Data Interpretation and Mental Ability for APPSC Group-II and AP Police

Data interpretation is the part of the mental-ability block that rewards patience more than talent. The numbers are small, the arithmetic is school level, and the same four chart types return in every paper: the table, the bar diagram, the line graph and the pie chart. What costs marks is not difficulty but habit: reading a question before reading the chart, mixing up percent and percentage points, comparing a bar in thousands with a bar in lakhs, or spending four minutes on a division that a rounded fraction would have settled in ten seconds.

This guide is written for two kinds of aspirant. The first is preparing for APPSC Group-II Services, where the Screening Test carries a 30-mark Mental Ability block. The second is preparing for AP Police recruitment, where the written test includes arithmetic and reasoning papers; check the current SLPRB notification for the scheme and syllabus of your post, and see our AP Police SI guide and AP Police Constable guide for what earlier notifications recorded. Every datasheet in this post is invented for practice and does not describe any real district, office or year. Every worked example and every MCQ answer was computed with a short script and the output checked against the text, so the arithmetic can be trusted. Cover the answer, attempt the question, then compare.

Arithmetic and the basic reasoning types (percentages, profit and loss, ratio, time and work, clocks, calendars, blood relations, direction sense) are covered in reasoning and arithmetic shortcuts for AP Police and Sachivalayam. We do not repeat them here. This post concentrates on what that guide leaves out: reading charts, caselets, data sufficiency, approximation, the item types that the Group-II syllabus names in its mental-ability block, and the logical-reasoning layer on top.

1. What the syllabus actually lists

For Group-II, the source is the Commission's own document, "Syllabus for Screening and Main Examination for the recruitment to the post of Group-II Services", signed 27 April 2023 (as on October 2026, as published on psc.ap.gov.in). The Screening Test is General Studies and Mental Ability for 150 marks, and Mental Ability is one of five blocks of 30 marks. The block has three paragraphs.

The first covers logical reasoning, described as deductive, inductive and abductive, with four named item types: statement and assumptions, statement and argument, statement and conclusion, and statement and courses of action. The second covers mental ability proper: number series, letter series, odd man out, coding-decoding, problems relating to relations, and shapes and their sub sections. The third covers basic numeracy: number system, order of magnitude, averages, ratio and proportion, percentage, simple and compound interest, time and work, and time and distance, followed by data analysis described as tables, bar diagram, line graph and pie-chart.

Three observations follow. First, the Group-II syllabus does not name caselets or data sufficiency. They are common in other exams and useful for practice because they train the same skill, but do not treat them as listed topics. Second, it does not name mean, median, mode or variance in the Mental Ability block. Those appear elsewhere: the Commission's Notification No. 21/2025 for Assistant Motor Vehicle Inspector lists "Basic data analysis (Summary Statistics such as mean, median, mode and variance)" in its General Studies and Mental Ability paper. If you are preparing for more than one exam, learn them; they take an afternoon. Third, "shapes and their sub sections" is the least explained phrase. The sensible reading is non-verbal counting and pattern items, such as counting triangles in a figure, and section 8 treats it that way.

For the police syllabus we do not state a scheme here. Our AP Police SI guide records an arithmetic and reasoning paper of SSC standard in the earlier notification, so the same preparation serves. Confirm against the notification that is current when you apply.

2. How a data set is built, and how to read it

Every data question has four layers: the title, the units, the axes and the numbers. Questions that go wrong, go wrong in the first two. A title such as "Applications received, in hundreds" turns a bar of 12 into 1,200. A pie chart titled "Percentage share of expenditure" has no totals unless the question gives one. A line graph with a vertical axis that begins at 40 and not at 0 makes a small rise look dramatic, although the percentage change is small.

Adopt a fixed reading routine of thirty seconds. Read the title and the units aloud in your head. Note the scale of each axis. Write the plotted values beside the chart in a small column on your rough sheet, as figures, because reading a bar against a gridline repeatedly invites error. Only then read the question. If a question asks for a percentage, mark whether the base is the first value, the last value or a total. The same two numbers produce different answers under different bases, and examiners rely on that.

A second habit is to sort the questions by cost. Questions that ask for a single reading from the chart ("how many in 2023?") cost seconds. Questions that ask for a comparison need two readings. Questions that ask for a percentage change over several periods or a weighted average need a short calculation. In a set of five, answer the cheap ones first, because a quick two or three marks protects you if the set turns out to be heavy. On a paper with negative marking, as many APPSC notifications specify, check your notification for the penalty before deciding how much to guess; our guide on negative marking and time management shows how to set a rule for yourself.

