46. Mean of n observations is š. If each of these observations is increased by 1, 2, 3, 4...n, respec-
tively, then what will be their mean?

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Explanation:

Consider the set

which has n items in it. We don't know anything about any individual item, but what we do know is that the mean is S.

So,

In short, the sum of the values is exactly nS, where S is the mean of the same set.

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Next, we add 1 to , add 2 onto , etc to form this new set

Let's find the mean like so

which is why S+0.5(n+1) is the answer

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Let's look at a concrete example.

Consider the set {1,7,10} which has n = 3 items in it.

Its mean is,

S = (1+7+10)/3 = 18/3 = 6

Now increase the items in {1,7,10} by adding on the values {1,2,3} to each element in the order shown

1+1 = 2

7+2 = 9

10+3 = 13

So {1,7,10} becomes {2,9,13}

The mean of the new set {2,9,13} is...

mean = (2+9+13)/3 = 24/3 = 8

Notice how,

newMean = S + 0.5(n+1)

newMean = 6 + 0.5(3+1)

newMean = 8

which helps us see how the formula works and partially confirm the answer. While this example isn't definitive proof, it's still handy to go over. I recommend forming other sets to check to see that this formula works. Not only do I recommend trying different values, but also a different number of values as well (i.e. increase n to some larger integer).

1) 2/5

2) zero=0/1

3) zero=0/2

4) -4/5

5) undefined=3/0