Data Interpretation

Before we open full charts, we practise the calculations on small data: a list of five or six numbers, or a two-column table. Every big DI set is just these small steps joined together, so this type is both your warm-up and your base.

Total and average. Add the values asked for, and for the average divide by how many values you added. The trap is in the counting. "Average of the given 6 days" means divide by 6, even if one day's value is small. A quick check: the average must lie between the smallest and the largest value.

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Ratio. Write both totals, then cancel. 240 : 320 are both divisible by 80, which gives 3 : 4. If the common factor is not obvious, cancel in steps: 1260 : 1540 → 126 : 154 → 9 : 11.

Percentage of, more than, less than. Fix the base first. In "Tuesday is what % more than Monday", Monday is the base. In "Monday is what % less than Tuesday", Tuesday is the base. Between the same two numbers, the percentage increase and the percentage decrease are never equal: 500 → 600 is a 20% rise, but 600 → 500 is a 16.67% fall.

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Overall percentage. When marks are given in several subjects, or students in several schools, the overall percentage = total obtainedtotal maximum × 100. Never take the simple average of percentages when the bases are different. Averaging 80% of 300 and 70% of 500 as 75% is wrong; the true figure is 240 + 350800 = 73.75%.

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Growth. Increase = new − old; percentage increase = increaseold × 100; average annual increase = last − firstnumber of gaps.

Radhika's test scores are 18, 22, 15 and 25, each out of 25. Her total is 80 out of 100, so her overall percentage is 80% and her average score is 20. Her best score (25) is 25 − 1515 × 100 = 66.67% more than her lowest (15).

Percent or percentage points? When a percentage itself changes, there are two different answers, and the wording of the question decides which one you give. A pass rate that moves from 40% to 50% has gone up by 10 percentage points, because 50 − 40 = 10. But it has gone up by 1040 × 100 = 25 percent, because the change is measured against the old value. Radhika's attendance rose from 80% to 92%: that is 12 percentage points, and 1280 × 100 = 15% more attendance. A difference of two percentages is in percentage points; a percentage change always divides by the old figure. When the percentages belong to different totals, first turn them into actual numbers and only then compare.

A table is the most direct form of data: rows, columns and exact numbers. Nothing has to be read off a scale, so the examiner makes the questions harder in other ways, with more values to add, two-step percentages and comparisons across rows and columns.

Read the table in two directions. A row usually belongs to one item (a household, a school, a salesperson) and a column to one period or category (a month, a year, a subject). Before the questions, decide which totals you will need. If the questions talk about "total consumption of H3", that is a row total; if they talk about "total consumption in January", that is a column total.

Write totals at the edge. Add a "Total" column and a "Total" row, on the rough sheet or in your head. In a 5 × 4 table this takes very little effort, and then three or four questions are answered almost directly.

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Two-step questions. Many table questions first make a new quantity and then compare it. Take "Students who failed in S4 are what percent more than those who failed in S1". First make failed = appeared − passed for both, then compare. Write the new column neatly instead of holding it in your head.

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Overall percentage from a table. If a table gives appeared and passed for several schools, the overall pass percentage is total passedtotal appeared × 100. The average of the five pass percentages is a different number, unless every school has the same number of students.

Radhika's coaching centre has two batches. Batch 1 has 40 students and 30 passed (75%). Batch 2 has 60 students and 54 passed (90%). Overall = 84100 = 84%, not 75 + 902 = 82.5%. The bigger batch pulls the overall figure towards its own percentage.

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Columns in different units. Some tables give one column in thousands, another in lakh or crore, and a careless addition mixes them. Convert everything to one unit before you add, compare or take a ratio. The conversions are 1 lakh = 100 thousand = 10⁵ and 1 crore = 100 lakh = 10⁷. A figure marked "in '000" is a number of thousands. Radhika's shop earns ₹2.4 lakh in March and ₹85 thousand in April. In thousands that is 240 and 85, so March is 240 − 85 = 155 thousand more, and April is 85240 × 100 ≈ 35.4% of March. Write the unit next to every number in the first pass; the percentages and ratios are the same in any unit, but the sums are not.

