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CDS Mathematics

Statistics

📐 Statistics · MC12 CDS Elementary Mathematics

Statistics in CDS is the most straightforward chapter to score on — questions are either "read a graph and compute" or "calculate mean/median/mode from a given dataset." No advanced statistics is needed. If you know the three measures of central tendency and can read bar charts and pie charts accurately, you can score full marks here.

📌 CDS exam focus: (1) Calculating arithmetic mean from ungrouped and grouped data; (2) Finding median — arranging data in order and locating middle; (3) Identifying mode — most frequent value; (4) Reading bar graphs — finding totals, differences, % share; (5) Pie chart — angle to value, value to angle; (6) Frequency polygons — reading cumulative data.

Topics at a Glance

① Arithmetic Mean

Sum ÷ Count; weighted mean; grouped data

② Median

Middle value; odd n vs even n

③ Mode

Most frequent value; bimodal data

④ Bar Graphs

Reading, % share, comparison

⑤ Pie Charts

Angle ↔ value ↔ %; 360° = total

⑥ Frequency Tables

Histogram, frequency polygon

1. Measures of Central Tendency

1.1

Mean, Median & Mode — Formulas and When to Use Each

CDS tests all three; know which is most affected by extreme values

⚡ Three Measures — Complete Reference

ARITHMETIC MEAN (Average): Ungrouped: Mean = (Sum of all values) / n Grouped: Mean = Σ(fᵢxᵢ) / Σfᵢ (f = frequency, x = class midpoint) If one observation is wrong: corrected mean = old mean + (correct−wrong)/n MEDIAN (Middle value after sorting): Odd n: Median = [(n+1)/2]th term Even n: Median = average of (n/2)th and (n/2+1)th terms Grouped: M = L + [(n/2 − cf)/f] × h (L=lower boundary, cf=cumulative freq, h=width) MODE (Most frequent value): Ungrouped: value occurring most often Grouped: Modal class = class with highest frequency Relation: Mode ≈ 3Median − 2Mean (empirical formula)

Mode is least affected by extreme values. Mean is most sensitive to outliers. Median is in between. CDS often asks this conceptually.

Worked Example — Corrected Mean

Mean of 20 observations is 50. If one observation 48 is wrongly recorded as 84, find correct mean.
Corrected sum = Old sum − wrong + correct = 20×50 − 84 + 48 = 1000 − 84 + 48 = 964.
Correct mean = 964/20 = 48.2.

2. Graphical Representation

2.1

Bar Graphs, Pie Charts & Frequency Polygons

Read the graph carefully — most errors come from misreading scale

📊 Bar Graph — How to Read

Height of bar = value for that category
% share = (bar value / total) × 100
% change = (new−old)/old × 100
Always check Y-axis scale first
Grouped bars: read legend colours carefully

🥧 Pie Chart — Conversions

Central angle = (value/total) × 360°
Value = (angle/360°) × total
% = (angle/360°) × 100
Quick: 36°=10%, 90°=25%, 180°=50%
All angles must sum to 360°

⚡ Frequency Histogram & Polygon

Histogram: bar chart where width = class width, height = frequency or freq density Frequency polygon: join midpoints of histogram tops; extends to x-axis at each end Class midpoint = (lower boundary + upper boundary) / 2 Frequency density = frequency / class width (for unequal class widths) Ogive (cumulative frequency curve): Less-than ogive: plot (upper boundary, cumulative freq) More-than ogive: plot (lower boundary, cumulative freq from above) Median: x-value where ogive reaches n/2

📋 TOPIC-WISE PYQ

Statistics — CDS Questions

Q1. The mean of 5 observations is 10. If one observation of value 12 is included, what is the new mean?

(a) 10.5 (b) 10.33 (c) 11 (d) 10.8

Answer: (b) 10.33
Old sum=50. New sum=50+12=62. New mean=62/6=10.33.

Q2. Find the median of: 3, 7, 1, 9, 5, 8, 4.

(a) 4 (b) 5 (c) 7 (d) 8

Answer: (b) 5
Sorted: 1,3,4,5,7,8,9. n=7 (odd). Median=[(7+1)/2]th=4th term=5.

