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Descriptive Statistics
Unit I Β· Lecture 8

BDS113 β€” Introduction to Statistics

Unit I Review & Problem-Solving Session

Formula Sheet Β· Concept Map Β· 8 Worked Exam-Style Problems Β· Common Mistakes Β· Comprehensive Quiz Β· Unit Wrap-Up


Unit I Β· Lecture 8 of 40 Β· B.Sc. Data Science, 1st Year, Semester II

Descriptive Statistics
Unit I Β· Lecture 8

The Review Journey


πŸ“‹
Complete Formula Sheet β€” Unit I
Every formula from Lectures 1–7 in one place
πŸ—ΊοΈ
Concept Map & Decision Guide
How all Unit I topics connect Β· When to use which measure
✏️
8 Worked Exam-Style Problems (Click to Reveal)
Covering every topic Β· Step-by-step solutions Β· Exam technique
πŸŽ“
Common Mistakes, Exam Tips, Quiz & Unit Wrap-Up
Pitfalls to avoid Β· 12-question quiz Β· Complete unit summary
Descriptive Statistics
Unit I Β· Lecture 8

πŸ“‹ Complete Formula Sheet β€” Unit I


Arithmetic Mean β€” Ungrouped (Lecture 1)

xΜ„ = Ξ£xα΅’ / n

Arithmetic Mean β€” Grouped / Frequency Distribution (Lecture 2)

xΜ„ = Ξ£fα΅’xα΅’ / Ξ£fα΅’  |  where xα΅’ = midpoint of iα΅—Κ° class  |  Ξ£fα΅’ = N

Combined Mean & Weighted Mean (Lecture 2)

x̄₁₂ = (n₁x̄₁ + nβ‚‚xΜ„β‚‚) / (n₁ + nβ‚‚)  |  xΜ„w = Ξ£wα΅’xα΅’ / Ξ£wα΅’

Median β€” Ungrouped (Lecture 3)

Sort data β†’ Median = value of ((n+1)/2)α΅—Κ° observation  |  If n even: average of (n/2)α΅—Κ° and (n/2+1)α΅—Κ°

Median β€” Grouped (Lecture 3)

Me = L + [(N/2 βˆ’ CF) / f] Γ— h  |  L = lower limit of median class, CF = cumulative freq before median class, f = freq of median class, h = class width

Mode β€” Ungrouped & Grouped (Lecture 3)

Ungrouped: Most frequent value  |  Grouped: Mo = L + [(f₁ βˆ’ fβ‚€) / (2f₁ βˆ’ fβ‚€ βˆ’ fβ‚‚)] Γ— h

Empirical Relation (Lecture 3)

Mean βˆ’ Mode β‰ˆ 3(Mean βˆ’ Median)  |  Mode β‰ˆ 3Median βˆ’ 2Mean

Quartiles, Deciles, Percentiles β€” Grouped (Lecture 4)

Qα΅’ = L + [(iN/4 βˆ’ CF) / f] Γ— h  |  Dβ±Ό = L + [(jN/10 βˆ’ CF) / f] Γ— h  |  Pβ‚– = L + [(kN/100 βˆ’ CF) / f] Γ— h

Range & Coefficient of Range (Lecture 5)

Range = H βˆ’ L  |  Coefficient of Range = (H βˆ’ L) / (H + L)

Quartile Deviation (Lecture 5)

QD = (Q₃ βˆ’ Q₁) / 2  |  Coefficient of QD = (Q₃ βˆ’ Q₁) / (Q₃ + Q₁)

Mean Deviation (Lecture 6)

MD = Ξ£|xα΅’ βˆ’ A| / n  |  A = Mean, Median, or Mode  |  MD from Median is always minimum

Variance & Standard Deviation (Lecture 7)

σ² = Ξ£fα΅’(xα΅’ βˆ’ xΜ„)Β² / N  |  Οƒ = βˆšΟƒΒ²  |  Shortcut: σ² = (Ξ£fα΅’xα΅’Β² / N) βˆ’ xΜ„Β²

Combined Standard Deviation (Lecture 7)

σ₁₂² = [n₁(σ₁² + d₁²) + nβ‚‚(Οƒβ‚‚Β² + dβ‚‚Β²)] / (n₁ + nβ‚‚)  |  dα΅’ = xΜ„α΅’ βˆ’ x̄₁₂

Coefficient of Variation (Lecture 7)

