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Compare Mean and Median in less than 20 seconds

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Compare Mean and Median in less than 20 seconds [#permalink] New post 22 Jan 2013, 09:50
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Compare Mean and Median in less than 20 seconds


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In a table analysis question that contains a long list, calculating mean through traditional method takes time. To save time, we will introduce two techniques that will allow you to logically deduce whether mean is greater than or less than the median.



To illustrate my point, let’s review OG 13/#24 question for discussion.

‘The table lists data on the 22 earthquakes of magnitude 7 or greater on the Richter Scale during a recent year. Times are given in hours, minutes, and seconds on the 24-hour Greenwich Mean Time (GMT) clock and correspond to standard time at Greenwich, United Kingdom (UK). Latitude, measured in degrees, is 0 at the equator, increases from 0 to 90 proceeding northward to the North Pole, and decreases from 0 to –90 proceeding southward to the South Pole. Longitude, also measured in degrees, is 0 at Greenwich, UK, increases from 0 to 180 from west to east in the Eastern Hemisphere, and decreases from 0 to –180 from east to west in the Western Hemisphere.’


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For each of the following statements, select Yes if the statement is true based on the information provided; otherwise select No.
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The focus of our discussion will be question # 24A.

STANDARD APPROACH


“For the 22 earthquakes; the arithmetic mean of the depths is greater than the median of the depths.”

In mathematical terms:

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The question asks us to compare mean and median. Our natural instinct will be to first calculate the mean and then the median.

Mean --> Mean involves handling 22 data points. Summing up all 22 data points and dividing the sum by 22 is quite time consuming. The mean comes out to be 112.56.

The median -->

1. Median is the value of the middle-most cell of depth column, when data points are arranged in ascending order.
2. In this dataset, there are 22 elements.
3. So middle most value = mean of 11th and 12th values
4. Median = \frac{(25+26)}{2}=25.5.

Now let’s review the two approaches.

APPROACH 1 – “LIMITED SET OBSERVATION”


Per the approach, we will at first calculate median, since median is less calculation intensive as compared to mean.

Median = 25.5 (Following the same approach as in “standard approach”)

Image


Image

Mean –

Fundamental principle
– In a series of all positive numbers, mean of the series is always greater than the sum of limited data divided by total # of data points in the set

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When we glance at the dataset, we find that the last value 641 is disproportionately high. This implies that mean of the all depths must be greater than 1/22 times 641. This equals to 1/22*(641) = 29.136 km. Now, 29.136 km itself is already greater than the median depth (25.5 km), so the actual mean of the depths must be greater than the median of the depths. So we could arrive at the answer without actually calculating the exact mean of the list.

This approach has also been used by OG.

APPROACH 2: THE SHERLOCK METHOD - LOGICAL DEDUCTION BY OBSERVATION


This approach is along the lines of the Approach 1- Limited set Observation approach, but there is no calculation involved. In fact this approach relies on data observation only.

Median- Instead of exactly calculating the median, we will make a few observations. After all to find the exact median, we did have to go through some effort (as shown in standard approach). We know that median is the middle most value.
Image
Just observing this data set, we can be confident that the median will fall between 25 and 31.

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Mean- Let us understand a property of mean in relation to ascending order listed dataset.

The given data set of depth is arranged in ascending order, & bottom cell values are disproportionately higher than other cell values. As all the data points lay equal importance for mean, this implies that mean will drift towards heavy values.

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So it is obvious that mean value will be more than the median. In fact it will be much higher than the median value in this specific question.

Just to recollect the question, “For the 22 earthquakes; the arithmetic mean of the depths is greater than the median of the depths.” So the answer to the question is YES.

The best thing about this approach is that we don’t need to do any calculation. Mere observation of data & use of basic concept is enough to determine the answer.

Table given below presents a few salient features of mean and median.

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OTHER SCENARIO



Few top cell values are disproportionately low and bottom cell values are disproportionately high

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In this scenario, we cannot infer that the mean is less or more than the median. The top 5 cells’ values will drift the mean towards a lower value, whereas the bottom 4 cells’ values will drift it towards a higher value. So it is difficult to infer by mere observation the way in which the mean will drift more.

So what to do in this scenario?

Well, GMAT is not going to test exactly this kind of scenario. Had this scenario been presented to you in the exam, there would have been some other concept applicable to deduce the answer. GMAT lays more importance on application of concepts than on long calculations.

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EXERCISE QUESTION


‘13 students from a school were rated for their proficiency in 4 sports - higher the rating, more the proficiency. Table Tennis and Basketball scores are out of 100 points, Lawn Tennis scores are out of 20 points, and badminton scores are out of 50 points.’
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For each of the following statements, select ‘Yes’ if statement is true based solely on the information provided in the table; otherwise select ‘No’.
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I have attached an excel file of this table. Please make sure that you only use sorting feature of Excel, that too in ascending order only.

