Concept 6 - Average

The average of a number is a measure of the central tendency of a set of numbers. In other words, it is an estimate of where the center point of a set of numbers lies.
The basic formula for the average of n numbers x1, x2, x3, … xn is

This also means A_n × n = total of the set of numbers.

The average is always calculated for a set of numbers.

Concept of weighted average: 

When we have two or more groups whose individual averages are known, then to find the combined average of all the elements of all the groups we use weighted average. 

Thus, if we have k groups with averages A1, A2 ... Ak and having n1, n2 ... nk elements then the weighted average is given by the formula:

Another meaning of average 

The average [also known as arithmetic mean (AM)] of a set of numbers can also be defined as the number by which we can replace each and every number of the set without changing the total of the set of numbers.

Properties of average (AM) 

The properties of averages [arithmetic mean] are:

Property 1: The average of 4 numbers 12, 13, 17 and 18 is:

Solution: Required average = (12 + 13 + 17 + 18)/4 = 60/4 = 15

This means that if each of the 4 numbers of the set were replaced by 15 each, there would be no change in the total. This is an important way to look at averages. In fact, whenever you come across any situation where the average of a group of ‘n’ numbers is given, we should visualise that there are ‘n’ numbers, each of whose value is the average of the group. This view is a very important way to visualise averages. This can be visualised as

Property 2: In Example 1, visualise addition of a fifth number, which increases the average by 1.

15 + 1 = 16

15 + 1 = 16

15 + 1 = 16

15 + 1 = 16

The +1 appearing 4 times is due to the fifth number, which is able to maintain the average of 16 first and then ‘give one’ to each of the first 4. Hence, the fifth number in this case is 20

Property 3: The average always lies above the lowest number of the set and below the highest number of the set.

Property 4: The net deficit due to the numbers below the average always equals the net surplus due to the numbers above the average.

Property 5: Ages and averages: If the average age of a group of persons is x years today then after n

years their average age will be (x + n). Also, n years ago their average age would have been (x – n). This happens due to the fact that for a group of people, 1 year is added to each person’s age every year.

Property 6: A man travels at 60 kmph on the journey from A to B and returns at 100 kmph. Find his

average speed for the journey.

Solution: Average speed = (total distance) / (total time)

If we assume distance between 2 points to be d. Then,

Average speed = 2d / [(d/60) + (d/100)] = (2 × 60 × 100)/ (60 + 100) = (2 × 60 × 100)/160 = 75

Average speed = (2S1 ◊ S2)/(S1 + S2) [S1 and S2 are speeds] of going and coming back respectively.

Short Cut The average speed will always come out by the following process:

The ratio of speeds is 60 : 100 = 3 : 5 (say r1 : r2)

Then, divide the difference of speeds (40 in this case) by r1 + r2 (3 + 5 = 8, in this case) to get one part. (40/8 = 5, in this case)

The required answer will be three parts away (i.e. r1 parts away) from the lower speed.

Check out how this works with the following speeds:

S1 = 20 and S2 = 40

Step 1: Ratio of speeds = 20 : 40 = 1 : 2

Step 2: Divide difference of 20 into 3 parts (r1 + r2) Æ = 20/3 = 6.66

Required average speed = 20 + 1 × 6.66

This process is essentially based on alligations.