Finally, the words matter. "Approximately" invites rounding. "Exactly" does not. "Between" and "from" with years can mean inclusive or exclusive; if the question says "growth from 2021 to 2025" it means the change over four intervals, not five. "Average annual" means the sum divided by the number of years listed, unless the question says "average annual growth rate", which is a different thing and rarely asked at this level.

3. Tables

A table is the plainest form and the most common. Use this practice table, which shows registrations of four vehicle types at a fictitious district office. The figures are invented.

Year	Two-wheelers	Cars	Goods vehicles	Buses	Total
2021	4,200	1,500	900	120	6,720
2022	4,800	1,650	960	150	7,560
2023	5,400	1,800	1,020	180	8,400
2024	6,000	2,100	1,080	210	9,390
2025	6,600	2,250	1,200	240	10,290

Question 1: What percentage of the 2025 total do two-wheelers form? Two-wheelers are 6,600 of 10,290. A quick estimate is 6,600 divided by 10,290, which is a little under two-thirds, and two-thirds is 66.7 percent. The script gives 64.14 percent, so "about 64 percent". Notice that the estimate overshoots because the denominator was rounded down to 9,900 in your head; knowing the direction of the error helps you choose between close options.

Question 2: By what percentage did car registrations grow from 2021 to 2025? Cars rose from 1,500 to 2,250, an increase of 750 on a base of 1,500. That is exactly 50 percent. The base is the first year, always, when the question says growth from a year.

Question 3: What is the ratio of goods vehicles to buses in 2023? The figures are 1,020 and 180. Divide both by 60: 17 and 3. The ratio is 17 to 3. Ratio questions reward dividing by the largest obvious common factor rather than by 2 repeatedly.

Question 4: In which year was the rise in the total over the previous year the largest, and by how much? The yearly rises are 840 (2022), 840 (2023), 990 (2024) and 900 (2025). The largest is in 2024, by 990. Note the trap: the percentage rise is not the same ranking. In percentage terms the rises are 12.5, 11.11, 11.79 and 9.58 percent, so 2022 has the highest percentage rise although 2024 has the highest absolute rise. A question that says "largest increase" without "percentage" asks for the absolute figure; a question that says "highest growth rate" asks for the percentage. Read the noun.

Question 5: What is the average number of cars registered per year? The five car figures sum to 9,300 and the average is 1,860. For averages of evenly spaced values the middle value is the average: 1,500, 1,650, 1,800, 2,100, 2,250 are not perfectly spaced, so the middle value (1,800) is close but not exact; the exact answer needs the sum. Use the middle value as an estimate to eliminate options and the sum to confirm.

Two more habits for tables. Add the row totals only if the question needs them; a given Total column is for use. And when a question mixes two rows, write the two numbers side by side before touching a calculator in your head.

4. Bar diagrams

A bar diagram shows the same sort of data as a table with the values encoded as lengths. In an exam the bars are labelled or readable against a grid, so the first job is to convert the bars into numbers on your rough sheet. A double bar adds a second series per category, and the questions almost always combine the two series into a percentage.

Take a double bar for five fictitious examination centres, A to E, showing the number of candidates who appeared and the number who qualified.

Centre	Appeared	Qualified
A	800	320
B	650	260
C	900	405
D	500	150
E	750	300

Which centre had the highest qualifying percentage? Compute qualified over appeared for each. A is 40 percent, B is 40 percent, C is 45 percent, D is 30 percent and E is 40 percent. The highest is centre C. The taller bar does not decide: C also had the highest absolute qualified number, but that is partly because it had the most candidates. Look at D, which has the smallest bars and the lowest rate. Now notice a more interesting point: A, B and E all share exactly 40 percent although their bars differ. When several options tie, the question will have been designed around the tie, so re-read it.

What percentage of all candidates qualified? Sum the appeared column: 3,600. Sum the qualified column: 1,435. The overall rate is 1,435 over 3,600, which is 39.86 percent, about 40 percent. It is not the average of the five centre percentages in general; here the simple average of 40, 40, 45, 30 and 40 is 39, so the two differ slightly. Averaging percentages without weights is a classic error, and the difference between 39 and 39.86 is exactly the size of gap between two neighbouring options in a well-set question.

What was the combined qualifying rate at centres A and C? Qualified at A and C is 320 plus 405, which is 725. Appeared at A and C is 800 plus 900, which is 1,700. The rate is 725 over 1,700, or 42.65 percent. Again the answer is not the simple average of 40 and 45, which would be 42.5.

What is the ratio of the qualified at B to the qualified at D? 260 to 150 reduces to 26 to 15.

When a bar diagram shows a stacked bar, remember that the top segment's value is the difference between the cumulative levels, not the level itself. When a bar diagram has two vertical axes, check which series uses which axis before reading anything.

