Htykuut

I took the new data as b and the data removed as a and calculated the new mean and used that to find the new mean and deviation in terms of old. But it gets too complicated and there is no way to get the relation looking at the terms.

Basically the question is, if after changing only one value of a data set, if the mean absolute deviation increases, will standard deviation always increase? Or is there any case where it can decrease too?

Here is an example using the definition of MAD implemented in R statistical
software: For the sample $X_i, dots, X_n,$
$$text{MAD} = 1.4826,text{Med}(|X_i - H|).$$
where $H$ is the median of the sample, and the constant multiple is intended
to put values on a scale so that MAD and sample standard deviation $S$ are
roughly comparable for large normal samples. So according to this definition
MAD is based on the Median of the absolute differences from the sample median.

Here is a sample of size $n = 20$ from $mathsf{Norm}(mu=100, sigma=15),$ along with its SD, R's version of the MAD, and a boxplot.

x = rnorm(20, 100, 15)

summary(x);  sd(x);  mad(x)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 

  60.01   84.13   98.49   98.67  111.50  138.14 

[1] 19.50935

[1] 20.83691



boxplot(x, horizontal=T, col="skyblue2", main="Boxplot of Original Sample")

So the two values are roughly the same. Now I sort the data, choose the 10th
order statistic, and replace it by the outlier 200.

x.sort = sort(x);  x.20 = x.sort[20];  x.20

[1] 138.1427

x.sort[20] = 200;  x.sort[20]

[1] 200

summary(x.sort);  sd(x.sort);  mad(x.sort)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 

  60.01   84.13   98.49  101.77  111.50  200.00 

[1] 28.79103

[1] 20.83691



boxplot(x.sort, horizontal=T, col="skyblue2", pch=20, 

   main="Boxplot of Modified Sample")

Notice that making this substitution has not changed the sample median (98.49 before and after) and noticeably increased the sample mean (from 98.67 to 101.77).
Also, the MAD was not increased (20.83691 before and after), but the sample SD
has increased noticeably (roughly, from 19.5 to 28.8).

One says that the sample median is a robust measure of the center of a sample and that the sample MAD is a robust measure of the dispersion of a sample.