Question

Explain, by taking a suitable example, how the arithmetic mean alters by (i) adding a constant k to each term, (ii) subtracting a constant k from each them, (iii) multiplying each term by a constant k and (iv) dividing each term by a non-zero constant k.

Answer 1

Let us take n observations.

Ifbe the mean of the n observations, then we have

(i) Add a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 natural numbers are 1, 2, 3, 4, and 5. Their mean is

Add 2 to each of the numbers. Then the numbers becomes 3, 4, 5, 6, and 7. The new mean is

Therefore adding a constant number to each observation the mean increased by that constant

(ii) Subtract a constant k from each of the observations.

Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 even natural numbers are 2, 4, 6, 8 and 10. Their mean is

Subtract 1 from each of the numbers. Then the numbers becomes 1, 3, 5, 7, and 9. The new mean is

Therefore subtracting a constant number to each observation the mean decreased by that constant

(iii) Multiply a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 even natural numbers are 2, 4, 6, 8 and 10. Their mean is

Multiply 2 to each of the numbers. Then the numbers becomes 4, 8, 12, 16, and 20. The new mean is

Therefore multiplying by a constant number to each observation the mean becomes the constant multiplied by the old mean

(vi) Divide a nonzero constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 even natural numbers are 2, 4, 6, 8 and 10. Their mean is

Divide 2 to each of the numbers. Then the numbers becomes 1, 2, 3, 4, and 5. The new mean is

Therefore dividing by a constant number to each observation the mean becomes the old mean divided by the constant