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mean, median and standard deviation

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by Anurag@Gurome » Thu May 31, 2012 9:33 am
shrutib wrote:...Specially the type where they ask whether mean and median are same?
The mean and median of a set are are same when,
  • 1. All the elements of the set are uniformly distributed (see note) around the mean. For example, {1, 2, 3, 4, 5} or {1, 2, 7, 12, 13} etc.
    2. The set contains only one element.
    3. All the elements of a set are same.
2 and 3 are special cases of 1.

Edited after the discussion below:

Note : The phrase "uniformly distributed around the mean" may create some confusion. I'm using the phrase "uniformly distributed around the mean" to mean that the first term and last term, second term and second last term, third term and third last term,... n-th term and n-th last term are equidistant from the mean.

For example, the terms of {1, 2, 7, 12, 13} are not equally spaced. But the first term 1 and the last term 13 are equidistant from the mean, i.e. 7. And so are the second term 2 and the second last term 12.
Last edited by Anurag@Gurome on Sat Jun 02, 2012 6:50 am, edited 1 time in total.
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by aneesh.kg » Thu May 31, 2012 9:20 pm
shrutib wrote:how to attack the Data Sufficiency questions regarding mean, median, range and Standard Deviation. Specially the type where they ask whether mean and median are same?

Thank you
Hi Shruti,

This is a very common doubt and an important concept.

Mean = Median when:

1. the set consists of evenly spaced numbers
2. if all the members of the set are equal
3. set has just one number
4. One more case

Let's discuss the point # 4.

We've already seen that for every set of evenly spaced numbers (or an Arithmetic Progression), Mean = Median.

BUT

If Mean = Median for a set of numbers, then the set of numbers need not be an AP.

Let me show some sets of numbers which are not in an AP but for which mean = median:

(i) 2, 3, 5, 7, 8
(ii) 3, 4, 4, 4, 5
(ii) 1, 4, 5, 6, 7, 8, 11

Your turn now! Can you show me a few such sets of numbers?

Hint: Take any AP. Tweak a few terms in it so that neither the Mean nor the Median change and it no longer remains an AP. :)
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by Gaurav 2013-fall » Sat Jun 02, 2012 1:56 am
aneesh.kg wrote:
shrutib wrote:how to attack the Data Sufficiency questions regarding mean, median, range and Standard Deviation. Specially the type where they ask whether mean and median are same?

Thank you
Hi Shruti,

This is a very common doubt and an important concept.

Mean = Median when:

1. the set consists of evenly spaced numbers
2. if all the members of the set are equal
3. set has just one number
4. One more case

Let's discuss the point # 4.

We've already seen that for every set of evenly spaced numbers (or an Arithmetic Progression), Mean = Median.

BUT

If Mean = Median for a set of numbers, then the set of numbers need not be an AP.

Let me show some sets of numbers which are not in an AP but for which mean = median:

(i) 2, 3, 5, 7, 8
(ii) 3, 4, 4, 4, 5
(ii) 1, 4, 5, 6, 7, 8, 11

Your turn now! Can you show me a few such sets of numbers?

Hint: Take any AP. Tweak a few terms in it so that neither the Mean nor the Median change and it no longer remains an AP. :)
Aneesh,
your point 4 is an extension of the 1 only. Isnt it?
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by aneesh.kg » Sat Jun 02, 2012 5:38 am
Gaurav 2013-fall wrote: Aneesh,
your point 4 is an extension of the 1 only. Isnt it?
Hi Gaurav,

Point 4 is a manipulation of Point 4, if that's what you mean by 'extension'. You can manipulate a set of evenly spaced integers such that its Median and Mean do not change as shown in the examples in the previous post.
Anurag@Gurome wrote: The mean and median of a set are are same when,
1. All the elements of the set are uniformly distributed around the mean. For example, {1, 2, 3, 4, 5} or {1, 2, 7, 12, 13} etc.
2. The set contains only one element.
3. All the elements of a set are same.

2 and 3 are special cases of 1.
Hi Anurag,
I would beg to differ with your point no. 1. In my opinion, the numbers need not be uniformly distributed about the mean for mean = median.
For example:
2, 9, 10, 12, 17 does not have terms uniformly distributed about the mean ( which is 10) and yet mean = median.
I think uniform distribution is just a special case. Please correct me if I am wrong.
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by Anurag@Gurome » Sat Jun 02, 2012 6:16 am
aneesh.kg wrote:I would beg to differ with your point no. 1. In my opinion, the numbers need not be uniformly distributed about the mean for mean = median.
Hi Aneesh,

I used the phrase "uniformly distributed around the mean" to mean that the first term and last term, second term and second last term, third term and third last term,... n-th term and n-th last term are equidistant from the mean. Note that, I didn't use the phrase "evenly distributed" or "equally spaced" or only "uniformly distributed".

I have given an example accordingly. The terms of {1, 2, 7, 12, 13} are not equally spaced. But 1 and 13 are equidistant from the mean, i.e. 7. And so are 2 and 12.
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