Data Interpretation – Made Easy

Ask an MBA aspirant about the types of questions the Data Interpretation (DI) section of CAT contains and most will answer correctly that it contains graphs and tables. Ask them a follow-up to this question, viz. “What kinds of skills are tested by this section?” Most students will now flounder. Some will say that it tests your ability to read graphs, while others will opine that it evaluates your capability to make quick calculations. Their answers often lack an incisive understanding of the real purpose of this section.

Likewise, ask students how good they are in the DI section. Most will reply that it is an easy section (compared to the dreaded Problem Solving section or the Verbal Ability and Reading Comprehension for some) but it takes time to do calculations. This kind of answer indicates the faulty approach most students often take while dealing with this section.

Listening to feedback from students about their performance in this section soon after CAT, projects a new dimension. Most students will claim that they have done fairly well while the reality may be otherwise. They get a shock when they come to know about the actual answers and blame their bad luck for their appalling performance.

Such questions and answers give a clear indication that most students have not, unfortunately, understood the purpose of this section and the approaches to develop proficiency in this section. Without such an understanding, attempting DI problems is like embarking on a journey without knowing your destination and the means of transport.

This 2-series article aims to explain the purpose of DI questions and provide a structured approach to speed-building.

Unraveling the Purpose of Data InterpretationData Interpretation section virtually puts a student in the shoes of a Business Manager, who is inundated with useful (and not so infrequently with useless) data and has to make some quick interpretation of the data to reach important decisions. Once presented with graphs or tables, the mental processor of a manager starts whirring at the top gear and selects, prunes, manipulates, and compares the data and reaches certain conclusions. Successful operations of these processes require that a manager possesses qualities like:

o An eye for detail,o An ability to focus on key issues quickly,o An ability to work on numbers in different ways,o An ability to see a trend,o An ability to identify exceptional situations,o An ability to work with logical relationships and o An ability to reach a conclusion using deductive logic.

Let us see how some of these qualities are tested using a set of questions picked up from CAT 2001.

Question 1 to 3There are six companies, 1 through 6. All of these companies use six operations, A through F. The following graph shows the distribution of efforts put in by each company in these operations.

A, 17.7 A, 18.5 A, 16.4 A, 18.5 A, 15.1 A, 15.6

B, 17.6 B, 16.7 B, 16.4 B, 10.3 B, 16.1 B, 17.6

C, 9.3 C, 10.9C, 8.2

C, 13

D, 15.7D, 7.4

D, 16.3

D, 11.2D, 7.7

D, 11.8

E, 23.5

E, 25.9E, 21.8

E, 28.6E, 28.6

E, 23.6

F, 15.7F, 22.2 F, 18.2 F, 23.4 F, 19.7 F, 17.6

C, 13.8C, 9.8

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6

Company

% D

istr

ibu

tio

n o

f E

ffo

rt

Q1. Suppose effort allocation is interchanged between operations B and C, then C and D, and then D and E and if companies are then ranked in ascending order of effort in E, what will be the rank of company 3?

1. 2 2. 3 3. 4 4. 5

Q2. A new technology is introduced in company 4 such that the total effort for operations B through F gets evenly distributed among these. What is the change in the percentage of effort in operation E?

1. Reduction of 12.3 2. Increase of 12.33. Reduction of 5.6 4. Increase of 5.6

Q3. Suppose the companies find that they can remove operations B, C and D and re-distribute the effort released equally among the remaining operations. Then, which operation will show the maximum effort across all companies and all operations?

1. Operation E in company 1 2. Operation E in company 43. Operation F in company 5 4. Operation E in company 5

Answer to Question 1After the first interchange, efforts of operation C become equal to that of the operation B and vice versa. In the subsequent interchange between C and D, the efforts of C, which in the question were of B, were allocated to D. In the last inter-change, efforts of E became equal to that of the D. Thus in effect, the efforts of operation E after 3 inter-changes are equal to that of the operation B before the start of inter-changes. In effect, the question requires ranking of Company 3 if efforts in operation B are arranged in an ascending order. The answer to the question is 3. The question is thus testing logical reasoning. The fact that a graphical representation has been used to test this skill is almost like a clever decoy placed by the examiner. Successful CAT takers learn the knack of quickly identifying the root of the problem, rather than getting caught up in unnecessary details.