5. Line graphs

A line graph shows change over time. The questions revolve around three ideas: which period had the steepest rise, which had a fall, and how a later value compares with an earlier one in percentage terms. Because the eye reads steepness, not percentage, line graphs are the chart where the visual impression misleads most.

Use this fictitious series of monthly online applications received by a recruiting office: January 1,200, February 1,500, March 1,350, April 1,800, May 2,160 and June 1,944.

Month	Applications	Change on previous month
January	1,200	none
February	1,500	+300 (25 percent)
March	1,350	-150 (-10 percent)
April	1,800	+450 (33.33 percent)
May	2,160	+360 (20 percent)
June	1,944	-216 (-10 percent)

In which month was the percentage increase the highest? April, at 33.33 percent over March. The largest absolute rise is also April, at 450, but be aware that May has a smaller percentage rise (20) with a large absolute change (360). A question can ask for either, and the two can differ.

What is the overall growth from January to June? June is 1,944 and January is 1,200. The change is 744, and 744 over 1,200 is 62 percent. Do not add the monthly percentages. Adding 25, -10, 33.33, 20 and -10 gives 58.33, which is wrong, because each monthly percentage has a different base. Chain the multipliers instead: 1.25 times 0.9 times 1.3333 times 1.2 times 0.9 equals 1.62.

What is the average monthly figure over the six months? The total is 9,954 and the average is 1,659.

The shortcut for repeated percentages is to convert each change to a multiplier. A fall of 10 percent is 0.9, a rise of 20 percent is 1.2, and a rise of 25 percent is 1.25. Notice that 1.25 times 0.8 equals exactly 1.00: a rise of 25 percent is cancelled by a fall of 20 percent, not 25. That fact appears again in section 9.

6. Pie charts

A pie chart divides a whole into sectors. The whole is 100 percent and 360 degrees, so one percent is 3.6 degrees. The questions come in three families: percentage to value, degrees to percentage, and comparisons between two pies. The chart carries no totals unless the question gives them, so a pie alone supports only ratios and percentages, never absolute numbers.

Take a fictitious office budget of 7,20,000 rupees, divided as follows: salaries 35 percent, fuel 20 percent, maintenance 15 percent, rent 10 percent, training 10 percent and other 10 percent. The shares sum to 100 percent.

Head	Share	Amount (rupees)	Central angle
Salaries	35 percent	2,52,000	126 degrees
Fuel	20 percent	1,44,000	72 degrees
Maintenance	15 percent	1,08,000	54 degrees
Rent	10 percent	72,000	36 degrees
Training	10 percent	72,000	36 degrees
Other	10 percent	72,000	36 degrees

How much more is spent on fuel than on rent? The difference in share is 20 minus 10, which is 10 percent, and 10 percent of 7,20,000 is 72,000. You never need the individual amounts: subtract the shares first, then take the percentage once.

What is the central angle for maintenance? 15 percent times 3.6 is 54 degrees. The shortcut is 15 times 3.6, which is 10 times 3.6 (36) plus half of that (18).

If salaries rise by 20 percent and nothing else changes, what is the new salary bill? 2,52,000 times 1.2 is 3,02,400 rupees. Be careful with the next step: the new total is 7,20,000 plus 50,400, which is 7,70,400, so the new share of salaries is not 42 percent but 3,02,400 over 7,70,400, which is about 39.25 percent. Whenever one sector changes, the whole changes too.

Now a pie given in degrees. A survey of 4,800 respondents is shown in a pie with sectors P 90 degrees, Q 72, R 54, S 108 and T 36, which sum to 360. The shares are 25, 20, 15, 30 and 10 percent. Sector S is 108 over 360 of 4,800, which is 1,440 respondents. Sector P is 1,200. The ratio of S to R is 108 to 54, which is 2 to 1, and you read it directly from the degrees without converting to percentages.

For comparisons between two pies, remember that equal percentages in two pies are equal amounts only if the totals are equal.

7. Caselets

A caselet is a paragraph of data with no chart. It is a table that you must draw yourself. The skill is to turn sentences into a grid in under a minute, and then to answer from the grid.

Here is a worked caselet. An organisation has 240 staff in three offices, X, Y and Z. The staff in X and Y are in the ratio 3 to 2, and Z has 25 percent of the total staff. Half of the staff in X, a quarter of the staff in Y and 40 percent of the staff in Z are women. How many women are there, and what percentage of the total do they form?

