Data Interpretation — SSC CGL Quantitative Aptitude Formulas and Traps

Reading Tables and Graphs

Totals and differences of totals from a table

Difference of two totals

\sum a_i - \sum b_i = \sum (a_i - b_i)

$a_i$ the entries of the first group
$b_i$ the matching entries of the second group

Reading a cumulative and-above table

Bands from a cumulative table

N_{a \le x < b} = N_{\ge a} - N_{\ge b} \qquad N_{< a} = N_{\text{total}} - N_{\ge a}

$N_{\ge a}$ students with a or more marks
$N_{\text{total}}$ everyone, the "0 and above" column

Common traps

Dropping the unit line

A table headed "in thousands" that gives a difference of 8.2 means 8,200, and the options are usually printed in full. Read the unit line before you read the numbers.

Adding a printed Total row again

Some tables print a Total row or column. It is already the sum of the entries above it, so adding it in with them doubles the answer.

Counting the row that equals the limit

"More than the average" or "more than 55" leaves out a row exactly equal to it. Only "at least" or "not less than" includes the equal row.

Reading second highest as the second row

Second highest is about size, not position. Build the full column of values first, then take the largest after the largest.

Reading a cumulative column as a band

"20 and above" includes everyone with 30, 40 or more. The number who scored from 20 up to 30 is "20 and above" minus "30 and above".

Looking up the percentage instead of the marks

The columns are headed in marks. A cut-off given as a percentage must first be turned into marks out of the paper's maximum.

Averages from Data

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Average of a row or a column

Average and the assumed average

\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} \qquad \bar{x} = A + \frac{\sum (x_i - A)}{n}

$n$ number of entries
$A$ any round value you choose

Comparing or subtracting two averages

Difference of averages over the same count

\bar{x} - \bar{y} = \frac{\sum x_i - \sum y_i}{n}

$n$ the common number of entries

Values above, below or closest to the average

Gap from the average

d_i = x_i - \bar{x} \qquad \sum d_i = 0

Combined average of two or more groups

Combined average

\bar{x}_{\text{combined}} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

$n_1, n_2$ sizes of the two groups
$\bar{x}_1$ , $\bar{x}_2$ their averages

Common traps

Dividing by the wrong count

Years from 2010 to 2013 are four entries, not three. Count the entries you added, not the gap between the first and last year.

Rounding the entries before adding

Add the entries exactly, divide, and only then compare with the options. Rounding each entry first can shift the average onto a neighbouring option when the options sit close together.

Comparing totals when the counts differ

A total of 300 over 4 years is a larger average than a total of 360 over 6 years. Totals stand in for averages only when the number of entries is the same.

Counting the entry that equals the average

An entry equal to the average is not more than it and not less than it. Leave it out of both counts unless the question says "at least" or "at most".

Taking the plain mean of two averages

A group of 10 averaging 30 and a group of 30 averaging 40 do not combine to 35. The larger group pulls harder: the combined average is 37.5.

Ratios from Data

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Ratio of two totals in lowest terms

Ratio in lowest terms

a : b = \frac{a}{h} : \frac{b}{h}, \quad h = \text{HCF}(a, b)

How many times, and ratios of averages

Ratio of averages over the same count

\frac{\bar{x}}{\bar{y}} = \frac{\sum x_i \div n}{\sum y_i \div n} = \frac{\sum x_i}{\sum y_i}

Finding each part before taking the ratio

Recovering a value from a ratio

E : I = p : q \;\Rightarrow\; I = E \times \frac{q}{p}

$E$ the value you know
$I$ the value you want

Common traps

Writing the ratio the wrong way round

"The ratio of A to B" puts A first. Both orders nearly always appear among the options, so the order decides the mark.

Stopping before lowest terms

180 : 225

and

4 : 5

are the same ratio, but the options print only the reduced form. Keep dividing until the two numbers share no factor.

Dividing averages that cover different counts

The ratio of averages equals the ratio of totals only when both are over the same number of entries. A 3-year average against a 5-year average needs each division done.

