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!!!