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Data Interpretation: Tables, Charts and Graphs Notes for GATE DA

Data Interpretation: Tables, Charts and Graphs notes for GATE DA: 12 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice q

data interpretation tables charts and graphs notes

Chapter Roadmap: Data Interpretation

.smart-card { font-family: system-ui, -apple-system, sans-serif; color: var(--color-text); background: var(--color-bg); padding: 1.5rem; border-radius: 1rem; border: 1px solid var(--color-border); } .roadmap-step { opacity: 0; animation: fadeInStep 0.5s forwards; margin-bottom: 1.5rem; padding-left: 1.5rem; border-left: 2px solid var(--color-primary); position: relative; } .roadmap-step::before { content: ""; position: absolute; left: -6px; top: 0; width: 10px; height: 10px; border-radius: 50%; background: var(--color-primary); } .roadmap-step:nth-child(1) { animation-delay: 0.1s; } .roadmap-step:nth-child(2) { animation-delay: 0.3s; } .roadmap-step:nth-child(3) { animation-delay: 0.5s; } @keyframes fadeInStep { to { opacity: 1; } } .step-title { font-weight: 700; font-size: 1.1rem; margin-bottom: 0.5rem; color: var(--color-primary); } .step-detail { font-size: 0.95rem; color: var(--color-muted); line-height: 1.5; margin: 0.25rem 0; } .mastery-goal { margin-top: 1.5rem; padding: 1rem; background: color-mix(in srgb, var(--color-primary) 10%, transparent); border-radius: 0.5rem; font-style: italic; font-size: 0.95rem; } Chapter Journey: Data Interpretation Step 1: Chart-Based Percentage and Average Interpretation Focus: Pie charts, bar graphs, and line charts. Core Skill: Converting visual proportions into exact percentages and weighted averages. Weightage: High (Foundation for all DI questions). Step 2: Grid and Matrix Data Counting Focus: 2D arrays, pixel intensity matrices, and tabular conditions. Core Skill: Systematic counting under multiple simultaneous constraints. Weightage: Moderate (Requires careful, methodical checking). Mastery Goal: You will learn to bypass visual deception and extract raw numerical truth from any chart or grid in under two minutes.

Topic Hero: The Core of Chart Interpretation

.smart-card { font-family: system-ui, -apple-system, sans-serif; color: var(--color-text); background: var(--color-bg); padding: 1.5rem; border-radius: 1rem; border: 1px solid var(--color-border); } .core-principle { display: flex; flex-direction: column; align-items: center; text-align: center; margin: 1.5rem 0; padding: 1.5rem; background: color-mix(in srgb, var(--color-primary) 8%, transparent); border-radius: 0.75rem; border: 1px solid var(--color-border); } .mapping-text { font-size: 1.1rem; font-weight: 600; color: var(--color-primary); margin: 0.5rem 0; } .mapping-list { text-align: left; width: 100%; margin-top: 1rem; font-size: 0.95rem; } .mapping-list div { margin-bottom: 0.5rem; padding-left: 1rem; border-left: 2px solid var(--color-muted); } .golden-rule { margin-top: 1.5rem; padding: 1rem; background: color-mix(in srgb, var(--color-text) 5%, transparent); border-radius: 0.5rem; font-size: 0.95rem; line-height: 1.5; } .golden-rule strong { color: var(--color-primary); } The Core Principle Every chart interpretation problem reduces to a single mapping: Visual Feature \leftrightarrow Numerical Value Pie Chart: Angle or Area \leftrightarrow Percentage of Total Bar/Line Chart: Height or Position \leftrightarrow Absolute Value or Frequency The Golden Rule: Always identify the denominator (the total or base) before calculating any percentage or average. Charts often omit the total, requiring you to deduce it from a known segment's value and its visual proportion.

Computing Averages from Frequency Charts

.smart-card { font-family: system-ui, -apple-system, sans-serif; color: var(--color-text); background: var(--color-bg); padding: 1.5rem; border-radius: 1rem; border: 1px solid var(--color-border); } .method-box { margin: 1rem 0; padding: 1rem; background: color-mix(in srgb, var(--color-primary) 8%, transparent); border-radius: 0.5rem; } .step-list { list-style: none; padding: 0; margin: 1rem 0; } .step-list li { opacity: 0; animation: fadeInStep 0.4s forwards; padding: 0.75rem; margin-bottom: 0.5rem; background: color-mix(in srgb, var(--color-text) 4%, transparent); border-radius: 0.375rem; font-size: 0.95rem; border-left: 2px solid var(--color-primary); } .step-list li:nth-child(1) { animation-delay: 0.2s; } .step-list li:nth-child(2) { animation-delay: 0.4s; } .step-list li:nth-child(3) { animation-delay: 0.6s; } .step-list li:nth-child(4) { animation-delay: 0.8s; } .step-list li:nth-child(5) { animation-delay: 1.0s; } @keyframes fadeInStep { to { opacity: 1; } } .step-num { font-weight: 700; color: var(--color-primary); margin-right: 0.5rem; } The Weighted Mean Method For a chart displaying values x_{i} with corresponding frequencies f_{i} : \overset{ˉ}{X} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{\sum _{i = 1}^{n} f _{i}} Step-by-step execution: 1. Create a mental or scratchpad table of x_{i} and f_{i} . 2. Compute the product f_{i} x_{i} for each row. 3. Sum the products to get the grand total of all values. 4. Sum the frequencies to get the total number of observations. 5. Divide the grand total by the total observations.

