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Hi folks,
As we experience, life is all about laying priorities & importance to the things that affect our life. ‘Importance’ in mathematical parlance is referred to as‘Weight’.
It depends on an individual ‘how much weight one places on making a choice?’
Be it choosing a car or a would-be-spouse, we assign different weights to different attributes. Hope you guys don’t mind if I discuss this aspect using an example of buying a car instead of selecting a would-be-spouse.
Say John wishes to buy a car. He lays importance to 2 attributes—Mileage & Power. He narrows down his choice to two brands of car.
Assume on a score of 10, Brand A scores 8 on mileage & brand B scores 6. While on a similar scale; Brand A scores 6 on Power & brand B scores 8.
It is obvious that if John weighs Mileage & Power equally, then it is difficult for him to choose one brand of car over the other. While Brand B scores better in case of Power, Brand A excels in terms of mileage.
Practically speaking, we seldom lay equal importance to all attributes. For people who really care for the weight of their pocket, mileage will be more important than power, while for adventure loving people, it will just be the reverse.
Our friend John is an adventure loving guy, who reckons Power to be relatively more important than Mileage. He lays 60% weight to Power and 40% to Mileage. So here, the scenario will look like this.
Now, in this case, John will clearly prefer Brand B over Brand A since Brand B scores more in terms of Power, which is a more important attribute for him.
But David, another friend of ours, shares a different opinion than John’s. Because he drives a lot on a daily basis, he sets his priority to the brand that gives him value-for-money.
He lays 40% weight to Power, & 60% to Mileage. So for him, the scenario will look like this.
It is quite obvious that David will buy brand A rather than brand B.
In this world, there are varied needs of individuals, hence there are multiple brands that co-exist due to different weights assigned by each individual to different needs.
So let’s see how John rates the two brands for cars mathematically. For brand A, contribution of mileage is 40% of 8, equaling to 3.2. Similarly, contribution of power is 60% of 6, equaling to 3.6. So John rates Brand A at 3.2 + 3.6 = 6.8.
For Brand B, contribution of mileage is 60% of 8, equaling to 4.8 and of power is 40% of 6, equaling to 2.4. So John rates brand B as 4.8 + 2.4 = 7.2.
Now, these final rates of both brands of cars are in fact called ‘Weighted Average’.
Well friends, life is not that easy. We frequently come across situations, where a choice may involve multiple attributes with varied importance.But the good news is that even with such plethora of choices, we can easily derive “Weighted Average” with the help of this formula.
Consider a dataset with n attributes with their values:
The weighted average would be calculated as follows.
Now let’s consider an example with more than 2 attributes. Say another person, Joe, considers 5 attributes to buy a car; Mileage, Power, Cost, Service, & Safety.
He assigns relative weights to each attribute as given in the table below (The table also mentions the ratings of 3 brands based on these attributes).
For Joe, the final rating for Brand A would be = (8 *20 + 6 *20+8 *15 + 5 *15+6*30 )/100= 6.55
Similarly, brand B rates 6.7 and brand C rates 6.1.
Let’s orient our thinking as to how GMAT tests this concept. Let’s discuss one dataset which uses Weighted Average concept in a slightly different way.
‘For its 4-year Bachelor of Engineering (BE) program, the college proposes to the Student Union two-options to determine final score. In these two approaches, the college wishes to weigh each year marks differently.
Given below are the two suggested options along with sample percentage marks of a student for each year.’
For each of the following statements, select ‘Yes’ if statement is true based solely on the information provided in the table; otherwise select ‘No’.
1. If sample marks are taken into account, then Option 1 is more favorable to the Student Union.
2. If sample marks are taken into account, then calculating the final score as the median of marks is more favorable to the Student Union than opting for Option 1.
3. Students who expect to score higher marks in their 4th Year as compared to other 3 years should prefer Option 1 over Option 2.
Understand the dataset
Let’s first understand the data set. When we look at the dataset, we should register following aspects in our mind.
