Average

What is Average?

The average of n quantities of the same kind is equal to the sum of all the quantities divided by the number of quantities.

How to find Average?

Formula Method

Average = $\frac{Sum \hspace{1ex} of \hspace{1ex} Quantities}{Number \hspace{1ex} of \hspace{1ex} Quantities \hspace{1ex} (n)}$

For example:
An average or an arithmetic mean of given data is sum of the given observations divided by number of observations.
Average = $\frac{Sum \hspace{1ex} of \hspace{1ex} Observations}{Number \hspace{1ex} of \hspace{1ex} Observations}$

Method of deviations

Instead of using the formula, we can use method of deviations to ease out our calculation.

For example, we need to find average of n quantities – $x_1$, $x_2$, ….. $x_n$

In this method:

• We assume some arbitrary number (A) as the average.
• After that we algebraically add the deviations (differences) of all the quantities from A, i.e. ($x_1$ – A) ± ($x_2$ - A) ± ….. ± ($x_n$ - A).
• Thereafter we take average of these differences. That is, we find $\frac{1}{n} \sum_{i=1}^{n} (x_i − 𝐴)$
• Then we algebraically add this average to A to get our answer.

So, the average of these n items, Average = A ± $\frac{1}{n} \sum_{i=1}^{n} (x_i − 𝐴)$

Let’s consider an example:

A student scored the following marks in five subjects: 38, 58, 44, 54 and 61.

Now, assume some arbitrary number (A) as the average, say 50.

• The deviations of the marks from 50 are: -12, +8, -6, +4 and +11
• The algebraic sum of these deviations is +5.
• Hence the average of the student’s marks = 50 + 5/5 = 51

If we take A = 55.

• The deviations of the scores from 55 are: -17, +3, -11, -1 and +6
• The algebraic sum of these deviations is -20.
• Hence the average of the student’s marks = 55 - 20/5 = 55 - 4 = 51.

Q. If a cricketer scored 98, 82, 66, 76, 80 and 90 runs in his six innings, then what is his average score?

Explanations :
1 2

Q. The average of 10 numbers is zero. Which of the following can be concluded about the numbers?

(a) At least 9 numbers are positive.
(b) At most 5 numbers are positive.
(c) At most 9 numbers are positive.
(d) At most 1 number is positive.

Explanation:

Properties of Averages

Addition

If all the numbers get increased by a, then their average must be increased by a.
E.g. If the average age of group of people is x years, then their average age after n years will be (x + n). This is because with each passing year, each person’s age increases by 1 and vice versa.

Subtraction

If all the numbers get decreased by a, then their average must be decreased by a.
E.g. If the average age of group of people is x years, then their average age n years ago would have been (x – n).

Multiplication/Division

If all the numbers are multiplied by a, then their average must be multiplied by a.
If all the numbers are divided by a, then their average must be divided by a. (a ≠ 0)

Multiples of a number

Average of n multiples of any number = $\frac{Number × (n+1)}{2}$

Impact of Group change on Average

Average may increase/decrease on an entity leaving or joining the group. That’s what we are going to focus on, in this section.

There are three probable scenarios.

Addition of a new item

Average of n quantities is equal to X and on adding a new entity/quantity to the group the average becomes Y (i.e. number of entities in the group increases by 1), then:

Value of new entity = Y + (Y - X)n OR X + (Y - X)(n + 1)

Q. The average age of a group of 8 boys is 12 years. Thereafter a new boy joins the group and their average age increases to 13 years. What is the age of the new boy?

Explanations :
1 2 3

Q. In a cricket tournament Ponting scored 135, 125, 150, 230 and 160 runs in 5 consecutive innings. If after his next innings his average score increased by 3 runs, then which of the following is correct?

(a) Runs scored in the next innings is 180.
(b) Runs scored in the next innings is 186.
(c) Runs scored in the next innings is 188.
(d) Runs scored in the next innings is 178.

Explanations :
1 2 3

Removal of an existing item

Average of n quantities is equal to X and on removing an entity/quantity from the group the average becomes Y (i.e. number of entities in the group decreases by 1), then:

Value of removed entity = Y + (X - Y)n OR X + (X - Y)(n - 1)

Replacement of an existing item with a new one

The average of n quantities is equal to X. When one quantity of value P is replaced with a new quantity having value Q, the average of quantities becomes Y (i.e. number of entities in the group remain the same), then

Value of new entity, Q = P + (Y - X)n

Q. A group containing 13 girls and 9 boys had an average amount of Rs. 750 with them. When one of the boys, who had Rs. 540 with him, was replaced with a girl, the average amount with the group increased to Rs. 790. What is the difference in the amounts held by the new girl and the boy who got replaced?

(a) Rs. 800   (b) Rs. 880    (c) Rs. 1,120   (d) Rs. 1,200

Explanations :
1 2 3