 25, 2009

### boboYO

heh, i remember my stats lecturer said that too.
an analogy can be drawn with why we take the squares of the sides (pythagoras) to work out the hypotenuse and not the absolute value.

5. Jun 25, 2009

### daviddoria

The squared distance is also used because it is continuous, where the absolute distance function has a discontinuity. This is a big problem in optimization.

6. Jun 25, 2009

Not really the case in statistics - the median, median deviation, and other procedures use the absolute value.

7. Jun 26, 2009

### HallsofIvy

The absolute distance function does not have a derivative at a point. There is no discontinuity.

8. Jun 26, 2009

### HallsofIvy

When you sum the absolute values, you should not have a square root.

There is a third used occasionally:
$$\frac{max |x-\mu|}{f}$$

The end of your last sentence seems to be missing!

9. Jun 26, 2009