Suppose I have the following hypothetical data:
- One thousand times value 15 (i.e., 15 occurs 1000 times)
- and a single outlier value - 115 (i.e., 115 occurs just once - an outlier)
Thus the mean is: $$((15 \times 1000) + 115)/1001 \Rightarrow (15000 + 115) / 1001 \Rightarrow 15.1$$
The standard deviation is: $3.16$.
Whereas, $Σ|x - x̄| / N$ (i.e., using the absolute deviation instead of squaring) I get: $0.20$.
I think the later value is more appropriate, but I have read in many places that one of the reasons we take the square for calculating the SD is because we want to give more weight to outliers. But I have always thought that outliers are supposed to be ignored!
What is the benefit of using the standard deviation formula (with squared deviations) which amplifies the effect of outliers (as compared to using absolute deviations)?