Unit 6: Descriptive Statistics - Free Study Resources | Boundless Maths
📊 Unit 6: Descriptive Statistics
Measures of Dispersion · Percentile & Quartile Rank · Correlation
Weightage: 12 Marks in School Exams
📚
Topics Covered in Unit 6
1. Measures of Dispersion
Range, Quartile Deviation, Mean Deviation (about mean & median), Variance & Standard Deviation for ungrouped and grouped data; Coefficient of Variation
2. Percentile & Quartile Rank
Definition, calculation and interpretation of percentile rank and quartile rank in a given data set
3. Correlation
Product moment correlation; Karl Pearson's coefficient of correlation (ungrouped & grouped data); Spearman's Rank Correlation coefficient; interpretation of results
Which of the following is NOT a measure of dispersion?
aRange
bStandard Deviation
cMedian
dQuartile Deviation
✓ Correct Answer: (c) Median
Explanation: Median is a measure of central tendency, not dispersion. Range, Standard Deviation and Quartile Deviation all measure the spread or scatter of data around a central value.
Question 2
The range of the data set {4, 7, 2, 15, 9, 11, 3} is:
a9
b11
c13
d15
✓ Correct Answer: (c) 13
Explanation: Range = Maximum value − Minimum value = 15 − 2 = 13.
Question 3
For a data set, Q₁ = 20 and Q₃ = 50. The Quartile Deviation is:
a30
b35
c15
d70
✓ Correct Answer: (c) 15
Explanation: Quartile Deviation (QD) = (Q₃ − Q₁) / 2 = (50 − 20) / 2 = 30 / 2 = 15. It is also called the semi-interquartile range.
Explanation: CV = (σ / x̄) × 100. It is a relative (unit-free) measure of dispersion expressed as a percentage. A lower CV indicates more consistency; a higher CV indicates more variability.
Question 6
Mean Deviation about the mean for n observations is defined as:
aΣ(xᵢ − x̄) / n
bΣ|xᵢ − x̄| / n
c√[Σ(xᵢ − x̄)² / n]
dΣ(xᵢ − x̄)² / n
✓ Correct Answer: (b) Σ|xᵢ − x̄| / n
Explanation: Mean Deviation about the mean = Σ|xᵢ − x̄| / n. Absolute values are used so that positive and negative deviations do not cancel each other out. Note: option (c) is standard deviation and option (d) is variance.
Question 7
The mean deviation about the mean for the data 2, 4, 6, 8, 10 is:
If the standard deviation of a data set is 6, its variance is:
a3
b12
c36
d√6
✓ Correct Answer: (c) 36
Explanation: Variance = (Standard Deviation)² = 6² = 36. Conversely, Standard Deviation = √Variance. These two measures always satisfy this relationship.
Question 9
If 5 is added to every observation in a data set, the standard deviation:
aIncreases by 5
bDecreases by 5
cIs multiplied by 5
dRemains unchanged
✓ Correct Answer: (d) Remains unchanged
Explanation: Standard deviation measures the spread of data around the mean. Adding a constant to every value shifts all values equally, so the mean also increases by the same constant. The deviations (xᵢ − x̄) remain the same, and therefore SD is unchanged. However, the mean and variance are affected: mean increases by 5, variance stays the same.
📊 Percentile & Quartile Rank
Question 10
A student scores at the 80th percentile. This means:
aThe student scored 80 marks
b80% of students scored above the student
c80% of students scored at or below the student's score
dThe student is in the bottom 20%
✓ Correct Answer: (c) 80% of students scored at or below the student's score
Explanation: The pth percentile is a value at or below which p% of the observations fall. So the 80th percentile means 80% of data values are at or below that score — the student has outperformed 80% of the group.
Question 11
The second quartile (Q₂) is equivalent to which percentile?
a25th percentile
b50th percentile
c75th percentile
d90th percentile
✓ Correct Answer: (b) 50th percentile
Explanation: Q₁ = 25th percentile, Q₂ = 50th percentile (also the Median), Q₃ = 75th percentile. Q₂ divides the data into two equal halves.
