Data Interpretation

A table, bar chart, pie chart, line graph, scatter plot or map-like grid displays numbers. A value is then read from the display, or computed from the values read: a growth rate, a share, a ratio, an average, or the category that satisfies a condition. The arithmetic is that of Ratios and Percentages; the skill specific to DI is reading the display correctly and choosing the right base for each percentage.

Every chart type has its own reading rule (a bar is read at its top against the axis, a pie slice is a share of 360 degrees, a line's slope is a rate of change), and its own trap. The material is a reading procedure, the formulas for the common derived quantities, and one data set shown in three charts, worked through question by question, plus a scatter plot.

A 3-D surface plot shows a third variable as height above an $x$-$y$ grid (often also colour-coded); a bubble chart shows it as bubble size (by convention, area). Read the third value from the labelled vertical axis, contour labels or legend, not from apparent height or size in a perspective drawing.

Derived quantities.

Rounding convention. Worked percentages are rounded to two decimal places at the last step; intermediate values are kept exact.

A factory makes two products, A and B. Production in thousand units per quarter:

Example 1 (growth). Percentage increase in total production from Q1 to Q4: $\frac{115 - 70}{70} \times 100 = \frac{45}{70} \times 100 \approx 64.29\%$.

Example 2 (largest step). Quarter-on-quarter increases in the total are $85 - 70 = 15$, $95 - 85 = 10$ and $115 - 95 = 20$. The largest increase is into Q4. On the line graph this is the steepest segment.

Example 3 (ratio). Ratio of A to B in Q3: $45 : 50 = 9 : 10$.

Example 4 (average). Average quarterly production of B: $165 / 4 = 41.25$ thousand units.

Example 5 (share and pie angle). A's annual production by quarter as shares of 200: Q1 $20\%$, Q2 $25\%$, Q3 $22.5\%$, Q4 $32.5\%$. The pie angles are these shares of $360^\circ$: $72^\circ$, $90^\circ$, $81^\circ$, $117^\circ$, which sum to $360^\circ$.

Example 6 (condition test). In which quarters did B produce more than A? Only Q3 (50 against 45). In Q4 B equals its Q3 value but A rose to 65.

Example 7 (percent versus percentage points). A's share of the quarterly total was $40/70 \approx 57.14\%$ in Q1 and $65/115 \approx 56.52\%$ in Q4. The share fell by about $0.62$ percentage points, even though A's own production rose by $\frac{65 - 40}{40} = 62.5\%$.

Example 8 (comparing across displays). "B's Q3 bar is taller than A's Q3 bar" and "A's Q3 slice is smaller than its Q4 slice" are both true, but a pie of B's production (total 165) could not be compared slice by slice with this pie of A's (total 200): equal angles in the two pies would mean equal shares, not equal quantities.

Example 9 (compound average growth). Total production rose from 70 to 115 over three quarter-steps. The compound growth rate per quarter is $(115/70)^{1/3} - 1 \approx 18.00\%$, while the simple average of the three step rates $\frac{15}{70}$, $\frac{10}{85}$, $\frac{20}{95}$ is about $18.08\%$. They are close here but not equal in general.

Six shops U, V, W, X, Y, Z report advertising spend $x$ and sales $y$, both in lakh rupees: U $(2, 10)$, V $(4, 14)$, W $(5, 20)$, X $(7, 21)$, Y $(8, 28)$, Z $(10, 30)$.

Example 10 (ratio from a scatter plot). Sales per lakh of spend: U 5, V 3.5, W 4, X 3, Y 3.5, Z 3. The highest is U. On the plot, U is the point whose line to the origin is steepest.

Example 11 (boundary trap). How many shops have sales more than three times their spend? U, V, W and Y: 4 shops. X and Z have exactly three times and do not count. A question saying "at least three times" would give 6.

Example 12 (centre). Mean sales: $(10 + 14 + 20 + 21 + 28 + 30)/6 = 123/6 = 20.5$ lakh. Median spend: the spends sorted are 2, 4, 5, 7, 8, 10, so the median is $(5 + 7)/2 = 6$ lakh.

Example 13 (ratio of percentage-of-whole values). A company's expenses were 800 lakh in 2023 and 1,000 lakh in 2024. Two pie charts show salaries as 25% of the 2023 total and 22% of the 2024 total.

Example 14 (percent of a percent). In a firm, 40% of employees are women. 25% of the women and 15% of the men are engineers. Take 100 employees: 40 women and 60 men; engineers $10 + 9 = 19$. So 19% of employees are engineers, and women are $\frac{10}{19} \approx 52.63\%$ of the engineers. Averaging 25% and 15% to get 20% ignores the unequal group sizes.

Example 15 (CAGR versus average growth). Revenue went 1,000, then 1,200, then 1,080, then 1,331 over three years.

Example 16 (when the gap is large). A value goes from 100 to 200 and back to 100. The yearly rates are $+100\%$ and $-50\%$; their average is $+25\%$, but the CAGR is $0\%$, since the value ended where it started. The simple average of rates overstates growth whenever rates vary.

Example 17 (map or table of regions). A district map is divided into four labelled regions, with population (thousands) and area (square kilometres) given in a table printed on the map:

Density is population divided by area. East has density $30/5 = 6$, North $40/8 = 5$, West $15/5 = 3$ and South $20/10 = 2$, so East is the densest. East's share of the total population is $30/105 = 2/7$. A colour band on a choropleth map would only place East in a range such as "5 to 7"; the table on the map is what gives the exact values.

Three-variable worked example. A bubble plot uses horizontal $x$ = advertising spend (thousand rupees), vertical $y$ = sales (thousand rupees), and bubble area = customers $z$. Plot A at $(1,2)$ with $z=100$, B at $(2,3)$ with $z=400$, and C at $(3,2)$ with $z=200$. The legend's 100-customer bubble has radius 1 plotting unit of size, so B has radius 2 and C radius $\sqrt2$; positions use the axis scales independently of this size scale. B represents 400 customers versus A's 100: four times the area, but only twice the radius. C represents 200 customers, twice A's count. Do not read $z$ from the y-coordinate or apparent perspective height.