A bar graph shows each value as the height of a bar. In SSC papers the value is usually printed on top of each bar, so you read the number directly. If a value is not printed, read it against the gridlines on the vertical axis: a bar ending halfway between 100 and 150 stands for 125.

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Simple bar graph. There is one bar for each item, such as the sales of each shop, the marks of each student or the production in each year. The questions ask for the highest and lowest, totals, averages, differences and percentages between two bars. The tallest bar is visible at a glance, so "which is the maximum" needs no calculation at all.

Double (multiple) bar graph. There are two bars for each item, in two shades, for example boys and girls in each school, or doctors and nurses in each hospital. First fix the legend, that is, which shade stands for which group. Then decide what the question wants:

• a value of one group in one item (just read it);
• the total of both bars of one item (add the two bars);
• the total of one group across all items (add one shade only);
• a comparison between the two groups in the same item (the difference or ratio of the two bars).

Say Radhika draws the number of story books (S) and comics (C) on her two shelves. Shelf 1 has S = 40 and C = 25; Shelf 2 has S = 30 and C = 45. Total books on Shelf 2 = 30 + 45 = 75. Total comics = 25 + 45 = 70. On Shelf 2, comics are 45 − 3030 × 100 = 50% more than story books.

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Bars that show a rate. Sometimes the bars are not plain counts but rates, such as interest rates, strike rates or profit percentages. The question then supplies the formula. We take these up in the mixed DI type.

A line graph joins the values with straight lines, so it shows the trend: the rise, the fall, the steepest jump. That is why line graphs are mostly drawn over time (days, months, years), and the questions are full of growth: "on which day was the increase the highest", "what is the percentage fall", "what is the average increase".

Reading the line. Each dot is a value, and the slope between two dots is the change. A steep upward segment means a big increase; a downward segment means a decrease. You can often spot the biggest absolute rise just by finding the steepest upward segment, but check the numbers if two segments look alike.

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Biggest rise or biggest percentage rise? The steepest segment gives the biggest absolute rise. The biggest percentage rise also depends on the starting value: a rise of 12 from 40 (30%) beats a rise of 20 from 80 (25%). For percentage questions, always divide by the starting (previous) value.

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Two-line graphs. Two lines stand for two groups: exports and imports, two companies, two students. Use the legend to fix which line is which (solid or dashed). Typical questions:

• In how many years was line 1 above line 2? (Count the points where it is higher.)
• In which year was the gap between the two lines the biggest? (Subtract at each point.)
• Where do the lines cross? That is where the order changes.

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Radhika plots her weekly study hours: 12, 15, 14, 18. The increases are +3, −1 and +4, so the highest rise is in week 4. The percentage rises are 25% in week 2 and 414 ≈ 28.57% in week 4. Here week 4 leads on both counts, though that will not always happen.

Units on the axis. "In hundreds" or "in '000" applies to every value on the graph. If the question wants the answer in actual numbers, multiply once at the end, not at every step.

A pie chart shows how one total is divided into parts. The whole circle is the total, so each part is always given as a share, either in percent (adding up to 100%) or in degrees (adding up to 360°). Your first job is to see which form the chart uses.

Percent pie chart. Value of a sector = (its %) × Total ÷ 100. Central angle = (its %) × 3.6°. If the total is 30,000, then 1% = 300, and a 15% sector is 15 × 300 = 4500.

Degree pie chart. Value of a sector = its angle360° × Total. Percentage share = its angle360 × 100. A 54° sector is 54360 = 320 = 15% of the total.

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When a part is given instead of the total. SSC uses this setting a lot. Suppose "the total number of students in P and Q is 660". First add their shares (100° + 32° = 132°). Then 132° = 660, so 1° = 5 students. Now every sector is simply its angle × 5, and the total is 360 × 5 = 1800. In a percent chart, do the same with 1%.

Find the value of 1% (or 1°) once, and the whole chart opens up.