CV = (Οƒ / xΜ„) Γ— 100  |  Lower CV β†’ more consistent  |  Use to compare datasets with different units or scales
Descriptive Statistics
Unit I Β· Lecture 8

πŸ—ΊοΈ Concept Map & Decision Guide


1 Start: You have a dataset β€” raw values or a frequency distribution.
Identify whether data is ungrouped (individual observations) or grouped (class intervals with frequencies). This determines which formulas to use.
2 Find the Centre β€” Central Tendency (Lectures 1–3):
β†’ Symmetric data, no outliers: Use Mean (most efficient, uses all data)
β†’ Skewed data or outliers present: Use Median (robust to extremes)
β†’ Categorical data or "most typical" value: Use Mode
β†’ Open-ended classes: Use Median (Mean cannot be computed)
3 Find Positional Values β€” Quartiles, Deciles, Percentiles (Lecture 4):
Use the same interpolation formula as the grouped median, but replace N/2 with iN/4, jN/10, or kN/100. These divide data into 4, 10, or 100 equal parts.
4 Measure Spread β€” Dispersion (Lectures 5–7):
β†’ Quick rough idea: Range (but very sensitive to extremes)
β†’ With Median as centre: Quartile Deviation or Mean Deviation
β†’ With Mean as centre: Variance / Standard Deviation (most common)
β†’ Comparing two datasets with different units/scales: Coefficient of Variation (CV)
5 Interpret:
State the value, what it means in context, and compare if multiple datasets are given. For CV: lower CV = more consistent / less variable.

πŸ’¬ Exam Shortcut: Most exam problems on Unit I follow this sequence: (1) build the frequency table with midpoints, (2) compute mean, (3) find median and mode, (4) compute SD using the shortcut formula, (5) interpret or compare using CV. Master this flow and you can solve any Unit I problem.
Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 1: Mean, Median & Mode (Ungrouped)


πŸ“ Find the mean, median, and mode for the following data: 12, 15, 18, 18, 20, 22, 25, 28, 30, 35

β–Ά Reveal

(a) Mean:

n = 10
Ξ£x = 12 + 15 + 18 + 18 + 20 + 22 + 25 + 28 + 30 + 35 = 223
xΜ„ = 223 / 10 = 22.3


(b) Median:

Data is already sorted. n = 10 (even).
Median = average of (n/2)α΅—Κ° and (n/2 + 1)α΅—Κ° values
       = average of 5α΅—Κ° and 6α΅—Κ° values
       = (20 + 22) / 2 = 21


(c) Mode:

18 appears twice; all other values appear once.
Mode = 18


Verification β€” Empirical Relation:

3Median βˆ’ 2Mean = 3(21) βˆ’ 2(22.3) = 63 βˆ’ 44.6 = 18.4 β‰ˆ Mode (18) βœ“
(Approximate β€” works best for moderately skewed data)

Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 2: Mean & Median (Grouped Data)


πŸ“ Compute the mean and median from the following frequency distribution:

β–Ά Reveal
Classfx (mid)fxCF
0 – 1055255
10 – 2081512013
20 – 30122530025
30 – 40103535035
40 – 5054522540
Total401220

Mean:

xΜ„ = Ξ£fx / N = 1220 / 40 = 30.5


Median:

N/2 = 40/2 = 20. CF just exceeding 20 is 25 β†’ Median class: 20–30
L = 20, CF = 13, f = 12, h = 10
Me = L + [(N/2 βˆ’ CF) / f] Γ— h
   = 20 + [(20 βˆ’ 13) / 12] Γ— 10
   = 20 + (7/12) Γ— 10
   = 20 + 5.83 = 25.83

Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 3: Mode (Grouped) & Empirical Check


πŸ“ Using the same data from Problem 2, compute the mode and verify the empirical relation.

β–Ά Reveal

Mode:

Modal class = class with highest frequency = 20–30 (f = 12)
L = 20, fβ‚€ = 8 (freq of class before modal class)
f₁ = 12 (freq of modal class), fβ‚‚ = 10 (freq of class after), h = 10

Mo = L + [(f₁ βˆ’ fβ‚€) / (2f₁ βˆ’ fβ‚€ βˆ’ fβ‚‚)] Γ— h
   = 20 + [(12 βˆ’ 8) / (2Γ—12 βˆ’ 8 βˆ’ 10)] Γ— 10
   = 20 + [4 / (24 βˆ’ 18)] Γ— 10
   = 20 + (4/6) Γ— 10
   = 20 + 6.67 = 26.67


Empirical Relation Check:

Mean = 30.5, Median = 25.83, Mode = 26.67

Mean βˆ’ Mode β‰ˆ 3(Mean βˆ’ Median)
30.5 βˆ’ 26.67 β‰ˆ 3(30.5 βˆ’ 25.83)
3.83 β‰ˆ 3(4.67) = 14.01  βœ—  (Does not hold well)

Alternative check: 3Median βˆ’ 2Mean = 3(25.83) βˆ’ 2(30.5) = 77.49 βˆ’ 61 = 16.49 β‰  26.67

Note: The empirical relation is approximate and works best for moderately skewed, unimodal distributions. This distribution is somewhat irregular, so the approximation is poor. This is expected and acceptable.

Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 4: Quartiles, Deciles & Percentiles


πŸ“ Using the grouped data from Problem 2, find Q₁, Q₃, D₇, and P₉₀.

β–Ά Reveal

Q₁ (First Quartile):
iN/4 = 1Γ—40/4 = 10. CF just exceeding 10 is 13 β†’ class 10–20
L = 10, CF = 5, f = 8, h = 10
Q₁ = 10 + [(10 βˆ’ 5)/8] Γ— 10 = 10 + 6.25 = 16.25

Q₃ (Third Quartile):
iN/4 = 3Γ—40/4 = 30. CF just exceeding 30 is 35 β†’ class 30–40
L = 30, CF = 25, f = 10, h = 10
Q₃ = 30 + [(30 βˆ’ 25)/10] Γ— 10 = 30 + 5.0 = 35.0

D₇ (7th Decile):
jN/10 = 7Γ—40/10 = 28. CF just exceeding 28 is 35 β†’ class 30–40
L = 30, CF = 25, f = 10, h = 10
D₇ = 30 + [(28 βˆ’ 25)/10] Γ— 10 = 30 + 3.0 = 33.0

P₉₀ (90th Percentile):
kN/100 = 90Γ—40/100 = 36. CF just exceeding 36 is 40 β†’ class 40–50
L = 40, CF = 35, f = 5, h = 10
P₉₀ = 40 + [(36 βˆ’ 35)/5] Γ— 10 = 40 + 2.0 = 42.0


Interpretation: 25% of observations fall below 16.25. 75% fall below 35.0. 70% fall below 33.0. 90% fall below 42.0.

Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 5: Range, Quartile Deviation & Mean Deviation


πŸ“ For the dataset: 8, 12, 15, 18, 20, 22, 25, 30, compute Range, QD, and MD from Mean.

β–Ά Reveal

Mean: xΜ„ = (8+12+15+18+20+22+25+30)/8 = 150/8 = 18.75

Range:
Range = H βˆ’ L = 30 βˆ’ 8 = 22
Coefficient of Range = (30βˆ’8)/(30+8) = 22/38 = 0.579

Quartile Deviation:
n = 8. Q₁ = value of (9/4)α΅—Κ° = 2.25α΅—Κ° item
   = 12 + 0.25(15βˆ’12) = 12.75
Q₃ = value of (27/4)α΅—Κ° = 6.75α΅—Κ° item
   = 22 + 0.75(25βˆ’22) = 24.25
QD = (24.25 βˆ’ 12.75)/2 = 5.75
Coefficient of QD = (24.25βˆ’12.75)/(24.25+12.75) = 11.5/37 = 0.311

Mean Deviation from Mean:
|xα΅’ βˆ’ 18.75|: 10.75, 6.75, 3.75, 0.75, 1.25, 3.25, 6.25, 11.25
Ξ£|deviations| = 44.0
MD = 44/8 = 5.5


Interpretation: The mean deviation (5.5) is less than the SD (computed in Problem 6), as SD penalises larger deviations more heavily due to squaring.

Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 6: Variance & Standard Deviation (Grouped)


πŸ“ Compute the variance and standard deviation for the grouped data in Problem 2 using the shortcut method.

β–Ά Reveal
ClassfxfxfxΒ²
0 – 105525125
10 – 208151201800
20 – 3012253007500
30 – 40103535012250
40 – 5054522510125
Total40102031800

σ² = (Ξ£fxΒ² / N) βˆ’ xΜ„Β²
xΜ„ = 1020/40 = 25.5
σ² = 31800/40 βˆ’ (25.5)Β²
   = 795 βˆ’ 650.25
   = 144.75

Οƒ = √144.75 = 12.03


Verification: σ² = 144.75 β‰₯ 0 βœ“ (A negative variance is impossible and would signal an arithmetic error.)

Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 7: Combined Mean & Combined SD


πŸ“ Section A: n₁ = 30, x̄₁ = 65, σ₁ = 8. Section B: nβ‚‚ = 50, xΜ„β‚‚ = 72, Οƒβ‚‚ = 10. Find the combined mean and combined standard deviation.

β–Ά Reveal

Combined Mean:

x̄₁₂ = (n₁x̄₁ + nβ‚‚xΜ„β‚‚) / (n₁ + nβ‚‚)
     = (30Γ—65 + 50Γ—72) / (30 + 50)
     = (1950 + 3600) / 80
     = 5550 / 80 = 69.375


Combined Variance:

d₁ = x̄₁ βˆ’ x̄₁₂ = 65 βˆ’ 69.375 = βˆ’4.375
dβ‚‚ = xΜ„β‚‚ βˆ’ x̄₁₂ = 72 βˆ’ 69.375 = 2.625

σ₁₂² = [n₁(σ₁² + d₁²) + nβ‚‚(Οƒβ‚‚Β² + dβ‚‚Β²)] / (n₁ + nβ‚‚)
     = [30(64 + 19.14) + 50(100 + 6.89)] / 80
     = [30(83.14) + 50(106.89)] / 80
     = [2494.2 + 5344.5] / 80
     = 7838.7 / 80 = 97.98

σ₁₂ = √97.98 = 9.90


Interpretation: The combined SD (9.90) is larger than either individual SD (8 or 10). This is because the two sections have different means, adding between-group variability to the total spread.

Descriptive Statistics
Unit I Β· Lecture 8

✏️ Problem 8: Comparing Consistency Using CV


πŸ“ Two batsmen scored in 10 innings each. A: xΜ„ = 45, Οƒ = 12. B: xΜ„ = 55, Οƒ = 15. Who is more consistent? Who is the better scorer?

β–Ά Reveal

CV for A: (Οƒ / xΜ„) Γ— 100 = (12/45) Γ— 100 = 26.67%
CV for B: (Οƒ / xΜ„) Γ— 100 = (15/55) Γ— 100 = 27.27%

Since CV_A (26.67%) < CV_B (27.27%),
Batsman A is more consistent.


Interpretation:

Consistency: A is slightly more consistent (lower CV), meaning his scores vary less proportionally around his mean.

Better scorer: B has a higher average (55 vs 45), so B is the better run-scorer overall.

Key insight: "More consistent" and "better" are different questions. Consistency (CV) measures predictability; mean measures performance level. A coach might prefer A for reliability in a chase, but B for overall run contribution.

Descriptive Statistics
Unit I Β· Lecture 8

⚠️ Common Mistakes in Unit I


Arithmetic Errors

1. Wrong Ξ£fx β€” always verify by adding twice or using a calculator.

2. Negative variance (impossible!) β€” signals a computation error in Ξ£fxΒ² or Ξ£fx.

3. Forgetting to square the mean in the shortcut formula: σ² = (Ξ£fxΒ²/N) βˆ’ xΜ„Β², not Ξ£fxΒ²/N alone.

4. Using N vs Nβˆ’1 β€” in this course, use N (population SD). Some textbooks use Nβˆ’1 (sample SD).

Conceptual Errors

1. Using Mean for skewed data with outliers β€” Median is more appropriate.

2. Confusing QD with IQR β€” QD = IQR/2, they are not the same.

3. Thinking SD = 0 means "no data" β€” it means all values are identical.

4. Comparing SDs of datasets with different means or units β€” use CV instead.

Grouped Data Errors

1. Using class boundaries instead of midpoints for fx β€” always use midpoints.

2. Wrong cumulative frequency β€” add frequencies cumulatively, don't skip.

3. Identifying wrong median/modal class β€” check which CF first exceeds N/2.

4. Forgetting class width h in the interpolation formula.


πŸ’¬ Exam Tip: After computing variance, always do a quick sanity check: (1) Is σ² β‰₯ 0? (2) Does xΜ„ fall within the data range? (3) Is Ξ£f = N? (4) For CV comparison, does the lower CV correspond to the more "clustered" dataset? These catches prevent most errors.
Descriptive Statistics
Unit I Β· Lecture 8

πŸ“ Exam Tips for Unit I


πŸ“Š

Always Draw the Table

For grouped data, build a table with columns: Class, f, x, fx, fxΒ², CF before computing anything. This prevents most errors.

