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A team of researchers measured each of ten subjects' reactio [#permalink]
27 Nov 2011, 10:39

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Difficulty:

75% (hard)

Question Stats:

18% (01:50) correct
82% (00:56) wrong based on 33 sessions

A team of researchers measured each of ten subjects' reaction time to a certain stimulus and calculated the mean, median, and standard deviation of the measurements. If none of the reaction times were identical and an eleventh data point were added that was equal to the mean of the initial group of ten, which of these three statistics would change?

(A) The median only (B) The standard deviation only (C) The mean and the median (D) The mean and the standard deviation (E) The median and the standard deviation

Re: mean median and std dev [#permalink]
27 Nov 2011, 13:39

The mean certainly will stay the same if you add a value to a set equal to the set's current mean. Because you are adding a new element which is of a distance of 0 to the mean, the standard deviation will go down (as long as the standard deviation wasn't 0 to begin with) because you will now have more elements clustered close to the mean.

It is impossible to tell what will happen to the median, however, so this is a badly designed question. Where is it from? If you have a set like the following (I'll use four elements instead of ten just for simplicity) :

1, 3, 5, 7

then inserting a new element equal to the mean, which is 4, will not change the median; the median will still be 4. However, if you have this set:

0, 2, 4, 10

then if we insert an element equal to the mean of 4, the median changes from 3 to 4. _________________

Nov 2011: After years of development, I am now making my advanced Quant books and high-level problem sets available for sale. Contact me at ianstewartgmat at gmail.com for details.

Re: mean median and std dev [#permalink]
28 Nov 2011, 07:49

yeah i agree that the question's design is not the best because the median may or may not change. this is why i posted it here. Thanks for confirming Ian! It is from Jeff Sackmann's challenge series. OA was B.

A team of researchers measured each of ten subject's [#permalink]
19 May 2014, 05:08

Hi Guys, could not find the solution to this anywhere on the forum: Please help:

A team of researchers measured each of ten subjectsíreaction time to a certain stimulus and calculated the mean, median, and standard deviation of the measurements. If none of the reaction times were identical and an eleventh data point were added that was equal to the mean of the initial group of ten, which of these three statistics would change? (A) The median only (B) The standard deviation only (C) The mean and the median (D) The mean and the standard deviation (E) The median and the standard deviation

The answer as per the hacks solution set is B - but I don't agree because:

1. if you add a term that is equal to the mean of the set, how would it change the SD? since SD is the squared distance from the mean, adding a term that is equal to the mean should not change the SD, right?

2. if you add a term that is equal to the mean of the set, it shouldn't change the mean either right? I tried this by using the mean of 2,3,4 and then 2,3,3,4

3. The median, could change, because with 11 terms, the median should become the 6th term; previously it was the average of the 5th and 6th term?

Please let me know if I am right, or if the hacks answer (B) is correct?

Re: A team of researchers measured each of ten subject's [#permalink]
19 May 2014, 06:04

Expert's post

brobeedle wrote:

Hi Guys, could not find the solution to this anywhere on the forum: Please help:

A team of researchers measured each of ten subjectsíreaction time to a certain stimulus and calculated the mean, median, and standard deviation of the measurements. If none of the reaction times were identical and an eleventh data point were added that was equal to the mean of the initial group of ten, which of these three statistics would change? (A) The median only (B) The standard deviation only (C) The mean and the median (D) The mean and the standard deviation (E) The median and the standard deviation

The answer as per the hacks solution set is B - but I don't agree because:

1. if you add a term that is equal to the mean of the set, how would it change the SD? since SD is the squared distance from the mean, adding a term that is equal to the mean should not change the SD, right?

2. if you add a term that is equal to the mean of the set, it shouldn't change the mean either right? I tried this by using the mean of 2,3,4 and then 2,3,3,4

3. The median, could change, because with 11 terms, the median should become the 6th term; previously it was the average of the 5th and 6th term?

Please let me know if I am right, or if the hacks answer (B) is correct?

Merging similar topics. Please refer to the discussion above. _________________

Re: A team of researchers measured each of ten subjects' reactio [#permalink]
19 May 2014, 06:06

Quote:

1. if you add a term that is equal to the mean of the set, how would it change the SD? since SD is the squared distance from the mean, adding a term that is equal to the mean should not change the SD, right?

Standard deviation is the squareroot of the sum of squared distance from the mean divided by the number of elements in the set. Because we have 11 data points instead of 10, we divide the sum of squared distances by 11 instead of 10. This changes the standard deviation.

Quote:

2. if you add a term that is equal to the mean of the set, it shouldn't change the mean either right? I tried this by using the mean of 2,3,4 and then 2,3,3,4

Correct, it does not change the mean.

Quote:

3. The median, could change, because with 11 terms, the median should become the 6th term; previously it was the average of the 5th and 6th term?

Correct, the median could change.

This is a poor question as it asks what would change. Either the standard deviation will change, or both the standard deviation and median will change. _________________

Re: A team of researchers measured each of ten subject's [#permalink]
19 May 2014, 06:10

Expert's post

brobeedle wrote:

Hi Guys, could not find the solution to this anywhere on the forum: Please help:

A team of researchers measured each of ten subjectsíreaction time to a certain stimulus and calculated the mean, median, and standard deviation of the measurements. If none of the reaction times were identical and an eleventh data point were added that was equal to the mean of the initial group of ten, which of these three statistics would change? (A) The median only (B) The standard deviation only (C) The mean and the median (D) The mean and the standard deviation (E) The median and the standard deviation

The answer as per the hacks solution set is B - but I don't agree because:

1. if you add a term that is equal to the mean of the set, how would it change the SD? since SD is the squared distance from the mean, adding a term that is equal to the mean should not change the SD, right?