Answer to Question 2Since operation E has the highest allocation, redistribution of efforts equally among operations B to F will reduce the allocation of operation E. Hence, answer choices 2 and 4 are eliminated. Now, suppose the correct answer is ‘reduction of 5.6’, then allocation to E as well as to operations B, C, D and F would be 23% each. This is not feasible. Hence the correct answer is ‘reduction of 12.3’. An alternative approach could have been that allocation to operation A is 18.5. Therefore, allocation to all other efforts is 81.5 (100–18.5). Dividing this equally among 5 operations allocates 16.3 to each activity. Hence the reduction is 12.3 (28.6-16.3). The question therefore requires not just logical reasoning but also the ability to work on numbers in different ways.

Answer to Question 3This question can also be handled with equal ease. Compare operation E with operation F in company 5. The effort allocation is higher for operation E than for F, and will remain so after redistribution of efforts of operation B, C and D equally among the remaining operations. Choice 3 is therefore eliminated. Let’s now compare the answer choices 2 and 4. In both cases, operation E requires 28.6% effort. By visual inspection, we can see that the combined efforts of operations B, C and D in company 5 are higher than that of in company 4. It implies that allocation will be more in company 5 than in the company 4 and thus choice 2 is also eliminated. Between operation E of company 1 and operation E of company 5, the latter has about 5% more allocation before redistribution. If the choice 1 is to be the correct answer, the cumulative allocation of B, C and D for company 1 should exceed that of company 5 by 15%. Since that is not so (once again by visual comparison of allocations), the correct answer is operation E in company 5. Dear CAT aspirant, which of the qualities listed above are thus tested here?

It must be evident to you now how certain qualities are tested in DI. The good news is that these are innate qualities in most people but the bad news is that they do not know how to use them. Quite unfortunately, the plethora of training and coaching institutes also have not understood the real reason for the DI section and inundate students with non-essential short-cuts and techniques.

Let’s now work step by step for unleashing these qualities and honing them to develop mastery on data interpretation.

Speed Builder-1: Master RatiosMost questions on DI require computation of ratios, percentages, rate of change, average, maximum, minimum, count or data sorting, of which, ratio is the most common and challenging operation. Some techniques for their quick estimation are presented below. It’s important to note that most questions do not require precise calculations. What is really required is an ability to estimate approximate values quickly.

Technique-1 – Effective if numerator > > denominator Round off numerator to nearest 1, 10 or 100 depending upon the magnitude of its value, find quotient and compare.

Example: Which ratio is greater

or

?

and its quotient = 46

and its quotient = 40

Hence, the first ratio is bigger.

Technique-2 – Effective if numerator < < denominatorInverse all fractions and use Technique-1. The fraction with lowest quotient is biggest and the one with highest quotient smallest.

Technique-3 – Effective if numerator < denominatorFind the differences between the numerators of two fractions and similarly between the denominators.If the difference between numerators is greater than or equal to the difference between denominators, then the fraction with higher value of numerator is bigger.

Example: Compare

with

.

Here the difference between the numerators of both fractions is 11 and between the

denominators is 10. Hence

is the higher of the two numbers.

Similarly, between

and

, the latter is bigger because differences between

numerators and denominators are 18 each.

Technique-4 – Significant difference between the numerator and denominator values of two fractionsMultiply numerator and denominator of fraction with small values with an appropriate number and then compare

Example: Compare

with

.

Multiply numerator and denominator of the second fraction by 10. After multiplication,

=

, which is less than the first fraction, hence the

second

fraction is smaller.

The above-mentioned techniques can also be applied to percentages because a percent is a fraction multiplied with 100.

