# What happens to the mean of a set of numbers when a zero is added to the set?

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When adding a zero to the numbers, the mean will get smaller.

Let us prove it by an example.

For example:

Let "2, 3, 5, 7, 9" be numbers that their mean is as follow.

m = (2+ 3 + 5+ 7+ 9) / total number

= ( 26 / 5) = 5.2

**The,n the mean = 5.2**

Now if we add zero, the numerator will remain the same while the denominator will increase by 1:

Then the denominator will be greater, then the mean is smaller:

Let us calculate:

m = ( 2+ 3 + 5+ 7+ 9 + 0) / 6

= ( 26)/6 = 4.333

**==> m = 4.33**

The relative value of the adding zero with respect to the existing average counts

If zero is greater than the former average , then by adding a zero the average creases.

If the former average itsef is zero , and now a zero is added then the average remains unaltered.

If the former average is positive , then zero is less than the former average. So adding a zero decreases the average.

Example:

a)

The temperature of a place on two days are -5 degreeeC and -7 degree. So the average of 2 days = (-5-7)/2 = -6 degree.

The temperature of the place on the third day is zero. So the average of 3 days = (-5-7+0 )/3 = -4. So the average temperature has increased from -6 deg C. to -4 deg C.

b)

The temperature of the place on 3 days are : -2 deg C , 2 deg C and zero deg C. Then the average for first two days is (-2+2)/2 = 0. And the average for the 3 days = (-2+2+0)/3 = 0. So the average temperature remianed same.

c)

If the temperatures of the city for the 3 dates is : 1 deg C, 3 deg C and zero deg C, then the average for the first 2 days is (1+3)/2 = 2 C. The average for the 3 days = (1+3+0)/3 = (4/3) deg C .

Both the previous answers are correct. But I am explaining this in terms of a general formula.

Let:

x = mean of original set

n = number of values or readings in original set

x' = Mean of the new set with one additional value of o.

Then:

x' = nx/(n + 1)

Please note that when:

x is greater than 0, x' will be less than x.

x is less than 0, x' will be more than x.

x is equal to 0, than x' will also be equal to 0.

We can generalize the above formula for any value of the additional reading, rather than only 0. In tn this case id the value of the additional reading is c:

x' = (nx + c)/(n + 1)