2. if you add a term that is equal to the mean of the set, it shouldn't change the mean either right? I tried this by using the mean of 2,3,4 and then 2,3,3,4

3. The median, could change, because with 11 terms, the median should become the 6th term; previously it was the average of the 5th and 6th term?

Please let me know if I am right, or if the hacks answer (B) is correct?

As for your questions:

1. The standard deviation will decrease.

The standard deviation of a set shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

So, if you add a value to a set equal to the set's current mean, the data will be less widespread, hence the SD will decrease (if the SD wasn't 0 for the initial set).

2. The mean will not change.

3. The median may or may not change.

So, as you see, the question is flawed and you can ignore it. _________________

Re: A team of researchers measured each of ten subject's [#permalink]
19 May 2014, 06:11

brobeedle wrote:

Hi Guys, could not find the solution to this anywhere on the forum: Please help:

A team of researchers measured each of ten subjectsíreaction time to a certain stimulus and calculated the mean, median, and standard deviation of the measurements. If none of the reaction times were identical and an eleventh data point were added that was equal to the mean of the initial group of ten, which of these three statistics would change? (A) The median only (B) The standard deviation only (C) The mean and the median (D) The mean and the standard deviation (E) The median and the standard deviation

The answer as per the hacks solution set is B - but I don't agree because:

1. if you add a term that is equal to the mean of the set, how would it change the SD? since SD is the squared distance from the mean, adding a term that is equal to the mean should not change the SD, right?

2. if you add a term that is equal to the mean of the set, it shouldn't change the mean either right? I tried this by using the mean of 2,3,4 and then 2,3,3,4

3. The median, could change, because with 11 terms, the median should become the 6th term; previously it was the average of the 5th and 6th term?

Please let me know if I am right, or if the hacks answer (B) is correct?

r

If the mean for first 10 terms in a, and the term added is a again, the new mean is (10a + a)/11 =a which means the mean remains same.

In case median is not equal to mean for the initial 10 numbers, the new median will be this number we added. Hence, median will change

The SD will also change as the number of terms increases from 10 to 11. Please note that the sum of squares will remain the same though. SD will therefore decrease.

Thus, the answer is E).

The answer can also be reached by taking 10 values as 1,2,3,4,5,6,7,8,9,12

The mean will be 5.7

Now, if a 11th term equal to mean is added, we get new mean again as 5.5

Median of 1,2,3,4,5,6,7,8,9,12 will be 5.5 but median of 1,2,3,4,5,5.7,6,7,8,9,12 is 5.7 which is different from earlier median

In case of SD we have to find sum of (xi - mean)^2. This sum is same is both cases as 11th term is equal to the mean and will give us 0. But N is 10 in first case, and 11 in second. Thus, SD will also change.

Re: A team of researchers measured each of ten subject's [#permalink]
19 May 2014, 06:20

Expert's post

mittalg wrote:

brobeedle wrote:

Hi Guys, could not find the solution to this anywhere on the forum: Please help:

A team of researchers measured each of ten subjectsíreaction time to a certain stimulus and calculated the mean, median, and standard deviation of the measurements. If none of the reaction times were identical and an eleventh data point were added that was equal to the mean of the initial group of ten, which of these three statistics would change? (A) The median only (B) The standard deviation only (C) The mean and the median (D) The mean and the standard deviation (E) The median and the standard deviation

The answer as per the hacks solution set is B - but I don't agree because:

1. if you add a term that is equal to the mean of the set, how would it change the SD? since SD is the squared distance from the mean, adding a term that is equal to the mean should not change the SD, right?

2. if you add a term that is equal to the mean of the set, it shouldn't change the mean either right? I tried this by using the mean of 2,3,4 and then 2,3,3,4

3. The median, could change, because with 11 terms, the median should become the 6th term; previously it was the average of the 5th and 6th term?

Please let me know if I am right, or if the hacks answer (B) is correct?

r

If the mean for first 10 terms in a, and the term added is a again, the new mean is (10a + a)/11 =a which means the mean remains same.

In case median is not equal to mean for the initial 10 numbers, the new median will be this number we added. Hence, median will change

The SD will also change as the number of terms increases from 10 to 11. Please note that the sum of squares will remain the same though. SD will therefore decrease.

Thus, the answer is E).

The answer can also be reached by taking 10 values as 1,2,3,4,5,6,7,8,9,12

The mean will be 5.7

Now, if a 11th term equal to mean is added, we get new mean again as 5.5

Median of 1,2,3,4,5,6,7,8,9,12 will be 5.5 but median of 1,2,3,4,5,5.7,6,7,8,9,12 is 5.7 which is different from earlier median

In case of SD we have to find sum of (xi - mean)^2. This sum is same is both cases as 11th term is equal to the mean and will give us 0. But N is 10 in first case, and 11 in second. Thus, SD will also change.

I hope it is clear now.

Kudos if you like the response!!!

The median could change but it does not mean that it will change in all cases.

If the set is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then adding a new element equal to the mean, which is 5.5, will not change the median. _________________

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