Z has 25 percent of 240, which is 60. So X and Y together have 240 minus 60, which is 180. Divide 180 in the ratio 3 to 2: X has 108 and Y has 72. Women in X are half of 108, which is 54. Women in Y are a quarter of 72, which is 18. Women in Z are 40 percent of 60, which is 24. The total is 54 plus 18 plus 24, which is 96 women, and 96 over 240 is exactly 40 percent.

Office	Staff	Share of women	Women
X	108	50 percent	54
Y	72	25 percent	18
Z	60	40 percent	24
Total	240	40 percent	96

The method for any caselet is the same. List the unknowns. Use the sentence with the most direct number first (here, the 25 percent for Z). Fill a grid. Check that the grid adds up to every total given in the paragraph before answering, because a mismatch tells you that you misread a sentence while there is still time to fix it.

Caselets also hide a trap in the wording: "X and Y in the ratio 3 to 2" refers to X and Y, not to the total of the three offices. Reading it as 3 to 2 of 240 would give 144 and 96 and then no room for Z.

8. Data sufficiency

Data sufficiency asks whether the given statements are enough to answer a question, not for the answer itself. The usual options are: statement I alone is sufficient; statement II alone is sufficient; both together are needed; each alone is sufficient; neither is enough even together. The question wording differs between papers, so read the options, but the thinking is the same: decide, for each statement alone and for both together, whether exactly one value of the unknown is determined.

Example A. What is the marked price of an article? I: The selling price after a 20 percent discount is 960 rupees. II: The profit on cost is 25 percent. From I alone, the selling price is 80 percent of the marked price, so the marked price is 960 over 0.8, which is 1,200. Statement II alone gives no price. So I alone is sufficient and II alone is not.

Example B. Is x even? I: x plus 3 is odd. II: x is a multiple of 4. From I, x plus 3 odd means x is even. From II, a multiple of 4 is even. Each statement alone is sufficient.

Example C. What is the speed of a train in kilometres per hour? I: The train is 150 metres long. II: It crosses a pole in 10 seconds. Alone, I gives a length but no time and II gives a time but no length. Together the train covers its own length of 150 metres in 10 seconds, which is 15 metres per second, and 15 times 3.6 is 54 kilometres per hour. Both together are needed.

Example D. What is the average age of five friends? I: The oldest is 40. II: The youngest is 20. Even together, the three middle ages are unknown, so the average cannot be fixed. Neither is sufficient, alone or together.

The checks that save marks are three. Do not solve the question beyond the point where you know whether it is solvable. Do not use outside knowledge to fill a gap; if a statement does not give the number, it does not. And when a statement gives a ratio but the question asks for an absolute number, it is almost always insufficient alone.

9. Percentage and ratio inside data questions

Most data questions reduce to three operations on percentages: finding a percentage of a number, finding what percentage one number is of another, and finding a change. The traps live in the distinctions.

The first distinction is percent versus percentage points. If a share rises from 32 percent to 40 percent, it has risen by 8 percentage points and by 25 percent (8 over 32). A question that says "increase in percentage share" usually means points; a question that says "percentage increase in the share" means relative change. Notice the preposition.

The second is the base. A 20 percent rise followed by a 10 percent fall is a net rise of 8 percent, not 10, because 1.2 times 0.9 is 1.08. A rise of 20 percent three times in a row is 1.2 cubed, which is 1.728, a rise of 72.8 percent, not 60. Always convert to multipliers when a question chains changes.

The third is the reverse change. If a price rises by 25 percent, consumption must fall by 20 percent to keep the expenditure unchanged, because 1 minus 1 over 1.25 equals 0.2. If the price rises by 20 percent, the cut is one-sixth, or 16.67 percent. If it rises by 50 percent, the cut is one-third. The pattern is: a rise of 1 part in n is cancelled by a fall of 1 part in n plus 1. Rise of 1 in 4 (25 percent) is cancelled by fall of 1 in 5 (20 percent). Rise of 1 in 5 (20 percent) is cancelled by fall of 1 in 6.

The fourth is the weighted average. When a question combines two groups, weight each rate by the group size, as in section 4. The overall rate always lies between the two group rates and is nearer the larger group.

Ratios are simpler but fail in the same way. A ratio compares two quantities of the same kind. To compare ratios across years, convert them to a common form. In the practice table, the two-wheeler to car ratio is 14 to 5 in 2021, 32 to 11 in 2022, 3 to 1 in 2023, 20 to 7 in 2024 and 44 to 15 in 2025. Cross-multiplying 14 over 5 (2.8) with 3 over 1 (3.0) shows the ratio was highest in 2023, and you need no decimals to see it: 14 times 1 is 14, 5 times 3 is 15, so 14 over 5 is less than 3 over 1.

10. Approximation and calculation tricks

Data questions rarely need exact long division. The exam options are usually spaced widely enough that a good approximation picks the right one. The tools are few and worth knowing by heart.