Speed Builder -2: Think SmartSome smart thinking can make solving DI problems very simple for the students taking the CAT. In my teaching at Career Plan, this is what I stress upon the most. I feel it is important to eliminate some myths about how to prepare oneself for this section. The best way to illustrate this approach is through an example taken from CAT 2002.

Question 4 and 5Answer the questions based on the pie charts given below:

Chart-1 shows the distribution by value of top 6 suppliers of MFA Textiles in 1995. Chart-2 shows the distribution by quantity of top 6 suppliers of MFA textiles in 1995. The total value is 5760 million Euros (the European currency). The total quantity is 1.055 million tonnes.

Q4. The country, which has the highest average price, is1. USA2. Switzerland 3. Turkey 4. India

Q5. The average price in Euro/ kg for Turkey is roughly 1. 6.20 2. 5.60 3. 4.20 4. 4.80

Answer to Question 4Conventional approach to solving the problem is to compute the average prices of all 4 countries and compare the values.

Pakistan, 12%

China, 15%

Turkey, 16%USA, 17%

Switzerland, 20%

India, 20%Pakistan, 16%

China, 17%

Turkey, 15%USA, 15%

Switzerland, 11%

India, 26%

USA =

Switzerland =

Turkey =

India =

You would note that all values contain a factor of . This factor can be ignored

because you are required to identify the country having the highest average value, not the

average value itself. Hence, you have to just compare

,

, and

. Obviously,

Switzerland gets the highest average price. Smart thinkers can easily solve this problem

by observation without writing even a single number on paper.

Answer to Question 5This question can also be answered by some quick mental calculations. The overall

average price for all countries is about 5.5 ( ). For Turkey, share of value to

share of quantity is

, which is slightly more than 1. Average price for Turkey should

be a multiplication of these two ratios. Hence, the answer will be roughly 5.60.

Clearly, the examiner is not testing how quick you are with a series of calculations, but wants to find out those who use a combination of common sense and smart thinking to quickly and efficiently arrive at the final answer. This is their way of finding future managers and leaders, who will always need to take the right decisions under severe time and resource constraints.

Question 6 to 7 (Source: CAT 2004)The profitability of a company is defined as the ratio of its operating profit to its operating income, typically expressed in percentage. The following two charts show the operating income as well as the profitability of six companies in the financial years (F.Y.s) 2001-02 and 2002-03.

Operating Income

0

50

100

150

200

250

300

350

A B C D E F

Company

Ope

ratin

g In

com

e (C

rore

s)

FY 01-02 FY 02-03

Q6. Which company recorded the highest operating profit in F.Y. 2002-03?1. A 2. C 3. E 4. F

Q7. The average operating profit in F.Y. 2002-03, of companies with profitability exceeding 10% in F.Y. 2002-03, is approximately:

1. 17.5 crore 2. 25 crore 3. 27.5 crore 4. 32.5 crore

Answer to Question 6Profit is operating income x profitability. Comparing company A with company F, both income and profitability of F are higher, hence Choice (1) is eliminated. Compare company F with company E, income of E is slightly lower but its profitability is almost twice. Choice (4) is therefore eliminated. Comparison between C and E requires a closer call. Income of E is only marginally lower than C, but its profitability is distinctively

Profitability

-5%

0%

5%

10%

15%

20%

25%

A B C D E F

Company

FY 01-02 FY 02-03

higher (Profitability of C 15% and of E 17%, hence % difference =

>

10%). Hence, company E has the highest income.

Answer to Question 7Companies with profitability higher than 10% are C and E. C’s profits are 200 × 15% Rs. 30 cr. The profit of E is higher than that of C. Hence average should be more than Rs. 30cr. Therefore the correct answer is (4).

What are some of the key lessons from these examples? How can this learning be used to build up your speed? Quite obviously, what is required is an ability to look at problems from different angles and an ability to simplify a seemingly difficult question. If you can develop these abilities, you performance will improve dramatically in DI. This ability is what we at Career Plan call “Think Without Ink” or TWI. The tools and principles of TWI are also available in the form of a book authored by Mr. K. Venkataraman, Founder of the TWI Foundation and a Director at Career Plan. I would strongly recommend this book to those students who have a mortal fear of numbers and a still want to do well in exams like the CAT!

