A student worked for 20 days....

[mailtopriyanka1]

A student worked for 20 days. For each of the amounts shown in the 1st row of the table, the 2nd row gives the no. of days at the student earned that amount. What is the median amount of money that the student earned per day for 20 days?

amt earned/day : 96 84 80 70 48
no. of days...... : 4 7 4 3 2


a) 96
b) 84
c) 80
d) 70
e) 48

How do i go about to calculate the median in this case..is there something called weighted median as well..please explain..

[thoppae.saravanan]

Arranging amount earned in ascending order with the number of days,

48-2
70-3
80-4
84-7
96-4

median = (amount earned in 10th day + amount earned in the 11th day)/2 = (84+84)/2 = 84.

[mailtopriyanka1]

thanks..

[StaceyKoprince]

Thanks, thoppae

priyanka, if it's confusing to look at the list in the way that thoppae wrote it, you can also just literally list them all out, from smallest to largest, and count your way to the middle. (And, of course, in this case, you actually need to average the middle two numbers beacuse you have an even number of terms.)

[pratik.munjal]

The only way to go about solving this question is to know the concept of median. I somehow forgot to apply that while attempting the question. (I got carried away by that caution-"Always arrange the numbers in ascending before calculating the median".)

But anyway-the way to solve it is this:

The total number of days are 20 (this is given in the question, but if you want, you can also add the number of days).

The "rule" says that if the number is even, the median will be the mean of the two middle terms. In this case the middle terms will be 10th and 11th terms. Now here comes the tricky part. Start by "adding" the days in a tabular format:

4+7=11 (these are the "first two"). This means that the median will "lie in this range" (sorry for the language-can't find any other phrase).

So the number that "lies in this range" is $84. And that, is the answer.

Perhaps Ron or Stacey can add more depth to this explanation. Also, they might explain why that "rule" of arranging numbers in ascending order is not used here (my guess is that the rule is not used because the "amount earned" has a "frequency" attached to it.)

[jnelson0612]

Perhaps Ron or Stacey can add more depth to this explanation. Also, they might explain why that "rule" of arranging numbers in ascending order is not used here (my guess is that the rule is not used because the "amount earned" has a "frequency" attached to it.)

Actually, Stacey did suggest listing out the values in ascending order. Please see what we have below. I would look at this and say to myself, "since there are twenty numbers the median will be the average of numbers 10 and 11. The three smallest pay numbers are two $48s, three $70s, and four $80s. That is nine numbers. Thus, I know that numbers 10 and 11 must be the next two numbers, which will be in the $84 category.

Day Salary
1 48
2 48
3 70
4 70
5 70
6 80
7 80
8 80
9 80
10 84
11 84
12 84
13 84
14 84
15 84
16 84
17 96
18 96
19 96
20 96

[krishnan.anju1987]

Perhaps Ron or Stacey can add more depth to this explanation. Also, they might explain why that "rule" of arranging numbers in ascending order is not used here (my guess is that the rule is not used because the "amount earned" has a "frequency" attached to it.)

Actually, Stacey did suggest listing out the values in ascending order. Please see what we have below. I would look at this and say to myself, "since there are twenty numbers the median will be the average of numbers 10 and 11. The three smallest pay numbers are two $48s, three $70s, and four $80s. That is nine numbers. Thus, I know that numbers 10 and 11 must be the next two numbers, which will be in the $84 category.

Day Salary
1 48
2 48
3 70
4 70
5 70
6 80
7 80
8 80
9 80
10 84
11 84
12 84
13 84
14 84
15 84
16 84
17 96
18 96
19 96
20 96

Hi,

I got this approach that you mentioned and I thought of the same. At the same time, I have a doubt. Why do we consider that the days given in the second line are consecutive days. For eg, the student could have earned 80$ on one day say the 10th and the next one on 12th. I am a little unsure of whether we can arrange the days consecutively since the question statement doesn't mention it anywhere. But otherwise, since the answer would be 80 and the number of days would be a data without purpose, that seems a little too easy and therefore doubtful.

Do I seem to make any sense or am I confusing myself here?

[tim]

they don't have to be literally consecutive days, but when we order them according to which day was worked, after we talk about the 10th day he worked, the next one must be the 11th day he worked. :) let me know if you have any other questions about this one..

[krishnan.anju1987]

Thanks for the explanation but what I don't understand is why do those two days have the wage of 84. They might also have wage of 80 and 96 and then the median might be 88.

[jnelson0612]

Thanks for the explanation but what I don't understand is why do those two days have the wage of 84. They might also have wage of 80 and 96 and then the median might be 88.

Hi Krishnan,
The median is the absolute middle value of a set of numbers. When we determine a median, we have to first order the numbers in order from smallest to largest. The median is either the middle number (if we have an odd number of numbers) or the average of the two middle numbers) if we have an even number of numbers.

I have listed out the values again below. Notice that I listed them from smallest to largest. That is absolutely necessary to be able to identify the median. You can't just have random numbers popping up in different positions. You must put the smallest number and every other identical number, then the next smallest (and every other identical number if there are any, etc.).

Please let us know if you need further clarification.

Day Salary
1 48
2 48
3 70
4 70
5 70
6 80
7 80
8 80
9 80
10 84
11 84
12 84
13 84
14 84
15 84
16 84
17 96
18 96
19 96
20 96

[krishnan.anju1987]

Hi Jamie,

Thanks for the clarification. That point is clear to me. We need to sort the elements in ascending order to be able to determine the median. However, I guess I failed to express my doubt clearly
In the below order
Days Salaries
1 48
2 48
3 70
4 70
5 70
6 80
7 80
8 80
9 80
10 84
11 84
12 84
13 84
14 84
15 84
16 84
17 96
18 96
19 96
20 96

How do we know the days are related to the salaries? I mean it doesn't mention anywhere in the question that the dates consecutive or the wages are earned in the order mentioned. So the order could as well be as mentioned below

1 48
2 80
3 84
4 70
5 48
6 80
7 80
8 84
9 70
10 84
11 80
12 70
13 84
14 84
15 84
16 84
17 96
18 96
19 96
20 96

or something like this. I know this would make the question ridiculously unsolvable and I solved it in a manner similar to the one you took up but have this doubt lingering in my mind.

[tim]

your concern does not have to do with the specifics of this problem; you seem to be fundamentally misunderstanding what the definition of a median is. put these numbers in order (without the numbers 1-20 if that's what's confusing you), and find what the middle number is. this is what a median is..