Learn the fraction to percent table. One half is 50 percent, one third is 33.33, one fourth is 25, one fifth is 20, one sixth is 16.67, one seventh is 14.29, one eighth is 12.5, one ninth is 11.11, one tenth is 10, one twelfth is 8.33 and one sixteenth is 6.25. From those, build the rest: three eighths is 37.5, five eighths is 62.5, seven eighths is 87.5, five sixths is 83.33, and the sevenths are 14.29 times the numerator, so three sevenths is 42.86 and six sevenths is 85.71. If a question asks for four sevenths of 2,100, divide first: 2,100 over 7 is 300, and times 4 is 1,200. The result matches the exact value.

Round to convenient numbers and track the direction of the error. 4.98 times 19.97 is close to 5 times 20, which is 100, and the exact product is 99.45; rounding both numbers up gave an answer slightly too high. 596 divided by 29.8 is exactly 20, because 29.8 times 20 is 596. 398 times 0.249 is about 400 times 0.25, which is 100, and the exact value is 99.10. For squares near a round number, use the identity that 49.7 squared is 2,500 minus a little: 2,500 minus 29.91 gives 2,470.09, so "about 2,470".

Use the difference method for percentage changes. For 1,299 times 0.0199 the quickest route is 1,300 times 0.02, which is 26, and the exact value is 25.85, so any option near 26 is the answer. When two options are within 1 percent of each other, approximation will not separate them, and that is the moment to compute exactly.

Two more tools. Add numbers in groups that make hundreds or thousands before adding the rest: 1,249 plus 2,751 plus 3,198 plus 1,802 is easily 4,000 plus 5,000, which is 9,000. And when a question asks for a percentage between two close numbers, express the difference as a fraction of the base and simplify before converting: a rise from 160 to 184 is 24 over 160, which is 3 over 20, which is 15 percent.

For general arithmetic shortcuts (squares, multiplication by 11, divisibility), use the arithmetic shortcuts guide and do not duplicate them here.

11. Mean, median, mode and variance

These statistics appear in the Commission's Assistant Motor Vehicle Inspector scheme for General Studies and Mental Ability, and they are the natural extension of the Group-II averages topic. The definitions are short. The mean is the sum divided by the count. The median is the middle value of the ordered list, or the average of the two middle values if the count is even. The mode is the most frequent value. The variance is the average of squared deviations from the mean.

Take the data set 6, 7, 7, 9, 11. The sum is 40 and there are 5 values, so the mean is 8. The ordered list has 7 in the middle, so the median is 7. The value 7 occurs twice, so the mode is 7. The deviations from the mean are minus 2, minus 1, minus 1, 1 and 3; their squares are 4, 1, 1, 1 and 9, which sum to 16. Dividing by 5 gives a population variance of 3.2. Dividing by 4 gives a sample variance of 4. The two definitions differ in the divisor. A question in an objective paper will say which it means, or the options will contain only one of the two values. If the question is unspecific, the population version (divide by n) is the one a school-level syllabus teaches first.

Keep three relationships in mind. Adding the same number to every value changes the mean and median by that number but leaves the variance unchanged. Multiplying every value by k multiplies the mean and median by k and the variance by k squared. In a skewed set the mean is pulled towards the long tail, so for the set 6, 7, 7, 9, 11 the mean (8) is above the median (7).

A second set, 10, 20, 30, 40, 50, has mean 30 and population variance 200, because the squared deviations 400, 100, 0, 100 and 400 sum to 1,000, and 1,000 over 5 is 200. The standard deviation is the square root, about 14.14. The 2022 version of the same scheme also listed the coefficient of variation, the standard deviation divided by the mean, as a percentage; here it is 14.14 over 30, or 47.14 percent. The 2025 notification no longer lists it, so confirm the current scheme.

12. Number series and letter series

Number and letter series are the first two named items of the mental-ability paragraph. Work through a fixed order of tests so that you do not wander: constant difference, constant ratio, differences that themselves form a pattern, squares and cubes with a small adjustment, products of consecutive numbers, and alternating series.

The series 4, 9, 19, 39, 79 follows "double and add one": 4 times 2 plus 1 is 9, 9 times 2 plus 1 is 19, and so on, so the next term is 79 times 2 plus 1, which is 159. The series 3, 8, 15, 24, 35 is n squared minus 1 for n from 2, so the next is 7 squared minus 1, which is 48. The series 2, 6, 12, 20, 30 is n times n plus 1, and the next is 6 times 7, which is 42. The series 1, 2, 6, 24, 120 multiplies by 2, 3, 4, 5, so the next is 120 times 6, which is 720, the factorial of 6. The series 5, 6, 8, 11, 15 has differences 1, 2, 3, 4, so the next difference is 5 and the next term is 20.

