A Mean Calculation

Subtracting 100 from each of the numbers

The numbers are all around 100, so the mean will be close to 100.

If the numbers were all 100 smaller, then their mean would be 100 smaller. So finding the mean of the same set of numbers, but all 100 smaller, and then adding 100, will give the mean.

Subtracting 100 from each of the numbers gives

-3   -3   -2   -1   -1   -1   0   0   1   1   3   4   4   5

These numbers add up to 7, and there are 14 of them, so their mean is 7$\div$14 = 0.5.

So the mean of the original numbers is 100.5.

Converting groups/pairs of numbers into 100s

Notice that 97 + 103 = 200 = 100 + 100, so finding the mean of 97, 103 and some other numbers will be the same as finding the mean of 100, 100 and the other numbers.

So finding the mean of 

97   97   98   99   99   99   100   100   101   101   103   104   104   105

is  the same as finding the mean of

97   97   98   99   99   99   100   100   101   101   103   104   104   105   and

100   100

Pairing off the 99s with the 101s, it is the same as finding the mean of 

97   97   98   99   99   99   100   100   101   101   103   104   104   105   and

100   100   100   100  100  100

97 and 98 are 3 and 2 less than 100, so will balance with 105 to leave three 100s. So we can find the mean of

97   97   98   99   99   99   100   100   101   101   103   104   104   105   and

100   100   100   100   100   100   100  100  100

99 and 104 do not balance, but can be replaced with 100 and 103, since 99 + 104 = 100 + 103.

So we can find the mean of 

97   97   98   99   99   99   100   100   101   101   103   104   104   105   and

100   100   100   100   100   100   100   100   100  100  103

which is 1407$\div$14.

Notice that 1400$\div$14 = 100, and 7$\div$14 = 0.5,

so 1407$\div$14 = 100.5