Statistics – NCERT Class 11 Mathematics Chapter 13 – Measures of Central Tendency, Dispersion, Variance, and Standard Deviation
Introduces the science of statistics focusing on the collection, analysis, and interpretation of data. Discusses measures of central tendency including mean, median, and mode; various measures of dispersion such as range, mean deviation, variance, and standard deviation; methods for ungrouped and grouped data; illustrative examples and exercises; and historical evolution of statistics with key contributors.
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Categories: NCERT, Class XI, Mathematics, Statistics, Central Tendency, Dispersion, Variance, Standard Deviation, Chapter 13
Tags: Statistics, Measures of Central Tendency, Mean, Median, Mode, Measures of Dispersion, Range, Mean Deviation, Variance, Standard Deviation, Data Analysis, Frequency Distribution, Grouped Data, Ungrouped Data, History of Statistics, NCERT Class 11, Mathematics, Chapter 13
Statistics: Class 11 NCERT Chapter 13 - Ultimate Study Guide, Notes, Questions, Quiz 2025
Statistics
Chapter 13: Mathematics - Ultimate Study Guide | NCERT Class 11 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Statistics Class 11 NCERT
Overview & Key Concepts
Chapter Goal: Measures of dispersion beyond central tendency. Focus: Range, mean deviation (ungrouped/grouped). Exam Focus: Calculations, tables, steps. 2025 Updates: Batsmen examples emphasized. Fun Fact: Karl Pearson (1857-1936) advanced stats. Core Idea: Scatter around center. Real-World: Cricket scores variability. Ties: Ch5 (mean/median). Expanded: Ungrouped to grouped, discrete/continuous from PDF.
Wider Scope: Variability via range/deviations (PDF covers intro, range, mean dev up to continuous).
Expanded Content: Examples, tables, steps.
13.1 Introduction
Statistics analyzes data; central tendency (mean/median/mode) insufficient alone. Need dispersion for scatter. Batsmen A (0-117, mean 53) vs B (46-60, mean 53): A scattered, B clustered (Figs 13.1-2). Dispersion: Single number for variability.
13.2 Measures of Dispersion
Range, quartile dev, mean dev, std dev (study all except quartile). Depend on central tendency.
13.3 Range
Max - Min. A:117, B:14. Rough scatter idea, ignores center.
13.4 Mean Deviation
Mean of absolute deviations from 'a' (mean/median). Sum |x_i - a| / n =0 if signed (from mean). MD(a) = $$\sum |x_i - a| / n$$. Common: From mean/median.
13.4.1 Ungrouped Data
Steps: Compute a, deviations, absolutes, mean. Ex1: Data 6,7,10,12,13,4,8,12; mean=9, MD=2.75. Ex2: 20 obs, mean=10, MD=6.2. Ex3: Median=9, MD=5.27.
Class intervals. Relevance: Ranges. Ex: Marks 0-10. Depth: Midpoints.
Cumulative Frequency
Running total f. Relevance: Median find. Ex: Table 13.2 cf=18. Depth: >=N/2.
MD about Mean
From $$\bar{x}$$. Relevance: Common. Ex: 2.3 Ex4. Depth: Weighted sum.
MD about Median
From M. Relevance: Robust. Ex: 4.97 Ex5. Depth: Middle obs.
Midpoint
Class center. Relevance: Continuous. Ex: 0-10=5. Depth: Assume uniform.
Variability
Spread degree. Relevance: Interpretation. Ex: A high, B low. Depth: Dispersion measures.
Quartile Deviation
(Q3-Q1)/2. Relevance: Mentioned. Ex: Not studied. Depth: Interquartile.
Standard Deviation
Sqrt variance. Relevance: Later. Ex: Not in chapter. Depth: Root mean sq dev.
Measure of Central Tendency
Representative value. Relevance: Base for dev. Ex: Mean 53. Depth: Rough center.
Tip: Absolute for distance; N=sum f. Depth: No negatives. Errors: Forget abs. Historical: Bowley quote. Interlinks: Ch5. Advanced: Variance. Real-Life: Scores analysis. Graphs: Figs 13.1-2. Coherent: Center → Scatter.
Additional: Sum devs from mean=0. Pitfalls: Signed sum vanishes.
30 Questions & Answers - NCERT Based (Class 11) - From Exercises & Variations
Based on chapter examples. Part A: 10 (1 mark short), Part B: 10 (4 marks medium), Part C: 10 (8 marks long). Answers point-wise, numerical stepwise with MathJax.
