To find the standard deviation of the sample of 14 measurements, we can use the following formula:
Standard deviation (s) = √((∑(x - x̄)^2) / (n - 1))
Where:
- ∑ represents the sum of the values
- x represents each individual measurement
- x̄ represents the mean of the sample
- n represents the number of measurements in the sample
Given that the original sample has a mean (x̄) of 24, a standard deviation (s) of 4.5, and the sample is enlarged to 14 measurements by including two additional measurements having a common value of 24 each, we can calculate the standard deviation of the new sample.
Step 1: Calculate the sum of the original measurements
Since we know the mean and the number of measurements in the original sample, we can calculate the sum of the original measurements.
Sum of original measurements = x̄ * n
Sum of original measurements = 24 * 12
Sum of original measurements = 288
Step 2: Calculate the sum of the new measurements
The two additional measurements have a common value of 24 each, so we need to add 24 twice to the sum of the original measurements.
Sum of new measurements = Sum of original measurements + (24 * 2)
Sum of new measurements = 288 + 48
Sum of new measurements = 336
Step 3: Calculate the sum of squares of the new measurements
We need to calculate the sum of the squares of the new measurements.
Sum of squares of new measurements = (∑(x - x̄)^2) + 2 * (24 - 24)^2
Sum of squares of new measurements = (∑(x - x̄)^2) + 2 * 0
Sum of squares of new measurements = (∑(x - x̄)^2)
Step 4: Calculate the standard deviation of the new sample
Now we can plug the values into the formula for standard deviation.
Standard deviation (s) = √((∑(x - x̄)^2) / (n - 1))
Standard deviation (s) = √((Sum of squares of new measurements) / (n - 1))
Standard deviation (s) = √(∑(x - x̄)^2 / (n - 1))
Therefore, the standard deviation of the sample of 14 measurements is equal to the standard deviation of the original sample, which is 4.5. In this case, adding the two additional measurements does not affect the standard deviation.