Calculation of SD of (x-a/b)

To calculate the standard deviation (SD) of (x-a/b), we need to understand the properties of the mean and standard deviation and how they are affected by arithmetic operations.

Mean (μ) and Standard Deviation (σ)
The mean, denoted by μ, represents the average value of a dataset. It is calculated by summing all the values and dividing by the total number of values.

The standard deviation, denoted by σ, measures how spread out the values in a dataset are around the mean. It quantifies the amount of variation or dispersion present in the dataset.

Properties of Mean and Standard Deviation
1. Constant Addition: Adding a constant to each value in a dataset does not change the mean but shifts the entire distribution by that constant. The standard deviation remains the same.
2. Constant Multiplication: Multiplying each value in a dataset by a constant does not change the mean but scales the distribution. The standard deviation is multiplied by the absolute value of the constant.

SD of (x-a/b)
Let's consider a dataset x with mean a and standard deviation b. We want to calculate the standard deviation of (x-a/b).

Step 1: Subtract the mean a from each value in the dataset x.
This step ensures that the new mean becomes 0, as subtracting the mean shifts the entire distribution to have a new center at 0.

Step 2: Divide each value obtained in step 1 by the constant b.
This step scales the distribution by a factor of 1/b. The original standard deviation b is multiplied by the absolute value of 1/b, which simplifies to 1/|b|.

Step 3: Calculate the new standard deviation of the transformed dataset obtained in step 2.
The new standard deviation is calculated using the formula:
σ_new = |1/b| * σ

Example:
Let's consider a dataset x with a mean of 10 and a standard deviation of 2. We want to calculate the standard deviation of (x-10/2).

Step 1: Subtract the mean 10 from each value in the dataset x.
This gives us a transformed dataset of (x-10).

Step 2: Divide each value obtained in step 1 by the constant 2.
The transformed dataset becomes (x-10)/2.

Step 3: Calculate the new standard deviation of the transformed dataset obtained in step 2.
Using the formula σ_new = |1/2| * 2, we find that the new standard deviation is 1.

Therefore, the standard deviation of (x-10/2) is 1.