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Question 1

Level 1: Warm-up

If the average value of a dataset X is 10, and a new dataset Y is created using the linear transformation Y = 2 X + 3, what is the average value of Y ?

Question 2

Level 1: Warm-up

If the average of a dataset X is \overset{ˉ}{X}, and a new variable Y is defined by the linear transformation Y = 4 X + 7, what is the average of Y ?

Question 3

Level 1: Warm-up

When interpreting a pie chart that explicitly shows the "share of valid votes", what is the correct base value to use for percentage calculations?

Question 4

Level 1: Warm-up

In an election data interpretation problem, if a pie chart represents the "share of valid votes", what is the correct base value to use when calculating a candidate's actual votes from their percentage?

Question 5

Level 1: Warm-up

When calculating the average number of patients per shift from a bar chart showing the frequency of different patient counts, what is the essential first step in the computation?

Question 6

Level 1: Warm-up

If the average of a dataset X is \overset{ˉ}{X}, and a new dataset Y is formed by the transformation Y = c X (where c is a constant), what is the average of the new dataset \overset{ˉ}{Y} ?

Question 7

Level 1: Warm-up

A pie chart represents the distribution of a dataset. If a specific category accounts for 15% of the total, what is the central angle of its sector in degrees?

Question 8

Level 1: Warm-up

A pie chart represents a complete dataset. What is the central angle in degrees for a sector that accounts for 30% of the total?

Question 9

Level 1: Warm-up

Which mathematical expression correctly represents the weighted average \overset{ˉ}{X} from a frequency chart, where x_{i} is the value and f_{i} is its frequency?

Question 10

Level 1: Warm-up

A frequency distribution shows value x_{1} = 5 with frequency f_{1} = 4, and value x_{2} = 10 with frequency f_{2} = 2 . What is the value of the numerator \sum f_{i} x_{i} required for the weighted average formula?

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Data Interpretation: Tables, Charts and Graphs Notes for GATE DA

Data Interpretation: Tables, Charts and Graphs notes for GATE DA: 12 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Data Interpretation

Chapter Journey: Data Interpretation

Step 1: Chart-Based Percentage and Average Interpretation

Focus: Pie charts, bar graphs, and line charts.

Core Skill: Converting visual proportions into exact percentages and weighted averages.

Weightage: High (Foundation for all DI questions).

Step 2: Grid and Matrix Data Counting

Focus: 2D arrays, pixel intensity matrices, and tabular conditions.

Core Skill: Systematic counting under multiple simultaneous constraints.

Weightage: Moderate (Requires careful, methodical checking).

Mastery Goal: You will learn to bypass visual deception and extract raw numerical truth from any chart or grid in under two minutes.

Topic Hero: The Core of Chart Interpretation

The Core Principle

Every chart interpretation problem reduces to a single mapping:

Visual Feature ↔ Numerical Value

Pie Chart: Angle or Area ↔ Percentage of Total

Bar/Line Chart: Height or Position ↔ Absolute Value or Frequency

The Golden Rule: Always identify the denominator (the total or base) before calculating any percentage or average. Charts often omit the total, requiring you to deduce it from a known segment's value and its visual proportion.

Computing Averages from Frequency Charts

The Weighted Mean Method

For a chart displaying values $x_{i}$ with corresponding frequencies $f_{i}$ :

\overset{ˉ}{X} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{\sum _{i = 1}^{n} f _{i}}

Step-by-step execution:

1. Create a mental or scratchpad table of $x_{i}$ and $f_{i}$ .
2. Compute the product $f_{i} x_{i}$ for each row.
3. Sum the products to get the grand total of all values.
4. Sum the frequencies to get the total number of observations.
5. Divide the grand total by the total observations.

Worked Example: Election Vote Distribution

Scenario

Total votes cast: 115,000

Invalid votes: 5,000

Candidate A's share of valid votes: 35%

Solution

Step 1: Establish the correct base.

Valid Votes = 115, 000 - 5, 000 = 110, 000

Step 2: Apply the chart percentage to the valid base.

Votes for A = 0.35 \times 110, 000

Votes for A = 38, 500

Key Takeaway: The chart's percentages are strictly bound to the valid subset, not the gross total.

