Question

Question 5 --/1 View Policies Current Attempt in Progress Consider the MAD statistic. What would happen if you did not take absolute values before calculating the sum of differences between group means? You will always get negative differences. You could always arrange the differences so that you could get a MAD statistic of zero. You won't be able to determine the mean of the differences. How you arranged the order of subtraction could change the value of the MAD statistic. You will always get differences of zero. You will always get positive differences.

Answer 1

You will always get difference zero.

Explanation

Sum of deviation from arithmetic mean is always zero.

So if we calculate sum of differences between group means first with out taking absolute value, we will get zero.

Mathematically,

If we have data $x_i,x_2,...x_n$ the MA D .statistic is given by

$MAD=\frac{1}{n}\sum_{i=1}^n|x_i-\bar x|$ where $\bar x|=\frac{1}{n}\sum_{i=1}^nx_i$ .

If we take sum of differences between group means first we will get

$\sum_{i=1}^n(x_i-\bar x)=\sum_i x_i-n\bar x=n \bar x-n \bar x =0$

Hence

$\frac{1}{n}|\sum_{i=1}^n(x_i-\bar x)|=\frac{1}{n}0=0$

On the other hand $MAD=\frac{1}{n}\sum_{i=1}^n|x_i-\bar x|$ will be sum of non negative quantities so that it will be non negative always if all the observations are nt same MAD will be always strictly positive.