Question 12
A student's score falls between Q₁ and Q₂. This means the student is in which quartile group?
aBottom 25% of the class
bBetween the 25th and 50th percentile
cTop 25% of the class
dBetween the 50th and 75th percentile
✓ Correct Answer: (b) Between the 25th and 50th percentile
Explanation: Q₁ corresponds to the 25th percentile and Q₂ corresponds to the 50th percentile. A score between Q₁ and Q₂ places the student in the second quartile — i.e., between the 25th and 50th percentile, which is the second-lowest group.
🔗 Correlation
Question 13
Karl Pearson's coefficient of correlation (r) always lies between:
a0 and 1
b−1 and 0
c−1 and +1
d−2 and +2
✓ Correct Answer: (c) −1 and +1
Explanation: −1 ≤ r ≤ +1 always. r = +1 → perfect positive correlation; r = −1 → perfect negative correlation; r = 0 → no linear correlation between the two variables.
Question 14
Karl Pearson's coefficient of correlation is defined as:
Explanation: r = Cov(X,Y) / (σₓ · σᵧ), where Cov(X,Y) = Σ(xᵢ−x̄)(yᵢ−ȳ)/n. This is Karl Pearson's product moment correlation formula. It measures the strength and direction of the linear relationship between two variables.
Question 15
Spearman's Rank Correlation formula is:
aρ = 1 − (6 Σd²) / (n³ − n)
bρ = 1 − (6 Σd²) / n(n² − 1)
cρ = 6 Σd² / n(n² − 1)
dρ = 1 + (6 Σd²) / n(n² − 1)
✓ Correct Answer: (b) ρ = 1 − (6 Σd²) / n(n² − 1)
Explanation: Spearman's ρ = 1 − [6Σdᵢ²] / [n(n²−1)], where dᵢ = difference in the ranks of the iᵗʰ pair and n = number of pairs. It is used when data is ordinal or ranks are given directly.
Question 16
Two judges ranked 5 contestants. The rank differences (d) are 1, 0, −1, 2, −2. The Spearman's rank correlation coefficient is:
If the Karl Pearson's coefficient of correlation between height and weight of students is r = 0.85, which of the following conclusions is most appropriate?
aHeight causes weight to increase
bThere is a strong positive linear relationship between height and weight
cHeight and weight are not related
dThere is a weak negative relationship between height and weight
✓ Correct Answer: (b) There is a strong positive linear relationship between height and weight
Explanation: r = 0.85 is close to +1, indicating a strong positive correlation — as height increases, weight tends to increase too. However, correlation does not imply causation; we cannot conclude that height causes weight to increase. Option (a) is incorrect because correlation only measures association, not causation.
Question 18
Two examiners marked 8 students' essays. Spearman's rank correlation between their rankings is ρ = −0.9. What does this indicate?
aThe two examiners are in near-perfect agreement
bThere is no relationship between their rankings
cThe two examiners are in near-perfect disagreement — when one ranks a student high, the other tends to rank them low
dOne examiner has made errors in ranking
✓ Correct Answer: (c) The two examiners are in near-perfect disagreement — when one ranks a student high, the other tends to rank them low
Explanation: ρ = −0.9 indicates a very strong negative rank correlation. This means the two examiners are ranking the students in almost opposite orders — a student ranked high by Examiner A tends to be ranked low by Examiner B and vice versa. ρ = −1 would be perfect disagreement.
📋 Assertion-Reason Questions
Statement I is called Assertion (A) and Statement II is called Reason (R). Choose the correct option:
(a) Both A and R are True and R is the correct explanation of A
(b) Both A and R are True but R is not the correct explanation of A
(c) A is True but R is False
(d) A is False but R is True
Assertion-Reason 1
Assertion (A): Standard deviation is always greater than or equal to zero.
Reason (R): Standard deviation is the square root of variance, and variance is the mean of squared deviations from the mean, which is always non-negative.
aBoth A and R are True and R is the correct explanation of A
bBoth A and R are True but R is not the correct explanation of A
cA is True but R is False
dA is False but R is True
✓ Correct Answer: (a) Both A and R are True and R is the correct explanation of A
Explanation: Variance = Σ(xᵢ−x̄)²/n ≥ 0 always, because squared deviations are non-negative. Standard Deviation = √Variance ≥ 0. Both statements are correct and R directly and completely explains why A is true.
