Introduction to Statistics

Unit II · Lectures 11–12

BDS113 — Introduction to Statistics

Median, Mode & Empirical Relationship

Positional Averages · Grouped & Ungrouped Data · Mean–Median–Mode Link

Unit II · Lectures 11–12 of 40 · B.Sc. Data Science, 1st Year, Semester II

Introduction to Statistics

Unit II · Lectures 11–12

The Journey — Lectures 11 & 12

📍

Lecture 11 — Median
Definition · Median for ungrouped data (odd & even n) · Median for grouped data · Formula & derivation · Graphical method (ogive)

🏆

Lecture 12 — Mode & Empirical Relationship
Definition · Mode for ungrouped & grouped data · Modal class · Empirical formula · Comparison of Mean, Median, Mode · When to use which

Introduction to Statistics

Unit II · Lecture 11

📍 Median — Definition

📖 Formal Definition

The median is the middle value of a dataset when the observations are arranged in ascending (or descending) order. It divides the data into two equal halves — exactly 50% of observations fall below the median and 50% fall above it. It is a positional average, denoted by M or Me.

💡 Analogy: The Centre Seat

Imagine 7 students standing in a line, arranged by height (shortest to tallest). The student standing in the 4th position (the middle) represents the median height. Three students are shorter and three are taller — the median person perfectly splits the group in half.

💬 Key Insight: The median is not affected by extreme values. If the tallest student's height doubles, the median student's position doesn't change. This makes the median robust against outliers — a major advantage over the mean.

Introduction to Statistics

Unit II · Lecture 11

📝 Median — Ungrouped Data (Odd n)

Formula — When n is ODD

Median = Value at position (n + 1) / 2

First arrange data in ascending order, then pick the middle value.

📝 Example 1: Odd number of observations

Find the median of: 23, 18, 45, 12, 31, 27, 38

📝 Step-by-Step Solution

Step 1: Arrange in ascending order:
12, 18, 23, 27, 31, 38, 45

Step 2: Count: n = 7 (odd)

Step 3: Position of median = (n+1)/2 = (7+1)/2 = 4th value

Step 4: The 4th value in the ordered list is 27.

∴ Median = 27

📊 Verification

Values below 27: 12, 18, 23 → 3 values (left half) ✓
Values above 27: 31, 38, 45 → 3 values (right half) ✓
Equal halves confirmed. The median is correct.

Introduction to Statistics

Unit II · Lecture 11

📝 Median — Ungrouped Data (Even n)

Formula — When n is EVEN

Median = Average of values at positions n/2 and (n/2 + 1)

There is no single middle value — take the average of the two middle values.

📝 Example 2: Even number of observations

Find the median of: 15, 22, 38, 27, 41, 33

📝 Step-by-Step Solution

Step 1: Arrange in ascending order:
15, 22, 27, 33, 38, 41

Step 2: Count: n = 6 (even)

Step 3: Positions = n/2 = 3rd and n/2 + 1 = 4th

Step 4: 3rd value = 27, 4th value = 33

Step 5: Median = (27 + 33) / 2 = 60 / 2 = 30

∴ Median = 30 (Note: 30 is not an actual observation!)

💡 Key Difference from Mean: The median may not be an actual data point — it can fall between two observations. The mean can also be a non-observed value. Neither is guaranteed to be in the dataset.

Introduction to Statistics

Unit II · Lecture 11

🎬 Visual: Find the Median on a Number Line

Select Odd or Even n, then click "Find Median":

Introduction to Statistics

Unit II · Lecture 11

📊 Median for Grouped (Frequency) Data

📖 Logic

For grouped data, we do not know individual observations. We know that N/2 observations lie below the median. The median lies in the class where the cumulative frequency first reaches or exceeds N/2. This class is called the median class.