Enjoy averaging. :)
Shalabh
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Table 2.xlsx [10.5 KiB]
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Last edited by egmat on 31 Jul 2013, 13:14, edited 4 times in total.
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Re: Compare Mean and Median in less than 20 seconds [#permalink] New post 24 Jan 2013, 02:01
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OA for exercise questions-
[Reveal] Spoiler:
Yes/Yes/No.
. Worth attempting questions. do give it a try. :)

-Shalabh
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Last edited by egmat on 25 Jan 2013, 04:38, edited 1 time in total.
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Re: Compare Mean and Median in less than 20 seconds [#permalink] New post 24 Jan 2013, 09:11
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Hi Shalabh,

Question 3 is the best among the questions mentioned. I solved this question based on a method that is slightly tough to explain. Anyway i will try to give it a shot.

APPROACH-1 - "LOGICAL METHOD"
No of students currently in the table (excluding 2 new students) = 13
If we observe closely, the current mean score is higher than 10 ( i.e. the current mean score is very close to the mean of 2 students.)
Mean score of the 2 students (only), whom we want to include = 12.

Question statement says that new mean score will increase by at least 2 points if Manish & Imran are included in the table.
This means that the combined score of 2 new students must be equal to ->
= [Current Mean score (excluding 2 new students) X No of new students] + [Required Increase in Mean score x No of current students]
= [Min 10 x 2] + [ Min 2 x 13]
= Min 20 + Min 26
= Min 46

Thus if want to increase the current mean score by least 2 points, then Manish & Imran must contribute a total of = 46 points or Higher
But the Actual total score of Manish & Imran is = 5 + 19 = 24 (A score which is lower than the required score 46)
i.e. Total Score of Manish & Imran (24) < Required total score (46)
This means that new mean score will be lower than old mean score.

Thus Manish & Imran's inclusion will decrease the new mean score rather than increase.
Answer is "NO"


APPROACH-2 - "NUMERICAL METHOD"
Current mean score of 13 students = 185 / 13 = 14.2
Required new mean score = 14.2 + min 2 = Min 16.2
New mean score if Manish & Imran are included = (185 + 5 + 19)/ 15 = Approximately 14 (Precise 13.93)

It is clearly evident that new score is less than the required mean score. Thus Answer is NO


APPROACH-3 - "OBSERVATION METHOD"
Arrange the Lawn Tennis scores in increasing order & you will see that the scores form an Arithmetic sequence 4,6,8,10,12,14,16,18,20 (except few repeated entries 18, 19, 20, 20)
Mean of any AP is = (Sum of 1st & Last entry)/2
= (4 + 20)/2
= 12
Because the remaining scores (scores that we have excluded from the calculation of Mean) are higher than the Calculated mean, we can safely assume that the existing mean score is higher than 12.
Required new mean score = 12 + min 2 = Min 14
The mean score Manish & Imran is = (5 + 19)/ 2 = 12

In order to raise the new mean score, Mean score of Manish & Imran (12) must be equal or greater than required mean score (14)

But Mean score of Manish & Imran < Require Mean score.
Thus Answer is NO



Hope these methods will be useful for many others to come.
If you like these approaches, indicate the appreciation through KUDOS.


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Last edited by fameatop on 29 Jan 2013, 06:42, edited 6 times in total.
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Re: Compare Mean and Median in less than 20 seconds [#permalink] New post 28 Jan 2013, 21:50
Great example .

I used a eye-ball method for solving , I think the best way is the see extremes . If you have extreme values then the mean and median will shift to that side of the value. It helped me for A and B .

However, for C I think it was more of a calculated guess .
I wrote it as (x + 24) / 15 = how much times x / 13 . where x is sum of all the scores .
Thus to have 2 points 24/15 should yield >= 2 ie 24 >= 30 which is false . Thus C is
[Reveal] Spoiler:
False
, rest both are
[Reveal] Spoiler:
True


If someone gets a good/faster way to solve C would appreciate .

Eski
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Re: Compare Mean and Median in less than 20 seconds [#permalink] New post 01 Feb 2013, 04:22
Expert's post
Hello all,

Its time announce the winner of 'Best Approach' competition. We received quite a few entries for this competition at a different forum. Clearly, Fameatop is winner.
The diligence shown by him is praiseworthy. Kudos, and congratulations to Fameatop!

Out of 3 approaches, experts chose Approach 3- Observation Method as the BEST approach. It scores in most of the parameters of selection.

1. Efficient - Since this approach is based on observation, logic, and inference, it is truly efficient.
2. Concept - This approach applies concepts of a. Observation of limited data set, and b. Inference of new mean
3. Least calculation intensive - The approach smartly tackles calculation of limited data set mean with an application of arithmetic sequence concept.

