A sample of 12 measurements has a mean of 30 and a standard deviation of 4.75. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 30 each.
A. Find the mean of the sample of 14 measurements.
Mean = 30
B. Find the standard deviation of the sample of 14 measurements.
Standard Deviation =
I need help calculating the standard deviation for this problem.
To calculate the standard deviation of the sample of 14 measurements, you can follow these steps:
Step 1: Find the sum of the existing measurements.
In this case, we have 12 measurements with a mean of 30. So, the sum of these measurements would be 12 * 30 = 360.
Step 2: Add the two additional measurements.
We are given that the two additional measurements have a common value of 30. Therefore, we need to add 30 + 30 = 60 to the sum.
Step 3: Find the new mean of the sample of 14 measurements.
Sum of measurements + Sum of the two additional measurements = 360 + 60 = 420
Mean = Sum / Number of measurements
Mean = 420 / 14 = 30
So, the mean of the sample of 14 measurements is still 30.
Step 4: Calculate the sum of squares of the differences from the new mean.
This step involves finding the sum of the squared differences between each measurement and the new mean. Let's denote each measurement as x_i and the new mean as μ.
Squared differences = (x_1 - μ)^2 + (x_2 - μ)^2 + ... + (x_14 - μ)^2
In this case, all the measurements have a value of 30. So, the squared differences would be (30 - 30)^2 + (30 - 30)^2 + ... + (30 - 30)^2 = 0.
Step 5: Calculate the variance.
Variance = Sum of squares of the differences from the new mean / (Number of measurements - 1)
Variance = 0 / (14 - 1) = 0 / 13 = 0
Step 6: Calculate the standard deviation.
Standard Deviation = sqrt(Variance)
Standard Deviation = sqrt(0) = 0
Therefore, the standard deviation of the sample of 14 measurements is 0.