Median Formula — Grouped Data

Median = L + [(N/2 − CF) / f] × h

L = lower limit of the median class
N = total frequency (Σf)
CF = cumulative frequency of the class preceding the median class
f = frequency of the median class
h = class width of the median class

💡 Analogy: Finding Your Seat in a Theatre

Imagine rows of seats (classes) with people (frequency). You are person number N/2. You count people row by row (cumulative frequency). When you find the row where you belong — that is your median class. Your exact seat within that row is the median value.

Introduction to Statistics

Unit II · Lecture 11

📝 Median — Grouped Data: Worked Example

📝 Problem

Find the median from the following distribution:

Class	10–20	20–30	30–40	40–50	50–60	60–70
f	5	8	12	10	7	3

📝 Step-by-Step Solution

Step 1: Calculate N = Σf = 5+8+12+10+7+3 = 45

Step 2: N/2 = 45/2 = 22.5

Step 3: Build the cumulative frequency (Less Than type):

Class	f	CF (Less Than)
10–20	5	5
20–30	8	13
30–40	12	25 ← first CF ≥ 22.5
40–50	10	35
50–60	7	42
60–70	3	45

Step 4: Identify the median class: 30–40 (first class where CF ≥ 22.5)

Step 5: Identify the values:
L = 30 (lower limit of median class)
CF = 13 (CF of the class before the median class, i.e., 20–30)
f = 12 (frequency of the median class)
h = 10 (class width)

Step 6: Apply the formula:
Median = L + [(N/2 − CF) / f] × h
Median = 30 + [(22.5 − 13) / 12] × 10
Median = 30 + [9.5 / 12] × 10
Median = 30 + 0.7917 × 10
Median = 30 + 7.917
Median = 37.92

💡 Interpretation: 50% of students scored below 37.92 marks, and 50% scored above. The median falls inside the class 30–40, approximately 79% of the way through that class.

Introduction to Statistics

Unit II · Lecture 11

🎬 Visual: Median from Grouped Data

Click "Find Median" to see the median class identified and the value calculated on a number line:

Introduction to Statistics

Unit II · Lecture 11

✅ Merits & ⚠️ Limitations of Median

✅ Merits

1. Not affected by extreme values — robust against outliers.

2. Can be used for open-end distributions — only needs position, not exact values.

3. Can be determined graphically — from the ogive (at N/2 on Y-axis).

4. Suitable for qualitative data — can find median of ordinal data (e.g., rating: Low, Medium, High).

5. Easy to understand — the "middle value" concept is intuitive.

⚠️ Limitations

1. Does not use all observations — only the middle value matters; extreme values are ignored.

2. Not algebraically tractable — cannot be used in further calculations like variance.

3. Not unique for grouped data — depends on the interpolation assumption.

4. Requires ordering — data must be arranged in order first (tedious for large datasets).

5. Affected by sampling fluctuations — the median varies more across samples than the mean.

Introduction to Statistics

Unit II · Lecture 12

— Lecture 12 —

Mode & Empirical Relationship

Most Frequent Value · Modal Class · Mean–Median–Mode Link

Introduction to Statistics

Unit II · Lecture 12

🏆 Mode — Definition

📖 Formal Definition

The mode is the value that occurs most frequently in a dataset — the observation with the highest frequency. It is the value around which the data tends to cluster most densely. Denoted by Z or Mo.

💡 Analogy: Most Popular Choice

If a class votes for their favourite colour and the results are: Red (8), Blue (12), Green (5), Yellow (3) — Blue is the mode because it was chosen most often (12 times). The mode tells you what is most popular or most common.

📖 Special Cases

Unimodal: One mode. E.g., 2, 3, 3, 5, 7 → Mode = 3
Bimodal: Two modes. E.g., 2, 3, 3, 5, 5, 7 → Modes = 3, 5
Trimodal: Three modes.
No mode: All values occur equally. E.g., 1, 2, 3, 4, 5 → No mode

📖 Real-Life Examples

• Most common shoe size sold in a shop
• Most frequent daily temperature in a month
• Most popular product in a store
• Peak traffic hour on a road
• Most common blood group in a population

Introduction to Statistics

Unit II · Lecture 12

📝 Mode — Ungrouped Data

📖 Method

For ungrouped data, the mode is found by inspection — simply count the frequency of each value and pick the one that occurs most often. No formula is needed.

