lastshot wrote:
Hi Karishma !!!
Will you please elaborate with an example how in AP or evenly spaced sequence Mean=Median --> just want to know reason or theory behind it
Thanks in advance
Regards
Last Shot
What is the mean of 43, 44, 45, 46, 47?
Arithmetic mean is the number that can represent/replace all the numbers of the sequence. Notice in this sequence, 44 is one less than 45 and 46 is one more than 45. So essentially, two 45s can replace both 44 and 46. Similarly, 43 is 2 less than 45 and 47 is 2 more than 45 so two 45s can replace both these numbers too.
The sequence is essentially 45, 45, 45, 45, 45.
Hence, the arithmetic mean of this sequence must be 45! (If you have doubts, you can calculate and find out.)
It makes sense, doesn’t it? The middle number in the sequence of consecutive positive integers will be the mean. The deviations of all numbers to the left of the middle number will balance out the deviations of all the numbers to the right of the middle number.
Once again, what is the mean of 192, 193, 194, 195, 196, 197, 198?
It is 195 since it is the middle number!
Ok, what about 192, 193, 194, 195, 196, 197? What is the mean in this case? There is no middle number here since there are 6 numbers. The mean here will be the middle of the two middle numbers which is 194.5 (the middle of the third and the fourth number). It doesn’t matter that 194.5 is not a part of this list. If you think about it, arithmetic mean of some numbers needn’t be one of the numbers.
What about 71, 73, 75, 77, 79? What will be the mean in this case? Even though these numbers are not consecutive integers, the difference between two adjacent numbers in the list is the same (it is an arithmetic progression). So the deviations of the numbers on the left of the middle number will cancel out the deviations of the numbers on the right of the middle number (71 is 4 less than 75 and 79 is 4 more than 75. 73 is 2 less than 75 and 77 is 2 more than 75). Hence, the mean here will be 75 (just like our first example).
Just to re-inforce:
102, 106, 110 –> Mean = 106
102, 106, 110, 114 -> Mean = 108 (Middle of the second and third numbers)
Now think, what is the median? It is the middle number! Hence, in an AP, mean = median.
Check out this post for an extended discussion of the concept of deviations:
http://www.veritasprep.com/blog/2012/05 ... eviations/ _________________
Karishma
Veritas Prep GMAT Instructor
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