Counting only the "only" row

Everyone who plays football includes those who play football and basketball. The row "football but not basketball" is only part of the football players.

One Value as a Percentage of Another

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One value as a percentage of another

A as a percentage of B

\frac{A}{B} \times 100\%

$A$ the value being described
$B$ the base, the value after "of"

A part as a percentage of the whole

Share of the whole

\text{Share} = \frac{\text{part}}{\text{whole}} \times 100\% \qquad \%\,\text{rest} = 100\% - \%\,\text{share}

Aggregate percentage of marks

Aggregate percentage

\text{Aggregate} = \frac{\sum \frac{p_i}{100} M_i}{\sum M_i} \times 100\%

$p_i$ percentage scored in paper i
$M_i$ maximum marks of paper i

Fractions and their percentage values

Fraction	Percentage	Fraction	Percentage
$\frac{1}{2}$	50%	$\frac{1}{8}$	12.5%
$\frac{1}{3}$	33.33%	$\frac{3}{8}$	37.5%
$\frac{2}{3}$	66.67%	$\frac{5}{8}$	62.5%
$\frac{1}{4}$	25%	$\frac{7}{8}$	87.5%
$\frac{3}{4}$	75%	$\frac{1}{9}$	11.11%
$\frac{1}{5}$	20%	$\frac{1}{11}$	9.09%
$\frac{1}{6}$	16.67%	$\frac{1}{12}$	8.33%
$\frac{1}{7}$	14.29%	$\frac{1}{16}$	6.25%

Multiply up for other numerators:

\frac{5}{16} = 31.25\%

,

\frac{5}{32} = 15.625\%

.

Common traps

Dividing by the wrong base

"A is what percent of B" divides by B. Swapping them gives a number on the other side of 100%, and that number is usually one of the options.

Mixing up one-sixth and one-seventh

\frac{1}{6}

is 16.67% and

\frac{1}{7}

is 14.29%. Options are often built from the neighbouring fraction, so reduce fully before you read the table.

Comparing parts instead of rates

A state with 920 qualified out of 9,200 in one year (10%) and 850 out of 7,000 in another (about 12.1%) did better in the second year, though fewer qualified. Compare part over whole for each row, never the parts alone.

Averaging percentages from papers of different sizes

70% in a 150-mark paper is worth more marks than 70% in a 50-mark paper. When the maximums differ, convert each percentage to marks before adding.

Judging the best subject by raw marks

The subject with the most marks may also have the largest maximum. "Highest percentage" means marks divided by maximum for each subject.

Percentage Change and Percent More or Less

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Percentage increase or decrease between two values

Percentage change

\%\,\text{change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100

Percent more than and percent less than

Percent more or less

\frac{A - B}{B} \times 100 \qquad A = B\left(1 + \frac{r}{100}\right) \Rightarrow B \text{ is } \frac{r}{100 + r} \times 100\% \text{ less than } A

Applying a percentage change to find a value

Applying a percentage change

\text{New} = \text{Old} \times \left(1 + \frac{r}{100}\right) \qquad \text{Old} = \frac{\text{New}}{1 + \frac{r}{100}}

Common traps

Dividing by the new value

A rise from 40 to 50 is 25%, not 20%. The change is always divided by the starting value; dividing by the later value gives a smaller, wrong answer that sits among the options.

Taking the biggest jump as the biggest percentage

A rise of 36 on a base of 60 is 60%, while a rise of 21 on a base of 28 is 75%. Compare change over base, not the changes alone.

Using the same percentage both ways

If A is 25% more than B, B is not 25% less than A. The gap is measured on different bases, so B is 20% less than A.

Picking the size before the direction

More/less questions print each size twice, once with "more" and once with "less". Decide which value is bigger before you divide.

Taking the percentage off the new value

If 32,400 is 8% more than last year, last year is

32400 \div 1.08 = 30000

, not

32400 - 8\%

of 32,400. The 8% belongs to last year's value.

