LESSON 13.5 — Data Interpretation

A. Standard Map

GATE AR DI reality: DI questions in the GA section use short datasets — a single small table or one pie chart. The difficulty is not in the data volume but in the base-selection and unit-matching traps embedded in the question. Speed and accuracy on these 1–2 mark questions come entirely from knowing which number is the denominator before computing.

B. Mechanism in Words

1. Scan the full dataset before reading questions. Note: what does each row and column represent? What are the units? Are there subtotals that differ from grand totals? Spending thirty seconds here prevents three minutes of re-reading mid-question.

2. Tag each question with its type: percentage of a total, absolute difference, percentage change, ratio between rows/columns, or index. The question type determines the formula and the base.

3. Identify the base before touching any number. Percentage change always divides by the old value. Percentage share always divides by the total for that year/category — not the grand total unless the question says so. Write the base down before computing.

4. Convert chart values to numbers before computing. If a pie chart shows 72° for a category, convert to percentage: 72/360 × 100 = 20%. If a bar chart shows a bar at “approximately 3.5 on a scale of 0–5,” do not use 3.5 without checking the axis label and unit.

5. For multi-pie or multi-chart questions: use the base total of each chart separately. Never apply one chart’s percentage to another chart’s total unless the question explicitly asks for a combined calculation.

6. Eliminate distractor options by order of magnitude. GATE DI distractors are calibrated to catch specific errors — one option corresponds to using the wrong base, one to using a % without converting to actual numbers, one to adding instead of subtracting. After computing, verify which error each wrong option represents.

C. Core Concept Explanations

C1. Tables — Row/Column Totals, Missing Cell

A table presents data across two dimensions (rows = categories, columns = time periods or attributes). The three operations tested are: reading a cell, computing a row/column total, and finding a missing cell.

Reading total vs subtotal: Exam tables often show a “Total” row that is a pre-computed sum. Verify it before using it — GATE occasionally presents tables where the displayed total includes a category not listed separately (i.e., total ≠ sum of visible rows). Read the table header carefully.

Missing cell: If a row total and all other cells are given, the missing cell = row total − sum of other cells. If a percentage and a total are given, the absolute value = percentage × total / 100.

Cross-row comparison: “Which region showed the highest growth from Year 1 to Year 2?” — compute absolute change and % change for each row, then rank. Do not eyeball; tables with similar values across rows are designed to make visual ranking unreliable.

Row-to-column ratio: “What fraction of Year X total was contributed by Category Y?” = Cell(Y, X) / Column Total(X). The denominator is always the column total for that year, not the grand total of the entire table.

GATE AR 2024 Q8 (table type): Investment amounts for two persons across three funds were given in a table; the unknown was the rate of return for Fund A. Setting up the equation required reading each cell correctly and using the difference condition to solve algebraically.

C2. Bar Charts — Grouped vs Stacked; Read Axis Scale

Simple bar chart: One bar per category per time period. Bar height = value on Y-axis.

Grouped bar chart: Multiple bars per time period, each representing a different sub-category, placed side by side. Each bar’s height is read independently from the Y-axis. Do not add bar heights to get a combined value unless the question asks for combined.

Stacked bar chart: Sub-categories are layered within a single bar. The full bar height = total for that period. Each segment’s value = the segment’s top Y-value minus its bottom Y-value. This is the most common misread: students subtract from zero instead of from the bottom of the segment.

Axis scale traps:
– Y-axis may not start at zero — the first interval may represent a compressed baseline. Check whether the axis starts at 0 or at some non-zero value. A visual difference of “double the height” may represent a 5% increase if the axis is zoomed in.
– Y-axis may use thousands or lakhs as units. “Bar reaches 120” means 1,20,000 if the axis label says “in thousands.”
– X-axis may skip years. A bar for 2019 and the next for 2022 means a 3-year gap, not a 1-year gap — relevant when computing rate of change.

What to compute from bar charts: absolute values (read directly), differences between bars (subtract), % share (bar value / total bar or sum × 100), and % change (new bar − old bar / old bar × 100).

C3. Line Graphs — Trend, Intersection, Slope Intuition

A line graph plots values of one or more variables over a continuous dimension (usually time). Slope = rate of change; steeper slope = faster change.