βœ…

Verify Totals

Check: Ξ£f = N. Ξ£fx = xΜ„ Γ— N. σ² β‰₯ 0. These catches prevent most arithmetic errors.

🎯

Interpret Every Answer

Don't just give xΜ„ = 65. Say: "The average marks are 65." Interpretation earns marks.

πŸ“

Know When to Use What

Symmetric β†’ Mean+SD. Skewed β†’ Median+QD. Comparing datasets β†’ CV. Open-ended β†’ Median.


πŸ“– High-Frequency Exam Questions β€” Unit I

1. Compute mean, median, and mode from a grouped frequency distribution. (Very common)
2. Find quartiles, deciles, or percentiles from grouped data. (Common)
3. Compute variance and SD using the shortcut method. (Very common)
4. Compare two datasets using CV and state which is more consistent. (Almost every exam)
5. Compute combined mean and/or combined SD for two groups. (Common)
6. Verify the empirical relation Mean βˆ’ Mode β‰ˆ 3(Mean βˆ’ Median). (Occasional)
7. Compute Range, QD, and MD from raw data. (Occasional)
8. Theory: "When is Median preferred over Mean?" or "Properties of a good average." (Theory question)

Descriptive Statistics
Unit I Β· Lecture 8

🧠 Comprehensive Quiz β€” Unit I (12 Questions)


Q1: Which measure of central tendency uses ALL data values?

A) Mean
B) Median
C) Mode
D) Range

Q2: For a positively skewed distribution:

A) Mean < Median < Mode
B) Mean > Median > Mode
C) Mean = Median = Mode
D) Mode > Mean > Median

Q3: The median class in a grouped distribution is identified by:

A) The class with the highest frequency
B) The class where cumulative frequency first exceeds N/2
C) The middle class in the distribution
D) The class with frequency equal to N/2

Q4: Variance can be:

A) Negative
B) Any real number
C) Zero or positive only
D) Always strictly positive

Q5: If every value in a dataset is multiplied by 3, the new SD is:

A) Same as before
B) 9 times the original
C) 3 times the original
D) 3 + original SD

Q6: Mean Deviation from _____ is always minimum:

A) Mean
B) Median
C) Mode
D) Midrange

Q7: CV is used to compare variability when datasets have:

A) Same means
B) Same units only
C) Different means, units, or scales
D) Same standard deviations

Q8: Quartile Deviation (QD) equals:

A) Q₃ βˆ’ Q₁
B) (Q₃ βˆ’ Q₁) / 2
C) (Q₃ + Q₁) / 2
D) Qβ‚‚ βˆ’ Q₁

Q9: For open-end classes (e.g., "60 and above"), which measure can still be computed?

A) Mean
B) Median
C) Variance
D) All of the above

Q10: If σ² = 0, then:

A) The dataset is empty
B) All values are identical
C) The mean is zero
D) The dataset has only one value

Q11: The shortcut formula for variance is:

A) σ² = Ξ£fxΒ² / N
B) σ² = (Ξ£fxΒ² / N) βˆ’ xΜ„Β²
C) σ² = xΜ„Β² βˆ’ (Ξ£fxΒ² / N)
D) σ² = Ξ£fxΒ² βˆ’ (Ξ£fx)Β²

Q12: Which is NOT a property of a good measure of central tendency?

A) Rigidly defined
B) Based on all observations
C) Affected by extreme values
D) Easy to understand and compute
Descriptive Statistics
Unit I Β· Lecture 8

Unit I β€” Complete!

Descriptive Statistics Β· Lectures 1–8 Β· Central Tendency & Dispersion

Lectures 1–2

Arithmetic Mean
Ungrouped & Grouped
Combined & Weighted Mean

Lecture 3

Median & Mode
Ungrouped & Grouped
Empirical Relation

Lecture 4

Quartiles, Deciles
Percentiles
Positional Measures

Lectures 5–6

Range & QD
Mean Deviation
Measures of Dispersion

Lecture 7

Variance & SD
Combined SD
Coefficient of Variation

Key Skills You've Developed

Mean Median Mode Quartiles Deciles Percentiles Range QD MD Variance SD CV

You have completed Unit I β€” Descriptive Statistics.

The concepts of mean, variance, and SD from this unit appear in every subsequent unit β€” probability distributions (Unit III), correlation and regression (Unit IV), and beyond. Master these formulas and you'll find the rest of the course much smoother.

Next up: Unit II β€” Probability!

BDS113 Introduction to Statistics Β· B.Sc. Data Science, Semester II Β· Unit I Review Complete