Speed Builder – 3: Logical ReasoningCAT questions increasingly require logical reasoning rather than the ability to perform some quick calculations and intense number crunching. Very often, the questions require students to understand the implications of the conditions imposed, observe the relationships between data, and use deductive logic. Two examples taken from recent CATs are presented here for illustration.

Question 8 to 10 (Source: CAT 2003)The table below provides certain demographic details of 30 respondents who were part of a survey. The demography characteristics are: gender, number of children, and age of respondents. The first number is each cell is the number of respondents in that group. The minimum and maximum age of respondents in each group is given in brackets. For example, there are 5 female respondents with no children and among these five, the youngest is 34 years old, while the oldest is 49.

Q8. The percentage of respondents aged less than 40 years is at least. 1. 10% 2. 16.67% 3. 20.0% 4. 30%

No of Children Male Female Total

0 1 (38, 38) 5 (34, 49) 61 1 (32, 32) 8 (35, 57) 92 8 (21, 65) 3 (37, 63) 113 2 (32, 33) 2 (27, 40) 4

Total 12 18 30

Q9. Given the information above, the percentage of respondents older than 35 can be at most.

1. 30% 2. 73.33% 3. 76.67% 4. 90%

Q10. The percentage of respondents that fall into the 35 to 40 years age group (both inclusive) is at least

1. 6.67% 2. 10% 3. 13.33% 4. 26.67%

Answer to Question 8The condition “number of respondents aged less than 40 years is at least” can be interpreted as:

If minimum and maximum age in a group <40, then all respondents in the group meet the condition

If minimum age <40 and maximum >40, then at least 1 respondent meets the condition

If both minimum and maximum ages >40, then no respondent meets the condition

Therefore, number of respondents meeting the above criteria is 9 as shown in the Table below. Hence 30% is the correct answer.

Answer to Question 9The condition “maximum number of respondents older than 35” can be interpreted as:

If minimum and maximum age <= 35, then no respondent meets the condition If minimum <= 35 and maximum > 35, then one less than the number of

respondents in the group meet the condition If minimum and maximum > 35 then, all respondents meet the condition

Therefore, correct answer is 76.67%.

Answer to Question 10The condition “minimum number of respondents that fall into the 35 to 40 years age group (both inclusive)” can be interpreted as:

No. of children

Male Female

Total = 9

0 1 11 1 12 1 13 2 1

No. of children

Male Female

0 1 41 0 72 7 33 0 1 Total = 23

If both minimum and maximum ages are < 35 or > 40 then no respondent meets the condition

If both minimum and maximum ages are 35 to 40 then all respondents meet the condition

If minimum < 35 and maximum between 35 to 40 then at least 1 respondent meets the condition

If minimum is between 35 to 40 and maximum > 40 then at least 1 respondent meets the condition

Lets look at one more set of questions, this time from CAT 2004.

Question 11 to 12 (Source: CAT 2004)In a Decathlon, the events are 100m, 400m, 100m hurdles, 1500m, High jump, Pole-vault, Long jump, Discus, Shot put and Javelin. The performance in the first four of these events is consolidated into Score-1, the next three into Score-2, and the last three into Score-3. Each such consolidation is obtained by giving appropriate positive weights to individual events. The final score is simply the total of these three scores. The athletes with the highest, second highest and the third highest final scores receive the gold, silver and bronze medals, respectively. The table below gives the scores and performances of nineteen top athletes in this event.