Trend questions: “In which period did Category X consistently increase?” — look for a segment where the line moves only upward between consecutive points. A single downward blip disqualifies a period from “consistent” increase.

Intersection questions: “In which year did Category X and Category Y have equal values?” — find where the two lines cross. If no point is labelled, estimate from the graph. GATE plots usually mark key values at intersection points.

Slope intuition (for GATE without calculus):
– Steeper segment → faster rate of change in that period
– Flat segment → no change (rate of change = 0)
– Downward segment → decline
– The segment with the steepest positive slope is where the percentage increase per unit time is maximum — but only if both the start and end values are given can this be computed exactly.

Maximum increase vs maximum value: “In which year did the value reach its maximum?” and “In which period did the value increase the most?” are different questions. Maximum value = highest point on the graph. Maximum increase = steepest upward segment = largest (Yₙ − Yₙ₋₁) value.

Multiple lines: When two or more lines appear on the same graph, questions may ask about the gap between them (absolute difference), the period when one surpassed the other (intersection), or the period when the gap was largest.

C4. Pie Charts — Degree ↔ %, Multiple Pies

Single pie chart conversions:
– Given degree measure → percentage: (degree / 360) × 100
– Given percentage → degree: (percentage / 100) × 360
– Given percentage and total → absolute value: (percentage / 100) × total

GATE AR 2024 Q5: Total votes = 1,15,000; invalid = 5,000; valid = 1,10,000. Pie chart showed candidates B and C at 25% and 20% respectively. B + C share = 45%. Combined valid votes = 1,10,000 × 0.45 = 49,500. The wrong-base trap: option (A) gave 45,000 = 45% of 1,00,000 (used a round number instead of the computed valid total).

Multiple pie charts (GATE AR 2023 Q3): Two charts for different groups of different sizes. Do not average the percentages — use weighted average.
– Step 1: Multiply each group’s % by its own total to get the absolute count for that category.
– Step 2: Sum the counts across groups.
– Step 3: Divide by the combined total and multiply by 100.
Simple average of percentages (e.g., (7% + 23%)/2 = 15%) is always wrong when groups are unequal in size. The correct answer was 22.7%, not 15%.

Relative size of segments: “Which two categories together account for more than half?” — sum the relevant percentages. If given only degrees, sum degrees and check if > 180°.

C5. Mixed DI — Two Charts, One Dataset

GATE GA occasionally provides a table and a pie chart (or a bar and a line graph) that describe different dimensions of the same dataset. The question requires information from both.

Attack sequence:
1. Identify what each chart presents: one usually gives totals or absolute values; the other gives percentages or distributions.
2. Map the shared variable (e.g., year, category) between charts before computing.
3. Never apply a percentage from Chart 2 to the total from Chart 1 unless both refer to the same base population in the same time period.

Common mixed-DI question pattern:
– Table gives total output by year.
– Pie chart gives the distribution of 2023 output by department.
– Question: “How many units did Department X produce in 2023?”
– Answer: Read 2023 total from table → multiply by Department X’s percentage from pie chart.
– Trap: Using an earlier year’s total with the 2023 pie chart percentages.

Cross-chart consistency check: If the table shows Total = 500 for 2023 and the pie chart shows four categories, the four segments should sum to 100% and their absolute values to 500. If they don’t match, re-read whether the pie chart refers to a subset (e.g., only three out of five departments).

C6. % Change and Index — Base Year, Absolute vs Relative Change

Percentage change formula: (New − Old) / Old × 100

The base is always the Old (earlier) value. This is the most violated rule in DI questions.

Absolute change vs relative change:
– Absolute change = New − Old (in original units)
– Relative (%) change = (New − Old) / Old × 100
These can give opposite rankings. Category A may have a larger absolute increase but a smaller percentage increase than Category B if A started from a much larger base.

Index numbers:
A value index sets a reference year (base year) = 100 and expresses all other years relative to it.

If index = 120, the value is 20% higher than base year. If index = 85, it is 15% lower.

Interpreting % change on a % value: If a pie chart category goes from 30% to 36% of total, the change in share is 6 percentage points (not 6%). The percentage change in share is (36 − 30)/30 × 100 = 20%. These are different quantities; read the question carefully to know which is asked.

