Statistics
~12 min read
- What: Statistics covers the three measures of central tendency — mean, median, mode — for raw and grouped data, plus range, frequency, and basic graphical interpretation (bar graph, pie chart, histogram).
- Why it matters: CDS averages 2–4 questions per paper on this chapter, with year coverage from 2000 to 2023.
- Key fact: The empirical relation: \(\text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean}\). Given any two of mean, median, mode, the third can be estimated. CDS occasionally tests this directly.
Statistics is the closing chapter of the CDS Maths syllabus and one of the most "formulaic". The mean, median, mode for both raw and grouped data are tight formulas — once memorised, the chapter delivers 2–4 marks per paper.
This page is built from CDS Previous Year Questions across 2000–2023, plus NCERT Class 10 Statistics and Class 8 Tales by Dots and Lines. Pair with Average (mean specifically) and Set Theory (graphical interpretation).
What This Topic Covers
CDS scope: (1) raw-data measures — mean, median, mode for ungrouped values; (2) grouped-data measures — frequency tables, class intervals, mid-values; (3) median formula for grouped data; (4) mode formula for grouped data; (5) range and dispersion (light); (6) graphical — bar, pie, histogram, ogive; (7) the empirical relation Mean–Median–Mode.
Why This Topic Matters
- 2–4 CDS questions per paper, formula-driven.
- Closes the syllabus with high-yield, low-effort content.
- Connects with Average (mean is just one measure).
Exam Pattern & Weightage
| Year / Paper | No. | Subtopics Tested |
|---|---|---|
| 2008-I/II | 2 | Mean of raw data, median |
| 2010-I/II | 3 | Grouped mean, mode |
| 2011-I/II | 2 | Median grouped, empirical relation |
| 2012-I/II | 3 | Mean, range, weighted |
| 2013-I/II | 3 | Median, mode formula |
| 2014-I/II | 3 | Mixed |
| 2015-I/II | 3 | Mean, median |
| 2016-I/II | 3 | Empirical relation |
| 2017-I/II | 3 | Bar/pie interpretation |
| 2018-I/II | 2 | Mode grouped data |
| 2019-II | 2 | Mean, median |
| 2020-I/II | 3 | Mixed |
| 2021-I/II | 3 | Median, empirical |
| 2022-I / 2023-I | 3 | Mixed |
The median of \(n\) ordered observations is the middle term if \(n\) is odd (the \((n+1)/2\)-th term), and the average of the two middle terms if \(n\) is even. Always sort the data first — CDS often gives unordered data to trap inattentive readers.
Core Concepts
Mean (Arithmetic Mean)
Where \(x_i\) is the mid-value of the \(i\)-th class interval and \(f_i\) is its frequency.
Median
For raw data: sort, then take the middle term (or average of two middle terms if \(n\) is even).
Where \(l\) is the lower boundary of the median class, \(F\) is the cumulative frequency before the median class, \(f\) is the frequency of the median class, and \(h\) is the class width.
Mode
The most frequent value. For raw data, just look for the most common observation.
Where \(l\) is the lower boundary of the modal class, \(f_1\) is its frequency, \(f_0\) and \(f_2\) are the frequencies of the classes before and after.
Empirical Relation
This empirical relation holds approximately for moderately skewed distributions.
Range
Range = (Maximum value) − (Minimum value). Simplest measure of spread.
For finding the median of grouped data, the median class is the one whose cumulative frequency first exceeds or equals \(n/2\) — not the class with the highest frequency (that's the modal class). Don't confuse the two.
Worked Examples
Example 1 — Mean of Raw Data (2008-I)
Q: Find the mean of 5, 7, 9, 11, 13.
- Sum \(= 5 + 7 + 9 + 11 + 13 = 45\). Count \(= 5\). Mean = \(9\).
Example 2 — Median Odd Count (2013-II)
Q: Find the median of 23, 17, 31, 12, 28.
- Sort: 12, 17, 23, 28, 31. Middle (3rd term) = 23.
Example 3 — Median Even Count (2019-II)
Q: Find the median of 4, 9, 11, 17, 21, 28.
- Already sorted. Two middle terms (3rd and 4th): 11 and 17.
- Median = \((11 + 17)/2 = 14\).
Example 4 — Empirical Relation (2016-II)
Q: If mean = 25 and mode = 19, find the median.
- Mode ≈ 3·Median − 2·Mean ⇒ 19 ≈ 3M − 50 ⇒ 3M = 69 ⇒ M = 23.
Example 5 — Mode of Raw Data (2015-I)
Q: Find the mode of 3, 4, 4, 7, 4, 8, 5, 4, 6.
- Frequencies: 3 → 1, 4 → 4, 5 → 1, 6 → 1, 7 → 1, 8 → 1.
