Question

A set of n values X1, X2, X3,.. Xn has standard deviation of 6. Knowing that k is a constant value that was added to the n values; XI +k, X2 +k, X3 4k.... Xn +k, determine the new standard deviation

Answer 1

Hi,

6

Explanation:

`\displaystyle \overline{x}=(1)/(n)*\sum_(i=1)^n x_i\\\overline{x'}=(1)/(n)*\sum_(i=1)^n (x_i+k)=(\sum_(i=1)^n x_i)/(n)+(nk)/(n) =\overline{x}+k\\\\var(x)=(1)/(n)*\sum_(i=1)^n (x_i-\overline{x})^2=(1)/(n)*\sum_(i=1)^n x_i^2 -\overline{x}^2\\var(x+k)=(1)/(n)*\sum_(i=1)^n (x_i+k-(\overline{x}+k))^2=(1)/(n)*\sum_(i=1)^n x_i^2 =var(x)\\`

To determine the new standard deviation after adding a constant value, k, to each value in a set, we can use the following steps:

1. Recall that the standard deviation measures the amount of variation or spread in a set of values. It is a measure of how far the values are from the mean (average) of the set.

2. Adding a constant value, k, to each value in the set does not change the spread of the values, as the differences between the values remain the same. However, it does shift the entire set by the same amount.

3. The mean of the new set (XI + k, X2 + k, X3 + k, ..., Xn + k) will be increased by the value of k, as each value has been increased by k.

4. Since the standard deviation is calculated using the differences between each value and the mean, adding a constant value to each value does not affect the standard deviation directly. The differences remain the same.

5. Therefore, the new standard deviation of the set (XI + k, X2 + k, X3 + k, ..., Xn + k) will still be 6, the same as the original set. The spread or variation of the values has not changed.

In summary, adding a constant value, k, to each value in a set does not change the standard deviation. The new standard deviation will remain the same as the original set, which is 6 in this case.