Compounded change: If a value grows at 10% per year for 3 years, the total growth is not 30% — it is (1.1)³ − 1 = 33.1%. Percentage changes do not add linearly when applied to a changing base.

D. Worked Examples and Practice Sets

Mini Dataset — Housing Completions by Region (in thousands of units)

Units: thousands of housing completions. All totals pre-verified.

Worked Question 1 — Percentage Change (East Region, 2019 to 2023)

Question: By what percentage did housing completions in the East region increase from 2019 to 2023?

Step 1 — Identify: Percentage change question. Base = 2019 value for East = 80.

Step 2 — Identify new value: 2023 value for East = 130.

Step 3 — Compute:
% change = (130 − 80) / 80 × 100 = 50 / 80 × 100 = 62.5%

Step 4 — Sanity check: East grew from 80 to 130, which is clearly more than 50% of 80 (= 40) but less than 100% of 80 (= 80). 62.5% is consistent.

Trap avoided: Using 130 as the base instead of 80 gives (50/130) × 100 = 38.5% — this would be the wrong-base distractor option.

Answer: 62.5%

Worked Question 2 — Trend Identification and Absolute Change

Question: Which region showed a decline in completions between 2021 and 2022, and by how much (in thousands)?

Step 1 — Compare each region’s 2021 and 2022 values:
– North: 2021 = 154, 2022 = 148 → 154 − 148 = 6 decline ✓
– South: 2021 = 210, 2022 = 225 → increase (no decline)
– East: 2021 = 100, 2022 = 112 → increase (no decline)

Step 2 — Identify the declining region: Only the North region declined.

Step 3 — State the decline: North fell by 154 − 148 = 6 thousand units.

Trap avoided: Do not compute percentage change unless asked — the question asks “by how much,” which means absolute decline. Answering “3.9%” (the percentage decline) would be wrong for this stem.

Answer: North region; declined by 6,000 units

Worked Question 3 — Percentage Share (East Region in 2023)

Question: What percentage of total housing completions across all regions in 2023 was contributed by the East region? (Round to one decimal place.)

Step 1 — Identify numerator: East in 2023 = 130 thousand.

Step 2 — Identify denominator: Total all regions in 2023 = 515 thousand. This is the correct base — the total for the same year.

Step 3 — Compute:
Share = (130 / 515) × 100 = 25.24…% ≈ 25.2%

Step 4 — Sanity check: East is clearly the smallest region; 25% of the total is plausible (roughly 1/4 of three roughly-equal regions, though South dominates).

Trap avoided: Using the grand total across all years (400+417+464+485+515 = 2,281) as the denominator gives (130/2,281) × 100 = 5.7% — wrong. The question specifies 2023, so denominator = 2023 total only.

Answer: 25.2%

E. Common Confusions

• % change vs percentage point change: If Category A’s share rises from 20% to 25%, that is a 5 percentage point increase. The percentage change in the share is (25 − 20)/20 × 100 = 25%. These are different quantities. “By how many percentage points” → subtract. “By what percentage” → use change/old × 100.

• Simple average vs weighted average of percentages: (7% + 23%)/2 = 15% is only valid when the two groups have equal size. When group sizes differ, each % must be weighted by its group size. This is exactly the trap in GATE AR 2023 Q3 — the correct answer was 22.7%, not 15%.

• Reading stacked vs grouped bars: In a stacked bar, each segment’s value is read from its top minus its bottom, not from the top to zero. In a grouped bar, each bar is read from its top to zero. Misidentifying the chart type causes systematic errors on all questions.

• Pie chart degree vs percentage: 72° = 20% of the total (72/360 = 0.2). Students sometimes use 72 directly as a percentage, leading to answers 3.6 times too large (72% instead of 20%).

• Wrong year’s total as base: In a table covering multiple years, “what % of 2023 total” requires using the 2023 column total as the denominator — not the average of all years, not the earliest year, not the grand total. The specific year cited in the question determines the denominator.

• Absolute value from index: If an index shows City A = 140 (base year = 100) and the base year value is Rs. 500 crores, the current value is 500 × 1.4 = Rs. 700 crores — not 140 crores or 500 + 140 = 640.