Fameatop is rewarded with an e-GMAT IR course, free for 6 months.

We will soon post 5 approaches from e-GMAT. You can also rate those approaches based on above 3 parameters.

Cheers!

e-GMAT team
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Re: Compare Mean and Median in less than 20 seconds [#permalink] New post 01 Feb 2013, 07:37
Thanks Shalabh & E-gmat team for the appreciation & the gift.

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Re: Compare Mean and Median in less than 20 seconds [#permalink] New post 01 Feb 2013, 20:14
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Hello all,

Here are few approaches that we think would be applicable to this particular question. Please pick best approach as per you, and why you think so.

Approach 1 - Computational Approach

Current Mean = Sum of all 13 data/13 = 185/13=14.23;
New Mean = (Sum of all 13 data + 5+19)/15 = 209/15= 13.53;
Mean decreases instead of increasing. Answer is NO.

Approach 2 - Smart Computational Approach

To get the value of current mean, we need to sum all 13 datasets, an approach that is time consuming.
Let’s make some observations. Observe Lawn Tennis column when arranged in ascending order.

Image

There are 9 data points- 4, 6, 8…..20 as highlighted in red color. These data points form a series with equal interval. So their sum is equal to middle-most value*9= 12*9= 108.
Now, Current Mean = Sum of all 13 data points/13 = (Sum of all 9 data points + 18+19+20+20)/13 = (108+77)/13=185/13=14.23;
New Mean = (Sum of all 13 data points + 5+19)/15 = 209/15= 13.53;
Mean decreases rather than increasing. Answer is NO.

Approach 3 - Logical computational approach

Calculation of current mean is done in same manner as is done in approach 2.

We find that current mean is 14.23, which is greater than the mean (12) of Manish and Imran [(5+19)/2=12].
What will be the new mean of after inclusion of Manish and Imran?

Mean fundamental

Mean of a dataset cannot be less than the least value (Here it is ‘12’) and more than the highest value (Here it is ’14.23’). It will lie in between (do not read- Mid value). This implies that the new mean after inclusion of Manish and Imran will be less than 14.23.
So mean decreases rather than increases.

Approach 4 - Logical approach

Mean of 2 new values added= (5+19)/2=12
Current Mean of 13 data points is not known.

We need to deduce whether the new mean increases ?

New mean > Current mean, only and only if current mean < 12 (Mean of the 2 new values). Because if current mean > 12, the lower value 12 will pull the current mean towards it, and in that scenario, new mean will be less than the current mean

So we need to find whether the current mean < 12 or not?

Say current mean = x.

To answer the question, we need to know if the current mean increases by at least 2 points or not when the two data points are included?

So Is New mean > x+2?

We can form a mathematical equation as below

Image

There are 13 data points & 2 new inclusions - so total 15 data points for new mean

Upon solving, we get x ≤ -3. This means to get an increment of 2 over the current mean, current mean has to be at the max. -3.
Well, -3 is an absurd value since the mean of given data set cannot be this value.

Mean fundamental

Mean of a dataset cannot less than the least value (Here it is ‘4’- Karren’s score), and more than highest value (Here it is ‘20’- Stacey’s score). So the answer is NO.

Approach 5 - Inference approach

Inference through New mean

For new mean to absorb an increment of minimum 2 points over the current mean, the current mean has to be much less than 12, because after an increment of 2 or more points, new mean MUST be less than 12.

Let’s observe what could be the probable value of current mean?

Image

Let’s look at the Lawn tennis column. From Karren till Joe, there are total 8 data points. Their mean would be equal to median, because they have an equal interval of 2 points successively.
Now, the median of these 8 data points is 11 [(10+12)/2]. So mean would also be equal to 11. Rest of the data points are much higher than 11, so current mean would be more than 11. We already concluded earlier that new mean MUST be less than 12. If current mean is more than 11, then after an increment of 2, it can never be less than 12. So the answer is NO.

Hope this helps build your fundamentals. Do attempt challenging MSR question posted today.

-Shalabh
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Re: Compare Mean and Median in less than 20 seconds [#permalink] New post 21 Mar 2013, 03:19
I like the 1st and the 3rd approach. ( I could not figure out the SMART way to compute the mean, I missed that data values are at regular interval for some values, It would have made calculations much easier & faster)

Though time consuming, they give the SURE SHOT answer and there is no need to check again.

I tried with other approaches but had to recheck my observations again and again to make sure I had not made any mistake. (this method took more time than the first method)

P.S: Since I can do calculations quite fast, I prefer the 2nd (SMART conventional) method.

-Nishant
Re: Compare Mean and Median in less than 20 seconds   [#permalink] 21 Mar 2013, 03:19
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