What is the correct way to calculate an average frame rate?

Question

The example is in R, but really it is a mathematics question.

I have a video. I have the times of a few successive frames (tms). They are in number of seconds after the video started.

tms <- c(0.06, 0.07, 0.26, 0.30, 0.33, 0.38, 0.47)

I want to calculate the average frame rate of the video.

I can think of two ways, and I cannot decide which one makes more sense. They do not give the same result, so surely one is inaccurate!

Method 1

mean(1/diff(tms))

32.45127

Method 2

1/mean(diff(tms))

14.63415

Which one is correct and why is the other one not?

Answer 1

Method 2 is correct. Framerate is typically expressed as FPS, that is:

Total number of frames / total number of seconds

We can see the equivalence with this:

tms <- c(0.06, 0.07, 0.26, 0.30, 0.33, 0.38, 0.47)

# total number of seconds
n_seconds <- sum(diff(tms))

# total number of frames
n_frames <- length(tms) - 1

# fps
n_frames / n_seconds
[1] 14.63415

Note that in the code above I set n_frames as length(tms)-1 because the diff function always produces a vector with one less entry than the original vector (it is a differencing tool). You could add 0 to your tms vector at the start.