Pie Charts and Shares of a Total

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From a percentage share to a value

Value of a share

\text{Value} = \frac{p}{100} \times \text{Total}

$p$ the sector's percentage

Degrees and central angles in a pie chart

Pie chart angles

\text{Value} = \frac{\theta}{360^\circ} \times \text{Total} \qquad \theta = \frac{p}{100} \times 360^\circ = 3.6p^\circ

$\theta$ the sector's central angle
$p$ the sector's percentage

From one known share to the total or another share

From a known share

\text{Total} = \frac{V}{p} \times 100 \qquad \text{Other share} = V \times \frac{q}{p}

$V$ the known sector's value
$p$ the known sector's percentage
$q$ the wanted sector's percentage

Comparing sectors without working out the total

Comparing sectors of one chart

\frac{\text{Sector } A}{\text{Sector } B} = \frac{p_A}{p_B} = \frac{\theta_A}{\theta_B}

Common traps

Comparing percentages across different totals

22% of 18,000 is more than 24% of 15,000. When each year has its own total, a smaller share can still be a bigger number, so convert to numbers before comparing.

Rounding each share before adding

Add the percentages, multiply once, and round only at the end. Rounding each sector's value first can land one unit away, on a neighbouring option.

Reading degrees as percentages

A sector marked 45° is 12.5% of the total, not 45%. Check the chart's heading: "values are in degrees" means divide by 360, not 100.

Using 100 where 360 belongs

An angle is a share of 360°. A 15% sector is

15 \times 3.6 = 54^\circ

, not

15^\circ

.

Dividing by the wanted share instead of the known one

1% of the total comes from the sector whose value you know. If 24% is ₹21,600, divide by 24, not by the 18% you are asked about.

Comparing sectors of charts with different totals directly

A 25% sector of the passed students and a 100° sector of the appeared students cannot be compared as shares, because the two charts stand for different totals. Convert each to a number of students first; only charts on the same total compare share to share.

Two-Step Share Problems

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A ratio or a percentage inside a share

A part inside a share

\text{Part} = \text{Total} \times \frac{p}{100} \times \frac{b}{a + b}

$p$ the group's percentage of the total
$a : b$ the ratio that splits the group; b is the wanted side

Working back to a missing total

Back to the total

\text{Total} = \frac{\text{known part}}{\frac{p}{100} \times \frac{b}{a + b}}

Common traps

Adding ratios from different groups

5 : 7

in one school and

2 : 3

in another do not combine to

7 : 10

. Work out the boys and girls in each school, add the counts, then form the ratio.

Applying the second percentage to the whole total

"30% of the employees in department D are women" means 30% of department D, not 30% of the company. Find the department first.

Undoing only one step of the chain

If 270 women are

\frac{5}{9}

of those who qualified, 486 is the number who qualified, not who appeared. Divide by the qualifying rate as well.

Reading 20% less as 20% of

"B is 20% less than M" makes B equal to 80% of M. Writing B as 20% of M, or M as 120% of B, gives a wrong pair.

Data Interpretation formulas

Reading Tables and Graphs

Totals and differences of totals from a table

Reading a cumulative and-above table

Common traps

Averages from Data

Average of a row or a column

Comparing or subtracting two averages

Values above, below or closest to the average

Combined average of two or more groups

Common traps

Ratios from Data

Ratio of two totals in lowest terms

How many times, and ratios of averages

Finding each part before taking the ratio

Common traps

One Value as a Percentage of Another

One value as a percentage of another

A part as a percentage of the whole

Aggregate percentage of marks

Fractions and their percentage values

Common traps

Percentage Change and Percent More or Less

Percentage increase or decrease between two values

Percent more than and percent less than

Applying a percentage change to find a value

Common traps

Pie Charts and Shares of a Total

From a percentage share to a value

Degrees and central angles in a pie chart

From one known share to the total or another share

Comparing sectors without working out the total

Common traps

Two-Step Share Problems

A ratio or a percentage inside a share

Working back to a missing total

Common traps

More SSC CGL Quantitative Aptitude formula sheets