Part A: 1 Mark Questions (10 Qs - Short from Illustrations & Variations)
1. What is dispersion?
1 Mark Answer:
Scatter around center
2. Range formula?
1 Mark Answer:
Max - Min
3. Why absolute in MD?
1 Mark Answer:
Signed sum=0
4. MD notation?
1 Mark Answer:
MD(a)
5. For ungrouped, n=?
1 Mark Answer:
Number of obs
6. Batsman A range?
1 Mark Answer:
117
7. Median for odd n?
1 Mark Answer:
$$(\frac{n+1}{2})^{th}$$
8. N in grouped?
1 Mark Answer:
Sum f_i
9. Continuous uses?
1 Mark Answer:
Midpoints
10. cf for median?
1 Mark Answer:
>= N/2
Part B: 4 Marks Questions (10 Qs - Medium from Illustrations)
1. Range for batsmen? (Intro)
4 Marks Answer (Step-by-Step):
Step 1: A max117 min0=117
Step 2: B max60 min46=14
Step 3: A more dispersed
Relevance: Quick variability.
2. MD mean for Ex1 data? (6,7,10,...)
4 Marks Answer (Step-by-Step):
Step 1: Mean=72/8=9
Step 2: Abs devs 3,2,1,3,4,5,1,3 sum22
Step 3: 22/8=2.75
Relevance: Ungrouped.
3. Why sum devs from mean=0?
4 Marks Answer (Step-by-Step):
Step 1: Positive=negative
Step 2: Balance to 0
Step 3: Use absolute
Relevance: Dispersion need.
4. MD median Ex3? (3,9,5,...)
4 Marks Answer (Step-by-Step):
Step 1: Arrange, M=9 (6th)
Step 2: Abs sum=58
Step 3: 58/11=5.27
Relevance: Odd n.
5. Discrete mean Ex4? (x2 f2,...)
4 Marks Answer (Step-by-Step):
Step 1: Sum fx=300, N=40, mean=7.5
Step 2: Sum f|d|=92
Step 3: MD=92/40=2.3
Relevance: Weighted.
6. Median Ex5 discrete? (x3 f3,...)
4 Marks Answer (Step-by-Step):
Step 1: N=30 even, 15-16th=13
Step 2: cf=18 at 13
Step 3: M=13
Relevance: cf method.
7. MD median Ex5 calc?
4 Marks Answer (Step-by-Step):
Step 1: Abs devs sum f|d|=149
Step 2: 149/30=4.97
Relevance: From M.
8. Continuous midpoint? (0-10 marks)
4 Marks Answer (Step-by-Step):
Step 1: Mid= (0+10)/2=5
Step 2: Use as x_i
Step 3: Proceed discrete
Relevance: Assumption.
9. Ex2 MD mean? (12,3,18,...)
4 Marks Answer (Step-by-Step):
Step 1: Mean=200/20=10
Step 2: Abs sum=124
Step 3: 124/20=6.2
Relevance: Large ungrouped.
10. Range limitation?
4 Marks Answer (Step-by-Step):
Step 1: Only extremes
Step 2: Ignores middle
Step 3: Use devs for full
Relevance: Rough only.
Part C: 8 Marks Questions (10 Qs - Long Detailed)
1. Batsmen intro dispersion full. (Tables/figs)
8 Marks Answer (Step-by-Step Numerical):
Step 1: Means/med=53 both
Step 2: Ranges 117 vs 14
Step 3: Plots clustered B vs scattered A
Step 4: Need dispersion beyond center.
2. Ex1 ungrouped MD mean detailed table.
8 Marks Answer (Step-by-Step Numerical):
Step 1: Data sum=72, n=8, mean=9
Step 2: Devs -3,-2,1,3,4,-5,-1,3
Step 3: Abs sum=22
Step 4: MD=22/8=2.75. Proof: Steps.
3. Why MD over signed? Proof sum=0.
8 Marks Answer (Step-by-Step Numerical):
Step 1: Signed devs balance pos/neg
Step 2: Ex1 signed sum=0
Step 3: Abs for distance
Proof: Property of mean.
4. Ex3 median MD full arrange/table.
8 Marks Answer (Step-by-Step Numerical):
Step 1: Arrange 3,3,4,5,7,9,10,12,18,19,21
Step 2: n=11 odd, M=9
Step 3: Abs devs sum=58
Step 4: 58/11=5.27.
