If each of the observation x1,x2,...,xn is increased by ' a ', where a is a negative or positive number, show that the variance remains unchanged.
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Step-by-Step Solution
Step 1: Define original mean and variance
Let's start by defining the mean and variance for the original set of observations x1,x2,...,xn. The mean, denoted by xˉ, is the sum of all observations divided by the number of observations n. The variance, denoted by σ2, is the average of the squared differences from the mean.
Step 2: Define new observations and their mean
Now, let's consider the new set of observations, y1,y2,...,yn, where each observation xi is increased by 'a'. So, yi=xi+a. We then calculate the mean of these new observations, yˉ.
Step 3: Simplify the new mean
We can simplify the expression for the new mean yˉ. By distributing the summation and recognizing that ∑i=1na=na, we find that the new mean is simply the original mean plus 'a'.
Step 4: Calculate the new variance
Now, let's calculate the variance for the new set of observations, σy2. This is defined as the average of the squared differences between each new observation yi and the new mean yˉ.
Step 5: Substitute and simplify the new variance
Substitute yi=xi+a and yˉ=xˉ+a into the variance formula. Notice that the 'a' terms cancel out inside the parenthesis. This leaves us with the exact same expression as the original variance.
Step 6: Conclusion
Since the formula for σy2 is identical to the formula for σ2, we have shown that the variance remains unchanged when each observation is increased by a constant 'a'.