- Most frequent: 4 (appears 4 times). Mode = 4.
Example 6 — Weighted Mean (2012-I)
Q: A student scores 80 in Maths (weight 3), 70 in Science (weight 2), 60 in English (weight 1). Find weighted mean.
- Weighted mean = \((80 \cdot 3 + 70 \cdot 2 + 60 \cdot 1)/(3 + 2 + 1) = (240 + 140 + 60)/6 = 440/6 \approx 73.3\).
Example 7 — Range (2017-I)
Q: Find the range of 12, 25, 8, 19, 31, 14.
- Max = 31, Min = 8. Range = \(31 - 8 = 23\).
How CDS Tests This Topic
Six recurring archetypes: (1) mean of raw or grouped data, (2) median (odd or even count) of raw data, (3) median of grouped data via the lower-boundary formula, (4) mode — raw or grouped, (5) empirical relation between mean, median, mode, (6) graphical interpretation (bar, pie, histogram).
Exam Shortcuts (Pro-Tips)
Shortcut 1 — Sort First for Median
Always sort the data before identifying the median. CDS frequently presents unordered data; sorting takes 10 seconds but saves the answer.
Shortcut 2 — Empirical Relation One-Liner
Given any two of mean, median, mode, the third is determined approximately by Mode ≈ 3·Median − 2·Mean. Memorise.
Shortcut 3 — Cumulative Frequency for Grouped Median
For grouped median, build the cumulative frequency column first. The median class is the first one where CF ≥ \(n/2\).
Shortcut 4 — Mid-Value for Grouped Mean
For each class interval, the mid-value is \((\text{lower} + \text{upper})/2\). Use mid-values as the "representative \(x\)" for that class.
Shortcut 5 — Range is Trivial
Range = Max − Min. The simplest measure; takes 5 seconds. Often a CDS distractor question.
Common Question Patterns
Pattern 1 — Mean of Raw or Grouped Data
Sum / count for raw. \(\sum f_i x_i / \sum f_i\) for grouped.
Pattern 2 — Median of Raw Data
Sort, take middle (or average two middles for even count).
Pattern 3 — Median / Mode of Grouped Data
Apply the standard formula with class boundaries and frequencies.
Pattern 4 — Mode of Raw Data
Most frequent value. May have multiple modes (bimodal, multimodal).
Pattern 5 — Empirical Relation Application
Given two of mean/median/mode, find the third.
Preparation Strategy
Week 1. Master the three measures for raw data. Drill 20 problems on mean, median, mode. Memorise the empirical relation.
Week 2. Grouped data. Practice median and mode formulas with class intervals. Cover range, weighted mean, and basic graphical interpretation.
Mock testing. Use CDS mock tests. Most slip-ups: forgetting to sort for median, or confusing median class with modal class.
Cross-train with Average (mean specifically) and Percentage (often appears in pie chart sectors).
Drill Statistics at Speed
CDS mocks with mean, median, mode (raw and grouped), and empirical relation problems. Six archetypes — five formulas — reflex.
Start Free Mock TestFrequently Asked Questions
What are the three measures of central tendency?
Mean (arithmetic average), median (middle value when sorted), and mode (most frequent value). Each represents the "centre" of a dataset in a different sense. For symmetric distributions, all three are equal.
How do I find the median of grouped data?
Build the cumulative frequency column. Identify the median class — the first class where CF ≥ \(n/2\). Apply the formula: Median = \(l + ((n/2) - F)/f \cdot h\), where \(l\) is lower boundary, \(F\) is CF before this class, \(f\) is this class's frequency, \(h\) is the class width.
What is the empirical relation between mean, median, and mode?
For moderately skewed distributions: Mode ≈ 3 · Median − 2 · Mean. Given any two, the third can be estimated. Holds approximately, not exactly.
What's the difference between median and mode?
Median is the middle value when data is sorted; mode is the most frequent. For symmetric unimodal data, they often coincide. For skewed data, they differ.
Can a dataset have more than one mode?
Yes. Datasets with two equally frequent values are bimodal; with more, multimodal. If all values appear once, the dataset has no mode. CDS occasionally tests this distinction.
How do I find the mode of grouped data?
Identify the modal class (highest frequency). Apply: Mode = \(l + (f_1 - f_0)/(2f_1 - f_0 - f_2) \cdot h\), where \(l\) is lower boundary, \(f_1\) is modal class's frequency, \(f_0\) and \(f_2\) are the frequencies of the classes before and after.
Which CDS Maths topics connect to Statistics?
Average — the mean is one measure of central tendency. Percentage — used in pie chart interpretation. Set Theory — Venn-diagram interpretation of data.