Statistical Analysis, Graphs and Diagrams

Why this unit matters

• BPSC Mains GS Paper I has a section on statistical analysis, graphs and diagrams. These questions carry high marks and are fully scoring if you practise them.
• BPSC Prelims and JPSC ask quick arithmetic and data questions.

Measures of central tendency

• Arithmetic mean: x̄ = (Σx) ⁄ n.
  • Weighted mean: x̄ = (Σwx) ⁄ (Σw).
  • Combined mean: x̄ = (n₁x̄₁ + n₂x̄₂) ⁄ (n₁ + n₂).
  • For a frequency table: x̄ = (Σfx) ⁄ (Σf), using class mid-points for grouped data.
• Median: the middle value after sorting.
  • For an odd n, it is the ((n + 1) ⁄ 2)th value.
  • For an even n, it is the average of the (n ⁄ 2)th and (n ⁄ 2 + 1)th values.
  • Grouped data: Median = L + ((n ⁄ 2 − cf) ⁄ f) × h, where L is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h the class width.
• Mode: the most frequent value.
  • Grouped data: Mode = L + ((f₁ − f₀) ⁄ (2f₁ − f₀ − f₂)) × h.
  • Empirical relation: Mode ≈ 3 Median − 2 Mean.
• Which to use:
  • The mean is affected by extreme values (incomes, for example).
  • The median is better for skewed data.
  • The mode suits categories (the most common crop, for example).

Measures of dispersion

• Range = largest value − smallest value.
• Mean deviation = Σ|x − x̄| ⁄ n.
• Variance: σ² = Σ(x − x̄)² ⁄ n.
  • Standard deviation: σ = √(variance).
• Coefficient of variation (CV) = (σ ⁄ x̄) × 100. Use it to compare consistency: the lower the CV, the more consistent the series.
• Quartiles: Q₁ and Q₃. The interquartile range is Q₃ − Q₁.

Percentages, ratios and growth

• Percentage change = ((new − old) ⁄ old) × 100.
• Successive changes of a% and b%: net = a + b + (ab ⁄ 100). Use a minus sign for a fall.
• Percentage points vs per cent: a rise in unemployment from 4% to 5% is 1 percentage point, but 25 per cent.
• Growth:
  • Decadal growth = ((P₂ − P₁) ⁄ P₁) × 100.
  • CAGR = (ⁿ√(final ⁄ initial) − 1) × 100, where n is the number of years. For example, doubling in 10 years means a CAGR of about 7.2%.
  • Rule of 70: doubling time ≈ 70 ⁄ (growth rate in %).
• Ratio and proportion: a : b = ka : kb. To split ₹x in the ratio a : b, the first share is x × a ⁄ (a + b).
• Index numbers: Index = (current value ⁄ base value) × 100.
  • Laspeyres uses base-year weights; Paasche uses current-year weights.
  • India's CPI and WPI are index numbers.

Diagrams and graphs

Type	Use	Drawing tips
Simple bar diagram	Compare one variable across categories (districts, years)	Equal widths and gaps; start the axis at 0; label values
Multiple (grouped) bar	Compare two or three variables (male and female literacy by district)	A key or legend; distinct shading
Sub-divided (stacked) bar	Parts of a whole across categories	Same order of parts in every bar
Percentage bar	Composition in per cent	Every bar is 100
Pie chart	Parts of one whole	Angle = (value ⁄ total) × 360°; at most 6–7 slices
Line graph	Trend over time	Time on the x-axis; mark the points; use a false origin (a "break") only when needed
Histogram	Continuous class intervals	Bars touch; area ∝ frequency
Frequency polygon or ogive	Shape of a distribution; cumulative frequency	The ogive gives the median and quartiles
Pictogram, cartogram	Simple presentation; data on a map	Show the scale

• Every diagram needs a title, labelled axes with units, a scale, a legend and the source (for example "Source: Census 2011"). Neatness carries marks in BPSC Mains.

Worked BPSC-style example

Question. The literacy rates (%) of four districts of Bihar are: Patna 70.7, Rohtas 73.4, Purnia 51.1 and Kishanganj 55.5 (Census 2011). Find the mean and the range, and name a suitable diagram.

Answer.

• Mean = (70.7 + 73.4 + 51.1 + 55.5) ⁄ 4 = 250.7 ⁄ 4 ≈ 62.7%.
• Range = 73.4 − 51.1 = 22.3 percentage points.
• Diagram: a simple bar diagram with districts on the x-axis and literacy (%) on the y-axis (scale 1 cm = 10%). Draw a horizontal line for the state average (61.8%) for contrast.

Data interpretation strategy

• Read the title, units and notes first. Is the figure in lakh or crore, per cent or absolute?
• Approximate wherever the options are far apart.
• Avoid common traps: confusing the base year, mixing up percentage and percentage points, or reading a stacked bar's segment as a total.

Exam tip: Practise one table, one bar chart and one pie chart each day for a month. In BPSC Mains, carry a scale, pencil and eraser, and draw diagrams in pencil with neat labels.