Data Interpretation: Tables, Charts and Graphs: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Quantitative Aptitude MCQ

If the average value of a dataset $X$ is 10, and a new dataset $Y$ is created using the linear transformation $Y = 2 X + 3$ , what is the average value of $Y$ ?

A.
20
B.
13
C.
26
D.
23

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct substitution question, recognisable because it gives the average of a variable and asks for the average after a linear transformation.

Step 1: Recall the property of linear transformation of averages. If $Y = a X + b$ , then the average of $Y$ is $\overset{ˉ}{Y} = a \overset{ˉ}{X} + b$ .

Step 2: Identify the given values. $\overset{ˉ}{X} = 10$ , $a = 2$ , and $b = 3$ .

Step 3: Substitute these values into the formula: $\overset{ˉ}{Y} = 2 (10) + 3$ .

Step 4: Calculate the result: $20 + 3 = 23$ .

Answer: D

Question 2 · Quantitative Aptitude MCQ

If the average of a dataset $X$ is $\overset{ˉ}{X}$ , and a new variable $Y$ is defined by the linear transformation $Y = 4 X + 7$ , what is the average of $Y$ ?

A.
$4ar{X} + 7$
B.
$4(ar{X} + 7)$
C.
$4ar{X} - 7$
D.
$ar{X} + 7$

Correct Answer:

Step-by-Step Solution

Insight: This is a direct application of the linear transformation rule for averages; apply the exact same transformation to the original mean.

Exam route: Substitute $\overset{ˉ}{X}$ into the equation: $\overset{ˉ}{Y} = 4 \overset{ˉ}{X} + 7$ .

Learning route: The transformation rule states that if $Y = a X + b$ , then the average transforms identically: $\overset{ˉ}{Y} = a \overset{ˉ}{X} + b$ . Here, $a = 4$ and $b = 7$ . Therefore, $\overset{ˉ}{Y} = 4 \overset{ˉ}{X} + 7$ .

Tempting wrong path: A student might select Option B, incorrectly distributing the multiplication over the addition as $4 (\overset{ˉ}{X} + 7)$ . This breaks because the transformation is $4 X + 7$ , not $4 (X + 7)$ . Generalization: For any linear transformation $Y = a X + b$ , the new average is strictly $a \overset{ˉ}{X} + b$ . Verification: If $X = 2$ , $Y = 15$ . If $\overset{ˉ}{X} = 2$ , $\overset{ˉ}{Y} = 4 (2) + 7 = 15$ .

Answer: $4 \overset{ˉ}{X} + 7$

Question 3 · Quantitative Aptitude MCQ

When interpreting a pie chart that explicitly shows the "share of valid votes", what is the correct base value to use for percentage calculations?

A.
Total votes cast minus invalid votes
B.
Total votes cast
C.
Total votes cast plus invalid votes
D.
The number of candidates

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition question, recognisable because it asks for the correct baseline (denominator) when a chart specifies "valid" data.

Step 1: Read the chart title carefully. It specifies "share of valid votes".

Step 2: The pie chart represents parts of the "valid votes" whole.

Step 3: Therefore, the correct base value for any percentage calculation from this chart is the total number of valid votes, which is calculated as Total votes cast minus invalid votes.

Answer: A

Question 4 · Quantitative Aptitude MCQ

In an election data interpretation problem, if a pie chart represents the "share of valid votes", what is the correct base value to use when calculating a candidate's actual votes from their percentage?

A.
Valid votes (Total votes minus invalid votes)
B.
Total votes cast
C.
Invalid votes
D.
Average of total and invalid votes

Correct Answer:

Step-by-Step Solution

Key idea: This is a valid data baseline question, recognisable because it involves a pie chart of "valid votes" and requires finding the correct base for calculation.

Step 1: Identify that the pie chart percentages apply only to valid votes.

Step 2: The correct base is the total number of valid votes, which is calculated as Total votes minus invalid votes.

Answer: A

Question 5 · Quantitative Aptitude MCQ

When calculating the average number of patients per shift from a bar chart showing the frequency of different patient counts, what is the essential first step in the computation?

A.
Add all unique patient counts and divide by the number of bars
B.
Identify the maximum patient count
C.
Multiply each patient count by its corresponding frequency (number of shifts)
D.
Subtract the minimum patient count from the maximum

Correct Answer:

Step-by-Step Solution

Key idea: This is a frequency chart average computation question, recognisable because it asks for the procedure to find an average from grouped frequency data.

Step 1: Recall that an average from a frequency chart requires a weighted sum.

Step 2: The essential first step is to multiply each unique value (patient count) by its corresponding frequency (number of shifts) to get the weighted values.