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Ratios and percentages between sectors need no total. A : B = (A's share) : (B's share), and "E is what percent of A" = E's shareA's share × 100. So in a degree chart, sectors of 96° and 120° give 96120 = 80% straight away.

Central angle questions. In a percent chart, multiply by 3.6. A difference of 6% between two sectors is a difference of 6 × 3.6 = 21.6° in their angles. Going the other way, a sector that is 36° bigger has a share 10 percentage points bigger, that is, 10% of the total more.

Radhika's monthly pocket money is shown in a pie chart: Books 25%, Food 40%, Travel 15%, Savings 20%. If she saves ₹600, then 20% = 600, so 1% = 30. Books = 25 × 30 = ₹750, and the central angle of Travel = 15 × 3.6 = 54°.

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Two pie charts. If two pie charts are given (say, for two years), do not compare their percentages directly unless the totals are the same. 20% of 50,000 is more than 25% of 30,000.

Here DI questions become a little more involved. The data is spread over two sources, or some values are missing, or the chart shows a rate and a formula is needed to get the actual numbers. The mathematics is still percentage, ratio and average; you only build the required number in two or three steps.

1. Missing-data tables. Some cells show "–", and the question gives a total that lets you fill them. "Shop X sold 2400 items in four days", with three days known, means the missing day = 2400 − (sum of the known days). Fill only the cells the question needs.

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2. Percentage + ratio (two-level) tables. A table may give the percentage of a group and then the ratio inside that group. Work from the outside in: first the group (a percent of the total), then its split (the ratio). In a town of 6000 where 25% are below the poverty line, 4500 are above it. If males : females above the line = 2 : 3, then females above the line = 35 × 4500 = 2700.

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3. Pie chart + table. The pie chart gives each department's share of the total, and the table gives the male : female ratio in each department. For any department, number = (its %) × total, and then split it by the ratio. When a question needs all the departments (say, "total male employees"), make a small working table once.

4. Formula-based charts. The bars show a rate, and the question supplies the formula:

• Strike rate = RunsBalls × 100 ⇒ Runs = Strike rate × Balls100
• Selling price = Cost price × 100 + Profit %100 (use 100 − Loss % for a loss)
• Simple interest = P × R × T100
• Mileage = DistanceFuel ⇒ Fuel = DistanceMileage
• Income = Expenditure + Savings

Write the formula, put in the values, and only then calculate.

A combined rate is not the simple average of the rates. The two match only when the balls (or the CPs) are equal. The combined strike rate of two batsmen = total runstotal balls × 100, and the overall profit percent on several articles = total SP − total CPtotal CP × 100.

Take a cricket match between Radhika and her friend. Radhika scores 30 runs off 20 balls (strike rate 150) and her friend scores 30 runs off 40 balls (strike rate 75). Their combined strike rate = 6060 × 100 = 100, not 150 + 752 = 112.5.

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A caselet is a DI set without a chart or a table. The data sits inside a short paragraph, and two to four questions follow. The mathematics is the same percentage, ratio and average you have used all through this chapter. The only new job is to turn the sentences into a table before you calculate anything.

Read twice, then build. On the first reading, find out what is being counted (students, rupees, items) and which groups it is split into. On the second reading, write each number next to the group it belongs to. A small grid is enough: groups as rows, categories as columns, and a Total at the edge.

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Fill the easy cells first. Start with the cell that a sentence gives you directly, and move on to the cells that depend on it. Take this caselet: "A class has 60 students, 40% of whom are girls. Of the girls, 25% play chess, and of the boys, 50% play chess." Girls = 40% of 60 = 24, so boys = 36. Chess players: 25% of 24 = 6 girls and 50% of 36 = 18 boys, so 24 students play chess and the other 36 do not. Every later question, such as "what percent of the chess players are girls" or "how many boys do not play chess", is now a single look at the table.

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Name the base of every percentage. "40% of the girls" and "40% of the class" are different numbers. In a caselet the base hides inside two or three words, so underline it while you read.