5. Ex4 discrete MD mean table 13.1 full.
8 Marks Answer (Step-by-Step Numerical):
Step 1: fx=300, N=40, mean=7.5
Step 2: |d| f|d| sum=92
Step 3: MD=92/40=2.3
Full: Table values.
6. Ex5 median discrete cf table 13.2-3.
8 Marks Answer (Step-by-Step Numerical):
Step 1: cf table, N/2=15 in cf18 x=13
Step 2: M=13
Step 3: f|d-M| sum=149
Step 4: 149/30=4.97.
7. Ex2 large ungrouped MD detailed list.
8 Marks Answer (Step-by-Step Numerical):
Step 1: Sum=200, n=20, mean=10
Step 2: Abs devs list sum=124
Step 3: MD=6.2
Verify: All calcs.
8. Continuous MD process for marks ex.
8 Marks Answer (Step-by-Step Numerical):
Step 1: Mids 5,15,25,35,45,55
Step 2: f 12,18,27,20,17,6 N=100
Step 3: Mean=sum fm/N
Step 4: MD=sum f|m-mean|/N.
9. Range vs MD comparison batsmen.
8 Marks Answer (Step-by-Step Numerical):
Step 1: Range A117 B14
Step 2: MD would show A higher
Step 3: Range rough, MD full
Ex: Compute MD for insight.
10. Grouped vs ungrouped diff.
8 Marks Answer (Step-by-Step Numerical):
Step 1: Ungrouped /n, grouped /N f weights
Step 2: Median cf vs arrange
Step 3: Continuous mids
Relevance: Data size.
Tip: Tables for grouped, steps for ungrouped in 8 marks.
Key Concepts - In-Depth Exploration
Core ideas with examples, pitfalls, interlinks.
Dispersion Need
Center alone insufficient. Deriv: Batsmen same mean diff scatter. Pitfall: Ignore var. Ex: Figs cluster. Interlink: Ch5. Depth: Single number.
Range
Extremes diff. Deriv: Quick. Pitfall: Outliers sensitive. Ex: 117 vs 14. Interlink: Basic. Depth: No center.
Mean Deviation
Abs mean devs. Deriv: Full scatter. Pitfall: Forget abs. Ex: 2.75 Ex1. Interlink: Absolute value. Depth: From mean/median.
Ungrouped Calc
Direct steps. Deriv: Simple lists. Pitfall: No arrange median. Ex: Ex3 odd n. Interlink: Basic data. Depth: Sum /n.
Grouped Discrete
f weights. Deriv: Large data. Pitfall: Wrong cf. Ex: Ex5 M=13. Interlink: Frequency. Depth: /N.
Continuous
Midpoints. Deriv: Intervals. Pitfall: No gaps assume. Ex: Marks mids. Interlink: Histograms. Depth: Uniform freq.
Advanced: Std dev sqrt. Pitfalls: Signed devs. Interlinks: Ch14 prob. Real: Sports stats. Depth: Median robust. Examples: Ex1-5. Graphs: Batsmen lines. Errors: Miss f in sum. Tips: Abs always; cf cumulative.
Solved Examples - Book Illustrations with Simple Explanations
NCERT Examples 1-5 solved step-by-step.
Example 1: MD mean ungrouped (6,7,10,...)
Simple Explanation: Basic steps.
Step 1: Mean=9
Step 2: Abs devs sum=22
Step 3: MD=2.75
Simple Way: List calcs.
Example 2: MD mean large (12,3,18,...)
Simple Explanation: Sum abs.
Step 1: Mean=10
Step 2: Abs sum=124
Step 3: MD=6.2
Simple Way: Direct /20.
Example 3: MD median (3,9,5,...)
Simple Explanation: Arrange first.
Step 1: Arrange, M=9
Step 2: Abs sum=58
Step 3: MD=5.27
Simple Way: 6th obs.
Example 4: Discrete MD mean (x2 f2,...)
Simple Explanation: Weighted.
Step 1: Mean=7.5
Step 2: f|d| sum=92
Step 3: MD=2.3
Simple Way: Table 13.1.
Example 5: Discrete MD median (x3 f3,...)
Simple Explanation: cf method.
Step 1: M=13 via cf
Step 2: f|d-M| =149
Step 3: MD=4.97
Simple Way: Tables 13.2-3.
Interactive Quiz - Master Statistics Dispersion
10 MCQs with MathJax; 80%+ goal. Range, MD, grouped.