Answer: C

Question 6 · Quantitative Aptitude MCQ

If the average of a dataset $X$ is $\overset{ˉ}{X}$ , and a new dataset $Y$ is formed by the transformation $Y = c X$ (where $c$ is a constant), what is the average of the new dataset $\overset{ˉ}{Y}$ ?

A.
$ar{X} + c$
B.
$car{X}$
C.
$rac{ar{X}}{c}$
D.
$ar{X}^c$

Correct Answer:

Step-by-Step Solution

Key idea: This is a linear transformation of averages question, recognisable because it asks for the new average after multiplying all data points by a constant.

Step 1: Recall the property of averages under scalar multiplication.

Step 2: If every data point in a set is multiplied by a constant $c$ , the average of the set is also multiplied by $c$ .

Step 3: Therefore, the new average is $\overset{ˉ}{Y} = c \overset{ˉ}{X}$ .

Answer: B

Question 7 · Quantitative Aptitude MCQ

A pie chart represents the distribution of a dataset. If a specific category accounts for $15%$ of the total, what is the central angle of its sector in degrees?

A.
36
B.
45
C.
54
D.
60

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct formula application for pie chart angle conversion.

Step 1: Identify the given percentage, which is $15%$ .

Step 2: Recall that a full circle is $36 0^{\circ}$ , representing $100%$ .

Step 3: Calculate the angle using the formula: $Angle = (\frac{15}{100}) \times 36 0^{\circ}$ .

Step 4: Compute $0.15 \times 360 = 5 4^{\circ}$ .

Answer: 54

Question 8 · Quantitative Aptitude MCQ

A pie chart represents a complete dataset. What is the central angle in degrees for a sector that accounts for $30%$ of the total?

A.
$7 2^{\circ}$
B.
$9 0^{\circ}$
C.
$10 8^{\circ}$
D.
$12 0^{\circ}$

Correct Answer:

Step-by-Step Solution

Insight: This is a direct pie chart angle conversion; multiply the given percentage by the standard multiplier 3.6.

Exam route: $30 \times 3.6 = 10 8^{\circ}$ .

Learning route: A full circle is $36 0^{\circ}$ , representing $100%$ of the dataset. To find the central angle for a given percentage, use the formula: $Angle = (\frac{Percentage}{100}) \times 36 0^{\circ}$ . For $30%$ , this is $\frac{30}{100} \times 36 0^{\circ} = 0.30 \times 36 0^{\circ} = 10 8^{\circ}$ .

Tempting wrong path: A student might multiply $30$ by $3$ instead of $3.6$ , getting $9 0^{\circ}$ (Option B). This breaks because $10%$ is exactly $3 6^{\circ}$ , so $30%$ must be $3 \times 3 6^{\circ} = 10 8^{\circ}$ . Generalization: Always use the exact multiplier 3.6 (or the fraction $\frac{360}{100}$ ) for percentage-to-angle conversion. Verification: $108/360 = 0.3 = 30%$ .

Answer: $10 8^{\circ}$

Question 9 · Quantitative Aptitude MCQ

Which mathematical expression correctly represents the weighted average $\overset{ˉ}{X}$ from a frequency chart, where $x_{i}$ is the value and $f_{i}$ is its frequency?

A.
$r a c \sum f_{i} \sum x_{i}$
B.
$\sum (x_{i} im es f_{i})$
C.
$r a c \sum (x_{i} im es f_{i}) \sum f_{i}$
D.
$r a c \sum x_{i} \sum f_{i}$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct formula recall question, recognisable because it asks for the standard mathematical definition of a weighted average from a frequency distribution.

Step 1: Recall the definition of a weighted average. It accounts for how many times each value occurs.

Step 2: The total sum of all values is found by multiplying each value $x_{i}$ by its frequency $f_{i}$ and summing these products: $\sum (x_{i} \times f_{i})$ .

Step 3: This total sum is divided by the total number of observations, which is the sum of all frequencies: $\sum f_{i}$ .

Step 4: Combine these to get the formula: $\frac{\sum ( x _{i} \times f _{i} )}{\sum f _{i}}$ .

Answer: C

Question 10 · Quantitative Aptitude MCQ

A frequency distribution shows value $x_{1} = 5$ with frequency $f_{1} = 4$ , and value $x_{2} = 10$ with frequency $f_{2} = 2$ . What is the value of the numerator $\sum f_{i} x_{i}$ required for the weighted average formula?

A.
$15$
B.
$30$
C.
$40$
D.
$60$

Correct Answer:

Step-by-Step Solution

Key idea: This is a weighted average computation question, recognisable because it asks for the numerator of the weighted mean formula given specific values and frequencies.

Step 1: Identify the formula for the numerator: $\sum f_{i} x_{i}$ .

Step 2: Multiply each value by its corresponding frequency: $(5 \times 4)$ and $(10 \times 2)$ .

Step 3: Sum these products: $20 + 20 = 40$ .

Answer: C