Name Country Final Score Score-1 Score-2 Score-3 100m High jump Pole-vaultEduard Hamalainen BLS 8802 491 5322 2989 10.74 2.08 4.8Michael Smith CAN 8855 174 5274 3407 11.23 1.97 4.9Tomas Dvorak CZE 8796 499 5169 3128 10.63 1.91 4.7Uwe Freimuth DDR 8799 441 5491 3124 11.06 1.97 4.8Torsten Voss DDR 8880 521 5234 2868 10.69 2.10 5.1Erki Nool EST 8768 408 5553 2808 10.71 1.99 5.4Christian Plaziat FRA 8775 563 5430 2781 10.72 2.10 5.0Jurgen Hingsen FRG 8792 451 5223 3033 10.95 2.00 4.9Siegfried Wentz FRG 8856 470 5250 3137 10.85 2.05 4.8Guido Kratschmer FRG 8861 575 5308 3064 10.58 2.00 4.6Daley Thompson GBR   582   3003 10.55 2.11 4.6Frank Busemann GER 8905 568 5392 2945 10.60 2.04 4.8Alexandr Apaichev SOV 8803 492 5370 3115 10.92 1.95 4.8Grigory Degtyarov SOV 8823 339 5196 3114 11.05 2.08 4.9Robert Zmelik TCH 8832 494 5455 2883 10.78 2.06 5.1Dave Johnson USA 8811 366 5370 3114 10.78 2.10 5.0Steve Fritz USA 8827 427 5163 3119 10.75 2.04 5.0Bruce Jenner USA 8846 483 5280 3200 10.94 2.03 4.8Dan O'Brien USA 8897 408 5331 3120 10.36 2.09 4.8

No. of children

Male Female

0 1 01 0 12 0 13 0 1 Total = 4

Q11. What is the least that Daley Thompson must get in Score-2 that ensures him bronze medal?

1. 5309 2. 5296 3. 5271 4. 5270

Q12. At least how many competitors (excluding Daley Thompson) must Michael Smith have out-jumped in the long jump event?

1. 1 2. 2 3. 3 4. 4

Answer to Question 11Highest, second highest and third highest scores are 8905, 8897 and 8880 respectively. Therefore Daley Thompson score should be just higher than 8880 to qualify him for the Bronze medal. The least he should get in Score–2 is 5296 (8881 – 582 – 3003).

Answer to Question 12The weightage assigned to High jump, Pole vault and Long jump events to compute Score–2 are not known. Therefore, if Michael Smith got higher score either in High jump or Pole Vault compared to other athletes, we can not say whether he out-jumped in the Long jump or not. However, if his score is lower than or equal to High jump and Pole Vault scores of some other athletes and yet he scored higher Score–2, we are sure that must have out jumped in Long jump. Therefore, this question can be inferred as: compared to how many athletes, Michael Smith has a higher Score-2 despite lower than or equal to score in both High jump and Pole vault. These athletes are Torsten Voss, Jurgen Hingsen, Grigory Degtyarov and Steve Fritz. The correct answer therefore is 4.

It should be clear to you with these examples that the examiner is using a lot of data to disguise the true intent of the question, viz. to determine your critical thinking ability under the pressure of time. You must spend more time developing this thinking rather than solving a thousand questions in the conventional mode of thinking.

Speed Builder-4: Understanding the structure of a question.Having learnt techniques of quick computation of ratios, approaches to smart thinking and importance of logical reasoning, the stage in now set for developing a deeper understanding of the structure of problems. This understanding enables the student to get to the root of the problem, assess its difficulty level and select the most effective way of reaching the right answer.

A question may typically require the following or permutations of similar tasks:

Selection of data meeting specified criteria

Some mathematical operation on selected data

Second stage selection or operation.

The criteria may be specified using mathematical operators like =, <, , >, , , not less than, not more than, etc. Use of logical operators like AND, OR & NOT is also very common.Let me illustrate the structure of a question using a set of problems.

Question 13 to 14 (Source: CAT 2002)The following table provides data on different countries and location of their capitals, (the data may not match the actual Latitudes, Longitudes). Answer the following questions on the basis of this table.

Q13. What percentage of cities located within 10˚ E and 40˚ E (10 – degree East and 40 degree East) lie in the southern hemisphere?