Alternating series interleave two rules. In 3, 5, 6, 10, 12, 20, the odd positions are 3, 6, 12 (doubling) and the even positions are 5, 10, 20 (doubling), so the seventh term is 24. When a series looks irregular, split it into odd and even positions before giving up.

For letter series, convert letters to positions, A as 1 to Z as 26, and treat them as numbers. The series A, C, F, J, O moves by 2, 3, 4, 5, so the next move is 6 and the letter is U. When the count runs past Z, wrap round: after Z comes A again.

A check habit: after finding a rule, test it on every term, not only the first two. The classic trap is a rule that fits the first three terms and fails at the fourth.

13. Coding-decoding, odd man out and relations

Coding-decoding questions give a rule through one example and ask you to apply it to another. Common rules are shifting each letter by a fixed number, replacing each letter by its opposite in the alphabet (A with Z, B with Y), reversing, or converting to place values.

If ROAD is coded by shifting each letter one place forward, it becomes SPBE. If CAT is coded as ECV, every letter has moved two places forward, so DOG is coded as FQI. Under the opposite-letter rule, BUS becomes YFH, since B pairs with Y, U with F and S with H. As a numeric code, BAG with A as 1 gives 2 plus 1 plus 7, which is 10. Always test a coding rule on the full given word before applying it.

Odd man out asks which item does not belong. First look at number properties: in 8, 27, 64, 100, 125 four are perfect cubes (2, 3, 4 and 5 cubed) and 100 is not, so 100 is odd. In 2, 3, 5, 9, 11 every number except 9 is prime. In 121, 144, 169, 196, 225, 250 five are perfect squares, so 250 is odd. For words, look at category and then at function; with several possible answers, choose the one that depends on the property common to the largest group.

Relation problems chain statements. A is the father of B, B is the sister of C, and C is the mother of D. C is a mother, so C is female, and B is her sister, so B is female. A is the father of both B and C, so A is D's mother's father, the maternal grandfather. Draw the family as a small tree for every question of this kind; speed comes from a standard drawing, not from clever reading. For more relation types, use the blood-relations section of the shortcuts guide.

14. Number system, order of magnitude, and shapes

The number-system questions in this block are short: unit digits, remainders, divisibility. The unit digit of powers repeats in cycles of at most four. For 7, the unit digits are 7, 9, 3, 1, and then the cycle repeats, so 7 to the power 45 has the same unit digit as 7 to the power 1, because 45 leaves a remainder of 1 on division by 4. The unit digit is 7. For divisibility by 11, subtract the sum of the digits in the even places from the sum in the odd places; if the result is 0 or a multiple of 11, the number is divisible. For 91729, the odd-place digits (from the right) are 9, 7, 9, which sum to 25, and the even-place digits are 2 and 1, which sum to 3. The difference is 22, so the number is divisible by 11. The script confirms that 91729 is exactly 11 times 8,339.

Order of magnitude asks which quantity is larger or how large a result is. Count digits and compare powers of ten before multiplying. Which is larger, 3 to the 20 or 2 to the 30? Write 3 to the 20 as 9 to the 10 and 2 to the 30 as 8 to the 10. Since 9 is more than 8, 3 to the 20 is larger; the exact values are 3,486,784,401 and 1,073,741,824. For decimals, 0.5 times 10 to the minus 3 is 5 times 10 to the minus 4, which is more than 4 times 10 to the minus 4. A product of 4.8 times 10 to the 5 and 2.1 times 10 to the 3 is about 10 times 10 to the 8, or 10 to the 9; the exact value is 1.008 times 10 to the 9.

Shapes and their sub sections is best read as counting questions. If the apex of a triangle is joined to three interior points on the base, the base has five points in all (two corners and three interior), and every pair of base points forms a triangle with the apex, so there are 5 choose 2, which is 10 triangles. A grid with 3 rows and 4 columns of unit cells contains 6 times 10, which is 60 rectangles of any size, because you choose two of the 4 horizontal lines and two of the 5 vertical lines. A 4 by 4 grid contains 16 plus 9 plus 4 plus 1, which is 30 squares. Learn these three formulas and you can answer most counting items without drawing.

15. Logical reasoning: statements and the three modes

The Group-II syllabus names three modes of reasoning: deductive, inductive and abductive. They differ in what the conclusion promises. In deductive reasoning, if the premises are true the conclusion must be true. In inductive reasoning, a general conclusion is drawn from specific cases, and it is probable, not certain. In abductive reasoning, you choose the best explanation for an observation, which is a likely cause and not a proof.

