Here is my data
X 90 86 67 89 81 75
Y 62 45 40 55 64 53
I have to find the Least Absolute Deviation of the data, however I am absolutely stomped. I picked (75,53) as my point after looking at the scatter plot on excel. From here on though I am completely confused as to how to proceed. If I could just get alittle push on where to go I'd be happy with that thanks in advance!
This is the first time I work with least absolute deviations regression; so, please, forgive me if I am off topic in this answer.
If I refer to Wikipedia - Least Absolute Deviations and assume that we look for a linear model $$y=a+ bx$$ we need to minimize, with respect to $a$ and $b$ $$F=\sum_{i=1}^6 |a+b x_i-y_i|$$ Quoting the Wikipedia page "unlike least squares regression, least absolute deviations regression does not have an analytical solving method; therefore, an iterative approach is required".
This is what I did using a minimizer (you could use Excel solver for it) using, as (poor) initial estimates the values obtained using a standard linear regression, that is to say minimizing $$G=\sum_{i=1}^6 (a+b x_i-y_i)^2$$ which would lead to $a=4.799$ and $b=0.595$.
Doing it, I ended with $a= -24.0869$ and $b=0.956521$.
In the table below, I reproduce the original values as well as those given by linear least squares $(LSQ)$ and least absolute deviations $(LAD)$. $$\left( \begin{array}{cccc} x & y & LSQ & LAD \\ 90 & 62 & 58.3206 & 62.0000 \\ 86 & 45 & 55.9419 & 58.1739 \\ 67 & 40 & 44.6429 & 40.0000 \\ 89 & 55 & 57.7259 & 61.0435 \\ 81 & 64 & 52.9684 & 53.3913 \\ 75 & 53 & 49.4003 & 47.6522 \end{array} \right)$$