Start with x = 100 and add four x values to make a sample of five data such that the standard deviation of these data equals 0
Since there is no deviation, all of the scores would be 100.
hmmmm....this is so confusing for me....
To create a sample of five data values such that the standard deviation is zero, all the values in the sample should be the same. Therefore, we need to find a value that, when added four times together with 100, results in a sample with a standard deviation of zero.
Let's solve this step by step:
1. Start with x = 100.
2. Assume the five data values are x1, x2, x3, x4, and x5.
3. The standard deviation of a sample is calculated using the formula:
standard deviation = sqrt( Σ((xi - x̄)^2) / (N-1) ),
where Σ denotes summation, xi is each data value, x̄ is the mean of the data, and N is the number of data points.
4. Since we want the standard deviation to be zero, the sum of squared differences between each data value and the mean of the data (x̄) must be zero.
Σ((xi - x̄)^2) = 0
5. Expanding the equation using the values x1, x2, x3, x4, and x5 and simplifying, we get:
(x1 - x̄)^2 + (x2 - x̄)^2 + (x3 - x̄)^2 + (x4 - x̄)^2 + (x5 - x̄)^2 = 0
6. Since the standard deviation is zero, the mean of the data (x̄) is equal to any of the data values (x1, x2, x3, x4, x5).
x̄ = x1 = x2 = x3 = x4 = x5
7. Substituting x̄ = x1 into the equation, we have:
(x1 - x1)^2 + (x2 - x1)^2 + (x3 - x1)^2 + (x4 - x1)^2 + (x5 - x1)^2 = 0
8. Simplifying the equation, we get:
(x2 - x1)^2 + (x3 - x1)^2 + (x4 - x1)^2 + (x5 - x1)^2 = 0
9. Since all the values are the same, we can simplify further by letting a = (x2 - x1).
(a)^2 + (a)^2 + (a)^2 + (a)^2 = 0
4a^2 = 0
10. To solve for 'a', we divide both sides of the equation by 4:
a^2 = 0
a = 0
11. With a = 0, we find that all the values in the sample must be the same as x1:
x1 = x2 = x3 = x4 = x5 = 100.
Therefore, to create a sample of five data values such that the standard deviation is zero, you add four values of 100 to the initial value of 100. The final sample would be [100, 100, 100, 100, 100].