You take 4 measurements of the mass of a ball: 2.96 kg, 3.32 kg, 3.31 kg, 3.17 kg. What is the uncertainty of the average mass to 95% confidence?
CL=95% N=4 CC=1.84
calculate the mean
calcuate the variance
calculate the standard deviation
all that can be done on your calculator.
then, 95percent confidence, or +-1.96 standard deviations from the mean
To calculate the uncertainty of the average mass to a 95% confidence level, you can use the following formula:
Uncertainty = CC * (Standard Deviation / sqrt(N))
Where:
- CC = Critical value for the chosen confidence level (95% confidence level corresponds to CC = 1.84)
- Standard Deviation = Standard deviation of the measurements
- N = Number of measurements
In this case, N = 4 (you have taken 4 measurements).
To calculate the standard deviation, you need to find the mean of the measurements first. The mean can be calculated by summing up all the measurements and dividing by the total number of measurements.
Mean = (2.96 kg + 3.32 kg + 3.31 kg + 3.17 kg) / 4 = 3.19 kg
Next, you calculate the variance of the measurements by taking the average of the squared differences between each measurement and the mean.
Variance = [(2.96 kg - 3.19 kg)^2 + (3.32 kg - 3.19 kg)^2 + (3.31 kg - 3.19 kg)^2 + (3.17 kg - 3.19 kg)^2] / 4 = 0.0167 kg^2
Finally, you calculate the standard deviation by taking the square root of the variance.
Standard Deviation = sqrt(0.0167 kg^2) = 0.129 kg
Now, you can substitute the values into the uncertainty formula:
Uncertainty = 1.84 * (0.129 kg / sqrt(4))
= 1.84 * (0.129 kg / 2)
= 1.84 * 0.0645 kg
= 0.1189 kg
So, the uncertainty of the average mass to 95% confidence is approximately 0.119 kg.