Question

You measure the oscillations of a pendulum four times, and find that it makes 15 complete swings in times of 8.29 s, 8.40 s, 8.30 s, and 8.41 s.

What is the mean of these measurements?

What is the mean deviation of these measurements?

What is the standard deviation of these measurements?

Your lab instructor asks you to find the period (time for one full swing) of the pendulum. You should write

Hint: It is conventional to use the standard deviation as the measure of uncertainty.

Answer 1

Hi,

You can find the mean by adding the measurements and dividing by the number of measurements in this case being 8.29 s+ 8.40 s +8.30 s +8.41 s /4 = 8.35 s is the mean.

The mean deviation of these is found by subtracting the mean from each measurement and then finding the mean of those differences.

8.29-8.35=-.06

8.40-8.35=.05

8.30-8.35=-.05

8.41-8.35=.06

Adding these differences gives you 0 so the mean deviation is 0.

The standard deviation is found then by subtracting the mean from each measurement then squaring this value, then adding these values up, dividing by the number of measurements and taking the square root.

(8.29-8.35)^2=-.06^2 = .0036

(8.40-8.35)^2=.05^2 =.0025

(8.30-8.35)^2=-.05^2 .0025

(8.41-8.35)^2=.06^2 =.0036

.0036 + .0036 + .0025 +.0025 = .0122

.0122/4 = .00305

Sqrt (.0035) = .055

To report the period you could take the average swing time and divide by the number of swings which gives you 8.35/15 =.556 secs

The uncertainty we have is for each measurement of 15 swings so we could divide this by 15 to get the uncertainty.

.055/15=.0036

You would then report .556 secs +/- .0036 secs.

I hope this helps. Let me know if you need any clarifications.