Mean

The mean or arithmetic mean of a set of numbers is obtained by adding the numbers and dividing the result by the 'number' of numbers. It is a key piece of information: if you randomly sample some numbers from the series, you expect them to fluctuate around the mean, and the larger the sample, the closer their average will tend to be to the mean of the series.

An alternative indicator of average is the median, the middle number if all numbers are ordered according to size. The mean and the median need not have the same value; in some cases, the difference can be quite considerable.

Example
Consider the following set of numbers:

{ 5, 4, 10, 3, 3, 4, 7, 4, 6, 5 }

First, you add the numbers to find their sum, that is:

5 + 4 + 10 + 3 + 3 + 4 + 7 + 4 + 6 + 5 = 51

Then you divide that by the number of numbers. In this case, there are 10, so the mean of this set of numbers is:

51 / 10 = 5.1

Properties
Note from the example above, that:


 * the order of the numbers in the set does not affect the mean;
 * if one of the numbers is increased, and another is decreased by an equal amount, the mean will stay the same;
 * if the set is divided into subsets of equal size, the mean of all the means of the subsets equals the mean of the set.

The above rules can often be used to calculate the mean of a given set faster. Note further that:


 * if one of the numbers is changed, the mean changes in the same direction;
 * unless all numbers are equal, there will always be at least one number larger than the mean, and at least one number smaller.

The last property implies that, for instance, it is not possible that all stocks on the stock market are outperformers.

Questions
Which of the following statements are true, and which are false?


 * 1) If 10 is added to each number in the set, but one of the numbers was negative, the mean will grow by less than 10
 * 2) If all the numbers in the set get multiplied by 2, the mean is doubled as well
 * 3) The mean of a set of even numbers will always be an even number
 * 4) The mean of a set of odd numbers will never be an odd number
 * 5) The mean of the first half of the set is generally closer to the mean of the full set than the mean of the second half
 * 6) Seeing one of the numbers provides more information on the mean of the set than seeing a second one
 * 7) Upon removing both the highest and the lowest number from the set, the mean is expected to stay the same