The data set is about a 4-year college program. As the table shows, marks are allocated each year. The college has suggested two approaches to determine the final score of students. The options vary in terms of the weightage assigned to marks each year to calculate the final marks of each student. We have also been provided with 1 set of marks for a student.
Let’s just view a few data points. Weight-age means importance given to a certain value. In the present example, we can infer that 1st year sample marks carries an importance of 10% only compared to 50% for 4th year marks as per option ‘1’. Note, both columns 3 and 4 individually add up to 100%. This means that for computation of final marks, first year contribution is 7.5(10% of 75), whereas 4th year contribution is 41(50% of 82).
Another interesting aspect is that we can expect students who score less in final year to prefer option 2 over option 1. It is because option 2 lays 10 percentage points more weightage than option 1 for the marks earned in the 4th year.
Question 1 Statement
If sample marks are taken into account, then Option 1 is more favorable to the Student Union.
Understand the Question
Let's understand the first question statement. We have to find whether this statement is true or false per the information in the dataset. Now this question statement does not directly provide us with the mathematical relationship. We need to contextualize this statement.
What will the Student Union or students favor more? It will be the option that helps them score higher final marks given their scores in each of the 4 years. So clearly, the answer to this question statement is YES if the final score per option 1 is greater than the final score per option 2. The mathematical equation is presented below.
Is Final Score|Option 1 > Final Score|Option 2 ?
And we need to calculate this final score using the sample % marks that we have been provided in the table.
Approach
1. Calculate relative weightage percentage points for option 1 over option 2
a. Column 5 =Column 3 - column 4
2. Calculate relative more/less Final Score of option 1 over option 2
a. Column 2 & 5, & Apply
3. If calculated value in step 2.a is more than 0, answer is YES, else NO.
Solve-Apply Approach
So now let’s come up with the approach to solve this question.
Let’s understand the question again; we need not calculate the final scores via option 1 & 2.
The question asks us ‘Is option 1 final score relatively more than final score via option 2?’
So we simply calculate how much option 1 weight-age is more or less than option 2 relatively. This will save our precious time of calculating final scores twice i.e. final score for both the options, & then compare their scores.
Let’s put our findings in column 5.
Relative more or less Final year score =
We could have excluded year 1 calculation, as the relative weight-age of option 1 over option 2 is 0. It will not have any impact on final score calculation.
0% means that while calculating final year marks, option 1 does not weigh more or less compared to option 2. Final scores via both the options are equal. Hence the answer to this question statement is NO.
Should you wish to look at an interactive audio visual solution of this question, & the subsequent 2 more questions, you may click at ‘START FREE TRIAL’.
We realize that it is the time to address the need of the test takers to attempt variety of IR questions that are true reflection of Official Guide questions.
e-GMAT has launched 15+ hours of interactive audio visual content on IR. It encompasses 35 concepts, 120+ original questions, and 2 full length Mock Tests.
We appreciate if you attempt the question, & provide us feedback. We at e-GMAT constantly strive for excellence in learning pedagogy.
Here we sign off with 2 questions for you to attempt ( Given in dataset).
Happy Reasoning!
-Shalabh Jain
Attachments
Wt. Average article.pdf [1.06 MiB]
Downloaded 986 times
Updated on: 24 Oct 2019, 06:48
Originally posted by egmat on 22 Jan 2013, 10:50.
Last edited by Sajjad1994 on 24 Oct 2019, 06:48, edited 5 times in total.
Last edited by Sajjad1994 on 24 Oct 2019, 06:48, edited 5 times in total.
Updated - Complete topic (144).
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Compare Mean and Median in less than 20 seconds
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.’
For each of the following statements, select Yes if the statement is true based on the information provided; otherwise select No.
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:
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”)
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
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.
Just observing this data set, we can be confident that the median will fall between 25 and 31.
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.
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.
OTHER SCENARIO
Few top cell values are disproportionately low and bottom cell values are disproportionately high
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.
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.’
For each of the following statements, select ‘Yes’ if statement is true based solely on the information provided in the table; otherwise select ‘No’.
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
Attachments
Table 2.xlsx [10.5 KiB]
Downloaded 964 times
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.
Fame
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.
Fame
