SD of x- a / b:
To find the standard deviation (SD) of the expression (x - a) / b, we need to understand the properties of SD and how it changes when we perform operations on a random variable.

Mean and SD of x:
Given that the mean of x is denoted by a and the standard deviation of x by b, it means that the average value of x is a and the spread of x is b.

Defining the expression x - a / b:
The expression (x - a) / b represents the standardized value of x. By subtracting the mean a from each value of x and then dividing it by the standard deviation b, we are scaling the values of x to a new distribution with a mean of 0 and a standard deviation of 1.

Effect of subtracting the mean:
When we subtract the mean a from each value of x, the new distribution will have a mean of 0. This is because the mean represents the average value, and subtracting it from each value will center the distribution around zero.

Effect of dividing by the standard deviation:
Dividing each value of x - a by the standard deviation b scales the distribution so that it has a standard deviation of 1. This is because the standard deviation measures the spread of the data, and dividing by it normalizes the spread.

Calculating the SD:
To find the standard deviation of the expression (x - a) / b, we need to consider how the standard deviation changes when we perform operations on a random variable. When we multiply or divide a random variable by a constant, the standard deviation is also multiplied or divided by that constant.

In this case, we are dividing (x - a) by b. Since b is a constant, the standard deviation of (x - a) / b is the standard deviation of (x - a) divided by b.

Formula for SD:
SD = √[Σ(x - μ)² / N]

Applying the formula:
Using the formula, the standard deviation of (x - a) / b can be calculated as follows:

SD[(x - a) / b] = √[Σ((x - a) / b - μ)² / N]

Since the mean of (x - a) / b is 0, the formula simplifies to:

SD[(x - a) / b] = √[Σ((x - a) / b)² / N]

This can be further simplified as:

SD[(x - a) / b] = √[Σ(x - a)² / (b² * N)]

Therefore, the standard deviation of (x - a) / b is the standard deviation of (x - a) divided by the constant b.

Conclusion:
The standard deviation of the expression (x - a) / b is obtained by dividing the standard deviation of (x - a) by the constant b. It represents the spread of the data after centering it around zero and scaling it by the standard deviation of x.