Four statement formats are named.

Statement and assumptions. An assumption is something taken for granted for the statement to make sense. Statement: "The office has asked staff to submit forms online to save time." Assumption I: online submission is faster than manual submission. Assumption II: all staff have internet access. Assumption I is implicit, because the purpose depends on it. Assumption II is not necessary for the statement to make sense, though it would be needed for success. The test is: if the assumption were false, would the statement lose its point?

Statement and argument. An argument is strong if it is relevant, specific and important to the statement; it is weak if it is vague, personal or unrelated. Do not substitute your own opinion on the policy; judge only whether the argument bears on the proposal.

Statement and conclusion. A conclusion follows only if it is a necessary consequence of the statement. Take "All officers are graduates. Some graduates are trainers." The conclusion "Some officers are trainers" does not follow. A short model shows why: officers {1}, graduates {1, 2}, trainers {2}. All officers are graduates, some graduates are trainers, yet no officer is a trainer. One consistent counter-model is enough to reject a conclusion.

Statement and courses of action. A course of action is a step that follows the problem. Judge whether it is practicable, responsive to the stated problem and not extreme.

A rule that serves in all four: use only the information given, no outside facts, no opinions. If two options feel right, one of them is probably adding something the statement did not say.

16. A fourteen-day practice plan

Days 1 and 2: learn the fraction-to-percent table and the multiplier method; practise ten table questions daily. Days 3 and 4: bar diagrams, with a stopwatch of two minutes per question; write every value on rough paper before reading the question. Days 5 and 6: line graphs and pie charts, with extra attention to absolute versus percentage change. Day 7: a mixed set of fifteen questions and a review of every error against its category (misread unit, wrong base, arithmetic slip, speed). Days 8 and 9: caselets and data sufficiency, with a rule to draw a grid every time. Days 10 and 11: series, coding, odd man out, relations. Day 12: number system, order of magnitude and counting shapes. Day 13: logical reasoning formats. Day 14: a timed 30-question mental-ability section, which is the size of the Group-II block, in 30 minutes, then a written error log.

The ratio of practice to reading should be 80 to 20. Only repetition produces speed, and only an error log shows which kind of mistake you actually make.

17. Common mistakes

• Reading the chart before the title and the units.
• Using the last year as the base in a growth question.
• Calling an 8-point rise in share an 8 percent rise.
• Adding successive percentages instead of multiplying.
• Taking the simple average of percentages with unequal groups.
• Over-rounding when options are close.
• Solving past the point of sufficiency in data sufficiency.
• Treating a counter-model as a quibble in conclusion questions instead of as the proof it is.

Key facts for exams

• APPSC Group-II Screening Test: General Studies and Mental Ability, 150 marks, five blocks of 30; Mental Ability is one block (syllabus signed 27 April 2023, as on October 2026).
• The Group-II mental-ability data analysis lists tables, bar diagram, line graph and pie-chart.
• The Assistant Motor Vehicle Inspector scheme (Notification No. 21/2025) lists mean, median, mode and variance in General Studies and Mental Ability.
• 1 percent of a pie is 3.6 degrees.
• Percentage change = change divided by the first value; use multipliers for chained changes.
• A rise of 25 percent is cancelled by a fall of 20 percent; a rise of 20 percent by a fall of 16.67 percent.
• Population variance divides by n; sample variance divides by n minus 1.
• Triangles from an apex joined to k interior base points: (k+2) choose 2.

10 practice questions

Use the practice table (registrations) for questions 1 and 2, the double bar (centres) for question 3, the line series (applications) for question 4 and the pie (7,20,000-rupee budget) for questions 5 and 6.

1. What was the total number of registrations in 2024? (a) 9,090 (b) 9,390 (c) 9,690 (d) 10,290

2. By what percentage did car registrations grow from 2021 to 2025? (a) 33.33 percent (b) 45 percent (c) 50 percent (d) 75 percent

3. Which centre had the highest percentage of qualified candidates among those who appeared? (a) A (b) B (c) C (d) E

4. In which month was the percentage rise in applications over the previous month the highest? (a) February (b) March (c) April (d) May

5. How much more is spent on fuel than on rent, in rupees? (a) 36,000 (b) 54,000 (c) 72,000 (d) 1,08,000

6. What is the central angle, in degrees, of the maintenance sector? (a) 36 (b) 54 (c) 72 (d) 90

7. Staff of 240 are in offices X, Y and Z as in the caselet in section 7 (X to Y is 3 to 2, Z has 25 percent of the total; women are half in X, a quarter in Y and 40 percent in Z). How many women are there? (a) 87 (b) 90 (c) 96 (d) 102