1. 15% 2. 20 % 3. 25% 4. 30%

Q14. The ratio of number of countries whose name starts with vowels and located in the Southern hemisphere, to the number of countries, the name of whose capital cities starts with a vowel in the table above is:

1. 3:2 2. 3:3 3. 3:1 4. 4:3

Answer to Question 13Task-1: Select countries using the criteria: (City with longitude > 10˚E) AND

(city with longitude < 40˚E)

Task-2: Count (No. of selected countries)

S.No Country Capital Latitude Longitude1 Argentina Buenos Aires 34.30 S 58.20 E2 Australia Canberra 35.15 S 149.08 E3 Austria Vienna 48.12 N 16.22 E4 Bulgaria Sofia 48.45 N 23.20 E5 Brazil Brasilia 15.47 S 47.55 E6 Canada Ottawa 45.27 N 75.42 E7 Cambodia Phnom Penh 11.33 N 104.55 E8 Ecuador Quito 0.15 S 78.35 E9 Ghana Accra 5.35 N 0.6 E

10 Iran Teheran 35.44 N 51.30 E11 Ireland Dublin 53.20 N 6.18 E12 Libya Tripoli 32.49 N 13.07 E13 Malaysia Kuala Lumpur 3.9 N 101.41 E14 Peru Lima 12.05 S 77.0 E15 Poland Warsaw 52.13 N 21.0 E16 New Zealand Wellington 41.17 S 174.47 E17 Saudi Arabia Riyadh 24.41 N 46.42 E18 Spain Madrid 40.25 N 3.45 W19 Sri Lanka Colombo 6.56 N 79.58 E20 Zambia Lusaka 15.28 S 28.16 E

Task-3: Select a subset of the above set using criteria (suffix in latitude = S)

Task-4: Count (No. of elements in the subset)

Task-5: Compute percentage of second count with respect to first count.

Answer to Question 14Task-1: Select set-1 using criteria (First letter of country name =vowel) AND

(Suffix in Latitude = S)

Task-2: Count-1 (No. of selected countries)

Task-3: Select set-2 using criteria (first letter of capital =vowel)

Task-4: Count-2 (No. of selected cities)

Task-5: Ratio of count-1 to count-2 Once the question structure is understood, answering a question becomes much easier. Isn’t it?

Knowing the question structure also helps in understanding the difficulty level of a question. An examiner can make questions difficult by using one or more of the following approaches:

o Using many conditions in defining the selection criteria.

o Using negative operators like , not less than, not more than, etc.o Using operators like ‘Closest’, which require judgment.o Using one or more logical operator like AND, OR & NOT.o Increasing the number of mathematical operations.o Using many sets of selection criteria.

Students must develop capabilities to understand the structure of questions and observe how questions are made difficult. In general, questions requiring selection of true or false statements are difficult because each answer choice may use a different selection criterion. Such questions may be skipped.

Once the structure is understood, students must first evaluate whether the criteria can be simplified without changing its meaning. Question 12 described earlier is a good example of this approach.

In the next stage, student should apply the simplified criteria to the data set to make a selection. If the criteria are made of two sub-criteria connected to each other by AND operator, apply that sub-criteria first, which is met by fewer elements of the data set. On the other hand if the sub-criteria are connected by an OR operator, use that sub-criteria first which is met by more elements of the data set.

Most operations, particularly Max, Min, Count, etc. can be conducted simultaneously while criteria are being applied to select data elements. This activity, of course, requires some degree of parallel processing in students’ minds, which most students unfortunately lack due to absence of any practice. Fortunately, this skill can be developed with some practice under the guidance of a good tutor.

Many readers who have read this article this far may ask, ‘Is it really necessary to go to this extent of insight to solve such questions quickly’. My answer is an unqualified “YES”!. This is a very effective approach, particularly for questions based on the tabular data. In the recent years, majority of the questions are based on tabulated data rather than on graphs. If you can develop the mindset that allows you to solve these problems in less than 30 seconds each, you can be assured of a high percentile score in the CAT. I will cover some other techniques for problem solving in the next few issues.