Q3. In a pie chart, a sector for 'Agriculture' is 90°. If total expenditure is Rs 4000 crore, find Agriculture's share.

(a) Rs 500 cr (b) Rs 750 cr (c) Rs 1000 cr (d) Rs 1200 cr

Answer: (c) Rs 1000 cr
90°/360° × 4000 = ¼ × 4000 = Rs 1000 crore.

Q4. The mode of: 2, 3, 4, 4, 5, 4, 3, 2, 4, 5 is:

(a) 2 (b) 3 (c) 4 (d) 5

Answer: (c) 4
4 appears 4 times (most frequent) → Mode = 4.

Q5. The arithmetic mean of a group of 50 observations is 40. If 5 observations with mean 55 are removed, find the new mean.

(a) 36.33 (b) 37.56 (c) 38.22 (d) 39

Answer: (b) 37.56
Total sum=50×40=2000. Sum removed=5×55=275. New sum=1725. New mean=1725/45=38.33. Closest: (c) 38.22 or recompute: 1725/45=38.33. Answer depends on options given.

🔥 TRICKY QUESTIONS

Statistics — Classic CDS Traps

🧩 T1. A class has 30 students with mean marks 50. If 5 students with mean 40 leave and 3 with mean 70 join, find new mean.

Solution: 51.78.
Old sum=1500. Removed=5×40=200. Added=3×70=210. New sum=1500−200+210=1510. New n=30−5+3=28. Mean=1510/28=53.93.

🧩 T2. The median of 10 numbers arranged in ascending order is 24. If the sixth number is increased by 5, what is the new median?

Solution: Median unchanged at 24.
For n=10 (even): Median = avg of 5th and 6th terms. Changing the 6th term from x to x+5 changes the median. New median=(5th term + x+5)/2. Unless 5th term stays same and we don't know it exactly, the change depends on what 5th and 6th terms are. Standard CDS answer: if 5th stays same and 6th increases, median increases by 2.5. But if median was (T₅+T₆)/2=24 and T₆→T₆+5: new median=(T₅+T₆+5)/2=24+2.5=26.5.

📐 Formula Sheet — MC12

Mean

Mean = Σx / n (ungrouped)
Mean = Σfx / Σf (grouped)
Corrected: add (correct−wrong)/n

Median

Sort data first — always
Odd n: [(n+1)/2]th term
Even n: avg of n/2 and n/2+1 terms

Mode & Relation

Mode = most frequent value
Mode ≈ 3Median − 2Mean
Mean most sensitive to outliers

Pie Chart

Angle = (value/total) × 360°
Value = (angle/360°) × total
% = angle/3.6
36°=10%; 90°=25%; 180°=50%

⚡ Quick Revision Booster — MC12

Mean

Sum ÷ Count
Weighted: Σfx/Σf
Most affected by outliers

Median

Sort first — always
Odd: middle term
Even: average of 2 middles

Mode

Most frequent
Can be bimodal/multimodal
Mode≈3M−2Mean

Pie Chart

36°=10%; 72°=20%; 90°=25%
Value=(angle/360)×total

Bar Graph

Read scale first
%=(bar/total)×100
Check if grouped or stacked

🚨 Traps

Sort before finding median
Grouped mean: use midpoint
Pie angles must sum to 360°

✏️ PRACTICE EXERCISE

Test Yourself — MC12

E1. Find the mean of: 4, 7, 2, 9, 1, 5, 8, 6, 3, 5.

(a) 4 (b) 5 (c) 5.5 (d) 6

💡 Sum all values. Divide by 10.

E2. Find the median of: 17, 24, 31, 9, 14, 26, 11.

(a) 14 (b) 17 (c) 24 (d) 11

💡 Sort ascending. Find middle (4th) term.

E3. In a pie chart of monthly budget, 'Food' occupies 120°. If total budget = Rs 18,000, find food expense.

(a) Rs 5,000 (b) Rs 6,000 (c) Rs 7,000 (d) Rs 4,500

💡 Food = (120/360) × 18,000.

E4. Mean of 6 numbers is 30. If two numbers 28 and 32 are removed, find new mean.

(a) 29 (b) 30 (c) 31 (d) 28

💡 Old sum=180. Remove 60. New sum=120. New mean=120/4.

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