F. Exam Traps

G. Answer-Writing Cues

H. PYQ Linkage Note

I. Mini-Check — Lesson 13.5

Use the following dataset to answer Q1–Q4. Q5 is standalone.

Dataset: Annual Output (Units) — Three Departments

| Pie charts showing 504 universities and 25951 colleges by grade. Grade III percentage? | GATE AR 2023 Q3 | MCQ: Pie charts showing 504 universities and 25951 colleges by grade. Grade… |
| In an election, the share of valid votes received by the four candidates A, B, C, and D… | GATE AR 2024 Q5 | MCQ: In an election, the share of valid votes received by the four… |
| Person 1 and Person 2 invest in three mutual funds A, B, and C. The amounts they invest… | GATE AR 2024 Q8 | MCQ: Person 1 and Person 2 invest in three mutual funds A, B, and C. The… |
| ‘If his latest movie had been a commercial success, the actor would have made enough… | GATE AR 2026 Q5 | MCQ: ‘If his latest movie had been a commercial success, the actor would… |

Q1. (MSQ) Which of the following statements about the dataset are correct? Select all that apply.

(A) The Design department’s output increased every year from 2020 to 2023.
(B) The Construction department’s output in 2021 was lower than in 2020.
(C) The Maintenance department had the highest percentage growth from 2020 to 2023 among all three departments.
(D) Total output across all departments in 2022 exceeded 140 units.

Answer: (A), (B), (D)

Explanation:
– (A) Design: 40 → 48 → 52 → 65. Each year is higher than the previous. ✓
– (B) Construction 2021 = 55 < 2020 = 60. ✓
– (C) Wrong. % growth 2020→2023: Design = (65−40)/40 × 100 = 62.5%; Construction = (75−60)/60 × 100 = 25%; Maintenance = (30−20)/20 × 100 = 50%. Design has the highest % growth, not Maintenance. ✗
– (D) Total 2022 = 147 > 140. ✓

Q2. (MCQ) What was the Construction department’s percentage decline in output from 2020 to 2021?

(A) 6.25%
(B) 8.33%
(C) 9.09%
(D) 5.00%

Answer: (B) 8.33%

Explanation:
% change = (55 − 60) / 60 × 100 = −5/60 × 100 = −8.33% (decline of 8.33%)

(A) 6.25% = 5/80 — wrong denominator. (C) 9.09% = 5/55 — uses the new value (55) as the base instead of the old value (60). This is the classic wrong-base distractor. (D) 5% = 5/100 — treats 60 as 100 for no reason.

Q3. (MCQ) In 2021, what percentage of total output was contributed by the Construction department? (Round to the nearest whole number.)

(A) 40%
(B) 44%
(C) 48%
(D) 50%

Answer: (B) 44%

Explanation:
Construction in 2021 = 55. Total in 2021 = 125.
Share = (55/125) × 100 = 44%

(A) 40% would mean Construction = 50 out of 125 — not the case. (D) 50% would mean Construction = 62.5 — wrong. (C) 48% would mean ≈ 60 out of 125 — this corresponds to the 2020 value, not 2021.

Q4. (MCQ) By how much did the Design department’s output grow in absolute terms from 2020 to 2023?

(A) 15 units
(B) 20 units
(C) 25 units
(D) 30 units

Answer: (C) 25 units

Explanation:
Absolute growth = 65 − 40 = 25 units.

This is a direct read from the table requiring no formula beyond subtraction. The trap is solving for percentage change (62.5%) and confusing it with absolute change. (A) 15, (B) 20, (D) 30 are not the difference between Design’s 2023 and 2020 values.

Q5. (MCQ) A pie chart shows four project phases with the following degree measures: Planning = 90°, Design = 72°, Execution = 162°, Closure = 36°. What percentage of the project effort does the Execution phase represent?

(A) 40%
(B) 45%
(C) 36%
(D) 162%

Answer: (B) 45%

Explanation:
First verify total: 90 + 72 + 162 + 36 = 360° ✓
Execution % = (162 / 360) × 100 = 45%

(A) 40% = 144°/360 — wrong. (C) 36% = 36°/360 — this is the Closure phase’s percentage, not Execution’s. (D) 162% is the raw degree value used directly as a percentage — the classic degree-as-percent trap.