Relationship caselets. Sometimes there is no percentage at all: "Radhika has 4 stamps more than Aman, and Kavya has three times as many as Aman. Together they have 49 stamps." Call the quantity that the others are compared with x (here Aman = x). Then Radhika = x + 4 and Kavya = 3x, and x + (x + 4) + 3x = 49 gives 5x = 45, so x = 9. Radhika has 13 and Kavya has 27.

Overlapping groups. If a caselet says "60% like tea, 45% like coffee and 25% like both", the groups overlap. Tea or coffee = 60 + 45 − 25 = 80%, so neither = 100 − 80 = 20%, and only tea = 60 − 25 = 35%. Add the two groups and subtract the overlap once. Adding the groups without the subtraction counts the common people twice.

Before you answer, run your eye over the paragraph once more and check that every sentence has been used in the table. A sentence left unused usually means a wrong cell.

A histogram is the graph of grouped data. The classes lie along the horizontal axis as bars that touch each other, because the classes run on without a gap (0–10, 10–20, 20–30 …). The vertical axis shows the frequency. When all the classes have the same width, the height of a bar is the frequency, so you compare the bars by height and read each frequency straight off the bar.

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What a histogram answers:

• the number in one class is the height of its bar (here 14 students scored 20–30);
• the total number N is the sum of all the bars (4 + 9 + 14 + 8 + 5 = 40);
• "scored 30 or more" means the bars from 30 onwards: 8 + 5 = 13;
• a percentage is a bar (or a group of bars) divided by N: 1440 × 100 = 35%.

Unequal class widths. If a class is wider, it naturally collects more observations, so the bar height can no longer be the frequency. It shows the frequency density = frequencyclass width, and the area of the bar = height × width gives the frequency.

Frequency polygon. Mark the point above the class mark (mid-value) of every class at the height of its frequency, and join the points with straight lines. To close the figure, bring the line down to the axis at the class marks of one empty class before the first class and one after the last (frequency 0). The polygon carries the same information as the histogram, and you read it like a line graph: the height at a class mark is the frequency of that class.

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Cumulative frequency (cf). This is a running total. For the less-than type, the cf against an upper limit is the number of observations below that limit. In the data above, the cf at 30 is 4 + 9 + 14 = 27, so 27 students scored less than 30. The last cf equals N. For the more-than type, the cf against a lower limit is the number of observations at or above it. The frequency of one class is the difference of two consecutive cf values: 27 − 13 = 14.

Ogive. Plot each upper limit against its less-than cf, starting from the lower limit of the first class at 0, and join the points. The curve rises from 0 to N and never falls. To find the median, go to N2 on the cf axis, move across to the ogive and then down to the axis. Here N2 = 20 gives 25. Reading the ogive at two values and subtracting gives the number of observations between them. The cf read at an upper limit is the number of observations below it.

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A radar chart (also called a spider or web chart) draws one spoke for every category, all starting from the centre and spread at equal angles. The distance of a point from the centre, measured along its spoke, is the value of that category. Rings around the centre act as the scale, just like the horizontal gridlines of a bar graph, and the values of the rings are noted beside the chart. The points on the spokes are joined into a closed shape.

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How to read it. To get the value of S3, follow the spoke S3 out to the dot and see where it falls against the rings. A dot exactly on a ring has that ring's value. A dot halfway between the rings 60 and 80 is 70. In exam charts the values are usually printed beside the dots, so check for them before you start estimating.

What the shape tells you. A spoke that is far out is a high category and a spoke close to the centre is a low category. The spokes are separate categories, so a radar chart is not a graph of change. Do not read a "slope" between two spokes, and do not compare the area of the shape by eye; take the numbers and calculate.

Two shapes in one chart. A second shape (dashed in the figure) shows another group: two years, two firms, two students. Fix the legend first. Then compare spoke by spoke: the gap between the two dots is the difference at that category, and the total of a shape is the total of its dots.

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After the values are read, the questions are the usual DI calculations: total, average, difference, percentage more or less, and ratio. Read all the values you need off the spokes into a small list first, then calculate from the list. The percentage change from X to Y at a spoke is Y − XX × 100, and the base is always X.