Quarter Wit, Quarter Wisdom: The Meaning of Arithmetic Mean

Quarter Wit, Quarter WisdomLet’s start today with statistics – mean, median, mode, range and standard deviation. The topics are simple but the fun lies in the questions. Some questions on these topics can be extremely tricky especially those dealing with median, range and standard deviation. Anyway, we will tackle mean today.

So what do you mean by the arithmetic mean of some observations? I guess most of you will reply that it is the ‘Sum of Observations/Total number of observations’. But that is how you calculate mean. My question is ‘what is mean?’ Loosely, arithmetic mean is the number that represents all the observations. Say, if I know that the mean age of a group is 10, I would guess that the age of Robbie, who is a part of that group, is 10. Of course Robbie’s actual age could be anything but the best guess would be 10.

Say, I tell you that the average age of a group of 10 people is 15 yrs. Can you tell me the sum of the ages of all 10 people? I am sure you will say that it is 10*15 = 150. You can think of it in two ways:

Mean = Sum of all ages/No of people

So Sum of all ages = Mean * (No of people) = 15*10

Or

Since there are 10 people and each person’s age is represented by 15, the sum of their ages = 10*15. Basically, the total sum was distributed evenly among the 10 people and each person got 15 yrs.

Now, let’s say you made a mistake. A boy whose age you thought was 20 was actually 30. What is the correct mean? Again, you can think of it in two ways:

New sum = 150 + 10 = 160

New average = 160/10 = 16

Or

You can say that there is an extra 10 that has to be distributed evenly among the 10 people, so each person gets 1 extra. Hence, the average becomes 15 + 1 = 16.

As you might have guessed, we will work on the second interpretation. Let’s look at an example now.

Example 1: The average age of a group of n people is 15 yrs. One more person aged 39 joins the group and the new average is 17 yrs. What is the value of n?

(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

Solution: First tell me, if the age of the additional person were 15 yrs, what would have happened to the average? The average would have remained the same since this new person’s age would have been the same as the age that represents the group. But his age is 39 – 15 = 24 more than the average. We know that we need to evenly split the extra among all the people to get the new average. When 24 is split evenly among all the people (including the new guy), everyone gets 2 extra (since average age increased from 15 to 17). There must be 24/2 = 12 people now (including the new guy) i.e. n must be 11 (without including the new guy).

Let’s look at another similar example though a little trickier. Try solving it on your own first. If not logically, try using the formula approach. Then see how elegant the solution becomes once you start ‘thinking’ instead of just ‘calculating’.

Example 2: When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Solution: What is the first thing you can say about the initial average? It must have been between 39 and 15. When a person aged 39 is added to the group, the average increases and when a person aged 15 is added, the average decreases.

Let’s look at the second case first. When the person aged 15 is added to the group, the average becomes (initial average – 1). If instead, the person aged 39 were added to the group, there would be 39 – 15 = 24 extra which would make the average = (initial average + 2). This difference of 24 creates a difference of 3 in the average. This means there must have been 24/3 = 8 people (after adding the extra person). The value of n must be 8 – 1 = 7.

If you use the formula instead, it would take you quite a while to manipulate the two variables to get the value of n. I hope you see the beauty of this method. Next week, we will discuss some GMAT questions based on Arithmetic mean!

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

7 Responses

  1. utkarsh says:

    Thanks a lot for taking out some time from your CFA prep schedule for your blog..
    And all the best

  2. Thanks for this Karishma!

    Would you mind explaining the second example in a bit more detail? I’m unable to grasp intuitively how the concept of a difference of 24 is arrived at. The same goes for the difference of 3 in the average. The mechanics of the example are simple enough but I’d rather understand this a bit better…

  3. Karishma says:

    @Utkarsh: Yeah, it’s a welcome break!

    @Avinash: Did you understand Example 1? It yes, then think of Example 2 in this way:

    In a group, there are n people. It was later realized that a boy whose age was recorded as 15 yrs was actually 39 yrs old. When the average was re-calculated, it was found that it has increased by 3 yrs. What is the value of (n-1)?

    Now think of 2 things:
    1. How will you solve this question?
    2. Is this question basically same as example 2?

  4. UFRAILMEMEMONEY says:

    To say I was impressed by this article, is to understate my bewilderment at the oat meal eating ease with which you explain this stuff.
    What an insightful way to explain this stuff! Wow!
    I knew this, conceptual solution to average problems, before reading this article but if you asked me to explain it, I would have said: “never mind.”

  5. Ajay says:

    Hey Karishma ,
    as soon as i read the 1st problem, i tried solving by my own
    (39+15n ) / (n+1) = 17
    n=11
    (using the standard average formula -> avg=sum / # )

    But, when i went through the 2nd , i was bowled . i m still not able to understand how it works. Could u arrive at the solution using the Average formula, because only then it would help me understand your methodology better..
    Thanks in advance

    • Karishma says:

      When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

      Say, original average age is x.

      Using the traditional approach, you need to make two equations and then solve them

      (nx + 39)/(n + 1) = x + 2
      (nx + 15)/(n + 1) = x – 1

      Solve for n by cross multiplying (nx will get canceled in both the equations) and solving simultaneously.

  6. irda says:

    Hi Karishma,

    Wonderful post as always!

    I did this problem in way completely inspired by yours, but with slight variation in the logistics.

    I am considering “x” as the average. To me actual average is that point where people away from the point either walking backwards or walking forwards.

    Now, when we place a person 39 unit position, and another person at 15th position.

    For the person at the 39 position his walk backwards get distributed to the average position earlier and that point moves forward.

    Similarly, the effect of the person at 15 results into average position moving backward.

    Both have antagonistic effect which is shared equally among all the persons. However both are real move in terms of displacement.

    Above is the story that conjures in my head.

    Regarding the algebra:

    Remember the average is X:

    n = number of people initially

    N is the number of people after addition = n + 1

    39 – x = total distance walked backwards

    x – 15 = total distance walked forwards

    Distribution of distance is added to each other since there is net distribution. This is an absolute value.

    (39 – x)/N + (x – 15)/ N = 2 + 1

    (39 – x + x – 15)/ N = 3

    24 / N = 3

    N = 8

    n + 1 = 8 ;hence, n = 7

    Does this make sense?

Leave a Reply

Spam protection by WP Captcha-Free