Assertion-Reason 2
Assertion (A): Spearman's rank correlation can be computed even when only the ranks of the observations are available, without knowing the actual data values.
Reason (R): Spearman's rank correlation uses only the rank differences (d) between paired observations to compute ρ.
aBoth A and R are True and R is the correct explanation of A
bBoth A and R are True but R is not the correct explanation of A
cA is True but R is False
dA is False but R is True
✓ Correct Answer: (a) Both A and R are True and R is the correct explanation of A
Explanation: Both statements are correct. The formula ρ = 1 − 6Σd²/n(n²−1) requires only the rank differences (d) — not the actual data values. This is precisely why Spearman's correlation can be applied directly to ordinal (ranked) data. R correctly explains A.
📐 All Formulas. All 8 Units. One Crisp PDF.
Dispersion, Percentile, Correlation — every formula organised topic-wise in the Class 11 Formula Deck
Batsman B: Mean = (50+70+65+45+80+60+70)/7 = 440/7 ≈ 62.86
Σ(xᵢ−62.86)² for B ≈ 164.3+50.9+4.5+318.4+296.2+8.2+50.9 = 893.4. σ_B = √(893.4/7) ≈ √127.6 ≈ 11.30. CV_B = (11.30/62.86)×100 ≈ 18.0%
Since CV_B (≈18%) < CV_A (≈59%), Batsman B has lower relative variability
Batsman B is more consistent (CV ≈ 18% vs ≈ 59%)
Question 4
For a data set of 50 observations arranged in ascending order, Q₁ = 28 and Q₃ = 52. Find: (i) Interquartile Range, (ii) Quartile Deviation, (iii) Coefficient of Quartile Deviation.
Karl Pearson's r ≈ 0.775 → Strong positive correlation between X and Y
Question 8
For the following data on advertising expenditure (X, in ₹ thousands) and sales (Y, in ₹ lakhs), r is calculated as 0.92. Interpret the result and state what conclusion can be drawn.
Month
Ad Spend X (₹000)
Sales Y (₹ Lakhs)
1
10
50
2
20
70
3
15
60
4
25
85
5
30
95
Solution:
Given: r = 0.92 (between advertising expenditure X and sales Y)
Interpretation: r = 0.92 is close to +1, indicating a very strong positive linear correlation between advertising expenditure and sales
Conclusion: As advertising expenditure increases, sales tend to increase substantially. The two variables move strongly in the same direction
Important note: Correlation does not prove causation. Other factors (season, pricing, competition) may also influence sales. However, the strong correlation suggests advertising expenditure is a reliable predictor of sales in this context
r = 0.92 → Very strong positive correlation. Higher advertising spend is strongly associated with higher sales.
🔗 Correlation — Spearman's Rank Correlation
Question 9
Two judges ranked 6 students in a competition. Calculate Spearman's rank correlation coefficient and interpret the result.
Student
Judge A
Judge B
1
1
2
2
2
3
3
3
1
4
4
5
5
5
6
6
6
4
Solution:
d = (Rank A − Rank B): 1−2=−1, 2−3=−1, 3−1=2, 4−5=−1, 5−6=−1, 6−4=2. Check: Σd = −1−1+2−1−1+2 = 0 ✓
Spearman's ρ ≈ 0.657 → Moderately strong positive agreement between the two judges' rankings
Question 10
Marks of 8 students in Mathematics (X) and Science (Y) are given below. Calculate Spearman's rank correlation and interpret.
X: 70, 65, 80, 55, 90, 75, 85, 60 | Y: 68, 60, 75, 50, 85, 70, 80, 65
Solution:
Ranks for X (highest score = Rank 1): 90→1, 85→2, 80→3, 75→4, 70→5, 65→6, 60→7, 55→8
So Rₓ in data order: 70→5, 65→6, 80→3, 55→8, 90→1, 75→4, 85→2, 60→7
Ranks for Y (highest score = Rank 1): 85→1, 80→2, 75→3, 70→4, 68→5, 65→6, 60→7, 50→8
So Rᵧ in data order: 68→5, 60→7, 75→3, 50→8, 85→1, 70→4, 80→2, 65→6
