Problem Solving

Hi guys,

Does anyone know what the formula is for calculating a change in mean? Thanks!

I assume you are talking about replacing numbers in a set? Then it would be:

[(Value of new numbers) - (Value of numbers being replaced)/ (numbers in set)] + old mean.

Let me know if you were looking for something different.

(Sum of old values + new value)/(the number of old values + increase in the number of value)=new mean

New mean-Old mean=Change in mean

example: the numbers 3,8,9,1,14 have mean (arithmetic average) (3+8+9+1+14)/5=7
new value 1 is added to the set of numbers above, the new mean will be (3+8+9+1+14+1)/6=6
Change in mean, 6-7=-1 (decrease by 1)

conceptual framework: if new number added to the set is less than the old mean (mean for the set containing old numbers), then the new mean will be less than the old mean. Alternatively, if new number is greater than the old mean, the new mean will be greater than the old mean.

Symmetry rule: let's assign the old numbers in ascending/descending order (order matters, not direction) 1, 3, 8, 9, 14. The mean is 7 and can be placed between the 3 and 8 on the ordered line of numbers. The sum of distances between 7 (mean) and each number to the left from mean should be equal to the sum of distances between 7 (mean) and each number to the right from mean.
(7-1)+(7-3)=10
(8-7)+(9-7)+(14-7)=10
To understand how the new number 1 added to the set of old numbers will affect the mean, 7, we notice that 1 must be placed to the left from mean, as 1<7 (ordered new numbers line) and then the sum of distances between 7 (mean) and each number to the left from mean will be (7-1)+(7-1)+(7-3)=16 which is greater than 10, the sum of distances between 7 (mean) and each number to the right. Therefore, our new mean should be shifted to the left to have symmetry in totals on each side. If new mean=6 we have
(6-1)+(6-1)+(6-3)=13
(8-6)+(9-6)+(14-6)=13

RSK wrote:Hi guys,

Does anyone know what the formula is for calculating a change in mean? Thanks!

I was doing In Action problems in the Manhatton Word Translations Guide. Question no. 10 on pg 113:

10. Matt gets a $1,000 commission on a big sale. This commission alone raises his average
commission by $150. If Matt's new average commission is $400, how many
sales has Matt made?

I know how to solve this problem. However, I was reading the solution and it says:

10.5: Before the $1,000 commission. Matt's average commission was $250; we can express this algebraically with the equation S = 250n. After the sale, the sum of Matt's sales increased by $1,000, the number of sales made increased by 1, and his average commission was $400. We can express this algebraically with the equation: S + 1,000 = 400(n + 1)
250n + 1,000 = 400(n + 1)
250n + 1,000 = 400n + 400
150n = 600 n=4
Before the big sale, Matt had made 4 sales. Including the big sale, Matt has made 5 sales.

Alternatively, you can solve this problem using the "Change to the Mean" formula.

In order to solve this problem quicker, I was wondering whether there is such a formula?
Thanks!

The Formula is simple

Change in Mean=(New Item-Old Mean)/ New number of items

Coming to ur problem
We need to find how many sales Matt has made i.e. The New number of items

Number of commissions=Number of Sales

New Item= $1000
Change in Mean= $150 (as the mean is increasing by $150)
Old Mean=400-150= $250

Substitute these values in the formula

150=(1000-250)/New number of items
New number of items=750/150= 5

Which is the number of sales or the number of commissions he has recvd. Hope this helps

I believe the formula given above doesnt work when more than 1 term is added or removed. I dont know how common it is in GMAT to see more than one term added or removed, but in this case, I think it is best to do the old fashion method without having to remember any formule.

Any comments?

The formula works. Just subtract the mean from each of the two new terms - add up the difference and divide by the total number of new terms. This will give you the change in mean. Then add it to the old mean to get the new mean.

Basically what you are doing here is that, you are calculating how much the new term is making the average shift from the old mean. So if the new term is 10 greater than the old mean, and there are 5 terms, the average shifts by 10/5=2 points. If there is another term 20 greater than the mean added to the same series, you have 10+20=30 more. 30/(5+2) = 4.28 greater than the old mean. So the new mean will be: old mean + 4.28.

Hope you can understand. Otherwise let me know.

Cheers!

RSK wrote:The formula works. Just subtract the mean from each of the two new terms - add up the difference and divide by the total number of new terms. This will give you the change in mean. Then add it to the old mean to get the new mean.

Basically what you are doing here is that, you are calculating how much the new term is making the average shift from the old mean. So if the new term is 10 greater than the old mean, and there are 5 terms, the average shifts by 10/5=2 points. If there is another term 20 greater than the mean added to the same series, you have 10+20=30 more. 30/(5+2) = 4.28 greater than the old mean. So the new mean will be: old mean + 4.28.

Hope you can understand. Otherwise let me know.

Cheers!

I am a bit lost. can someone explain this adding of 2 terms with respect to the above Q of Matt who got commission of $1000. So if we were to assume that he got 2 more commissions, than how would the mean change & entire computation would be done?

Thanks in advance!

RSK wrote:The formula works. Just subtract the mean from each of the two new terms - add up the difference and divide by the total number of new terms. This will give you the change in mean. Then add it to the old mean to get the new mean.

Basically what you are doing here is that, you are calculating how much the new term is making the average shift from the old mean. So if the new term is 10 greater than the old mean, and there are 5 terms, the average shifts by 10/5=2 points. If there is another term 20 greater than the mean added to the same series, you have 10+20=30 more. 30/(5+2) = 4.28 greater than the old mean. So the new mean will be: old mean + 4.28.

Hope you can understand. Otherwise let me know.

Cheers!

I am a bit lost. can someone explain this adding of 2 terms with respect to the above Q of Matt who got commission of $1000. So if we were to assume that he got 2 more commissions, than how would the mean change & entire computation would be done?

Thanks in advance!