Statistcs help

Question

I don’t know I don’t know if the statistics is done on this site however here is my problem and I don’t not have answered this. My booklet says

Listed below are the errors between the predicted temperature and the actual temperature of a certain city. Find the mean and median for each of the two samples. Do the means and medians indicate that the temperature predicted one day in advance are more accurate than those predicted five days in advance, as we might expect?
(actual high)-(predicted high one day earlier) 0,0,0,3,2,2,0,0,-2,0
(actual high)-(predicted high 5 days earlier) 1,1,1,-6,-6,1,1,3,-5,6
the main difference between actual high and the predicted high one day earlier is?

Answer 1

Hello,

This is a pretty tricky question. If we find the mean and median for these errors, we end up with the following:

One day:
Mean = 0.5
Median = 0

Five days
Mean = -0.3
Median = 1.0

Since these are errors, if we look at the mean, we would conclude that the five-day prediction has less error. But if we look at the median, we would conclude that the one-day predictions were less error-prone. However, this is not how we typically evaluate errors. To see why, let's say we have temperature data for two days. On day one, the error was 1 degree; on day two, it was -1 degree. If we average those together, we would get 0 because one is positive and the other is negative. That means that, despite that fact that each day was off by 1, we're saying there was 0 error. To get around this, we typically square the error terms first. So in my example, we would have 1^2 and -1^2. This would give us an average error of 1 which makes more sense.

Let's apply that to your example. First, we square each of those numbers. Then we 1) take the average of the squared numbers and 2) take the median of the squared numbers (this is somewhat unusual to do for a median, but the question calls for it).

For one day predictions
Mean = 2.1
Median = 0
One last step is to take the square root (since I squared the errors earlier, I have to take the square root so that the units are correct). This gives us:
Mean = 1.449
Median = 0

For five day predictions
Mean = 14.7
Median = 5
Again, we have to take the square root. This gives us:
Mean = 3.384
Median = 2.236

When we do it this way, both the mean and the median say that one-day predictions had less error than five-day predictions.