8. For the data 6, 7, 7, 9, 11, what is the population variance? (a) 2.8 (b) 3.2 (c) 4 (d) 8

9. What is the next term in 5, 6, 8, 11, 15, ...? (a) 18 (b) 19 (c) 20 (d) 21

10. The price of an item rises by 25 percent. By what percentage must consumption fall to keep expenditure unchanged? (a) 15 percent (b) 20 percent (c) 25 percent (d) 33.33 percent

Answers:

1. (b) 9,390. 6,000 plus 2,100 plus 1,080 plus 210 is 9,390.
2. (c) 50 percent. The increase is 750 on a base of 1,500.
3. (c) C. Its rate is 45 percent against 40, 40, 30 and 40 for the others.
4. (c) April. The rise over March is 450 on 1,350, which is 33.33 percent.
5. (c) 72,000. The share difference is 10 percent of 7,20,000.
6. (b) 54 degrees. 15 percent of 360.
7. (c) 96. The offices have 108, 72 and 60 staff, with 54, 18 and 24 women.
8. (b) 3.2. The squared deviations sum to 16 and are divided by 5.
9. (c) 20. The differences are 1, 2, 3, 4 and next is 5.
10. (b) 20 percent. One minus 1 over 1.25 is 0.2.

Frequently asked questions

Does the Group-II Screening Test have data interpretation? The syllabus signed 27 April 2023 lists data analysis with tables, bar diagram, line graph and pie-chart inside the Mental Ability block of 30 marks. The number of questions the block carries is not stated in the syllabus document, so check the notification for the paper.

Are caselets and data sufficiency in the Group-II syllabus? They are not named. They are useful practice because they train the same reading and calculation skills.

Do I need to learn variance for Group-II? The Group-II mental-ability paragraph does not name it. The Assistant Motor Vehicle Inspector scheme in Notification No. 21/2025 does, so learn it if you plan to attempt both.

How fast should a data interpretation question be? Aim for 60 to 90 seconds per question once you have practised, and treat a question that needs more than two minutes as one to skip and return to.

Which is more important, speed or accuracy? Accuracy first, because wrong answers can cost marks where the notification prescribes negative marking, and speed grows with repetition. Check your notification for the penalty.

Is this the same as the AP Police arithmetic syllabus? There is overlap in numeracy, but the police test scheme is set by the SLPRB notification of the year. Treat this guide as shared preparation and confirm the scheme in the current notification.

Where do I find arithmetic shortcuts for percentages and ratio? In our guide on reasoning and arithmetic shortcuts.

Are the data sets in this post real? No. All datasets are invented for practice and do not describe real offices or years.

Related reading

• Reasoning and arithmetic shortcuts for AP Police and Sachivalayam
• APPSC Group 2 (Group-II Services): scheme, syllabus and preparation guide
• APPSC Group-II prelims 90-day plan
• AP Police SI notification guide
• AP Police Constable guide
• Negative marking, time management and mock strategy
• APPSC Assistant Motor Vehicle Inspector and the Transport Department
• Telugu version of this post
• Andhra Pradesh state guide: history, geography, government and every AP exam
• Telangana state guide (same syllabus for the general-studies sections)
• Telangana government exams 2026

Practise on pareeksha.in

After the fourteen-day plan, test yourself with a mock test, where the platform offers 10 Lakh+ MCQs, 100+ Exams and 5,000+ Full-Length Mocks. Time a 30-question mental-ability section, keep an error log by category, and confirm the current scheme in the official notification.

Sources and verification

Pages and documents actually opened for this post:

• Andhra Pradesh Public Service Commission, Syllabus for Screening and Main Examination for the recruitment to the post of Group-II Services, signed 27-04-2023 (psc.ap.gov.in), Mental Ability block read in full.
• Andhra Pradesh Public Service Commission, Notification No. 21/2025 dated 24-09-2025, Assistant Motor Vehicle Inspector in A.P. Transport Service, Annexure-II scheme and syllabus (psc.ap.gov.in), for the data-analysis wording in Paper-I.
• Andhra Pradesh Public Service Commission, Notification No. 12/2022 dated 30-09-2022, same post, Annexure-II (psc.ap.gov.in), for the earlier wording that included the coefficient of variation.
• Our own guides, linked above, for the arithmetic content, which is not repeated.

All datasets, examples and answers were computed with Python scripts written for this post; they are invented and carry no real-world source. Not verified: the number of Mental Ability questions and the negative-marking rule for Group-II (check the notification for your exam), and the current SLPRB scheme for the police written tests, which we have not restated here. Check the latest notification before relying on any scheme.