📝 Example 1: Single Mode

Data: 4, 6, 2, 6, 8, 6, 3, 5, 6, 7

Value	Frequency
2	1
3	1
4	1
5	1
6	4
7	1
8	1

∴ Mode = 6 (occurs 4 times — highest frequency)

📝 Example 2: Bimodal

Data: 3, 5, 3, 7, 5, 8, 3, 5, 9

Value	Frequency
3	3
5	3
7	1
8	1
9	1

∴ Modes = 3 and 5 (bimodal — both occur 3 times)

💡 Important: Mode can be found for qualitative (nominal) data too. E.g., if fruits sold are: Apple (50), Banana (80), Orange (30) — Mode = Banana. This is something the mean and median cannot do for nominal data.

Introduction to Statistics

Unit II · Lecture 12

📊 Mode for Grouped (Frequency) Data

📖 Identifying the Modal Class

For grouped data, the modal class is the class interval with the highest frequency. The mode is then estimated by interpolation within this class.

Mode Formula — Grouped Data

Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

L = lower limit of the modal class
f₁ = frequency of the modal class
f₀ = frequency of the class preceding the modal class
f₂ = frequency of the class succeeding the modal class
h = class width

💡 Analogy: The Peak of a Hill

Imagine the frequency distribution as a landscape of hills. The modal class is the tallest hill. The mode formula finds the exact peak of that hill by looking at how steeply the hill rises from the left (f₀) and falls to the right (f₂).

Introduction to Statistics

Unit II · Lecture 12

📝 Mode — Grouped Data: Worked Example

📝 Problem

Find the mode from the following distribution:

Class	0–10	10–20	20–30	30–40	40–50	50–60
f	5	8	15	12	7	3

📝 Step-by-Step Solution

Step 1: Identify the class with the highest frequency:
20–30 has f = 15 (highest) → Modal class = 20–30

Step 2: Identify the values:
L = 20 (lower limit of modal class)
f₁ = 15 (frequency of modal class)
f₀ = 8 (frequency of the class before modal class, i.e., 10–20)
f₂ = 12 (frequency of the class after modal class, i.e., 30–40)
h = 10 (class width)

Step 3: Apply the formula:
Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
Mode = 20 + [(15 − 8) / (2×15 − 8 − 12)] × 10
Mode = 20 + [7 / (30 − 20)] × 10
Mode = 20 + [7 / 10] × 10
Mode = 20 + 7
Mode = 27

💡 Interpretation: The most common marks range is 20–30, and the estimated peak frequency occurs at 27 marks. This is where the data clusters most densely.

Introduction to Statistics

Unit II · Lecture 12

🎬 Visual: Finding the Mode — The Peak

Click "Find Mode" to see the modal class identified and the mode value calculated:

Introduction to Statistics

Unit II · Lecture 12

✅ Merits & ⚠️ Limitations of Mode

✅ Merits

1. Easy to understand — "most frequent value" is intuitive.

2. Not affected by extreme values — outliers don't shift the mode.

3. Can be used for qualitative data — can find mode of nominal data (favourite colour, most common name).

4. Can be determined by inspection for ungrouped data.

5. Represents the "typical" value well in highly skewed distributions (e.g., most common income).

⚠️ Limitations

1. Not always defined — no mode if all values occur equally, or ambiguous if multiple modes exist.

2. Not based on all observations — only the most frequent value matters.

3. Not algebraically tractable — cannot be used in further calculations.

4. Not unique — a dataset can have multiple modes.

5. Affected by class width — for grouped data, changing the class intervals can change the modal class.

Introduction to Statistics

Unit II · Lecture 12

🔗 Empirical Relationship: Mean, Median & Mode

📖 The Karl Pearson Empirical Formula

For a moderately skewed distribution (not too far from bell-shaped), Karl Pearson discovered an approximate relationship between the three measures of central tendency:

Karl Pearson's Empirical Formula

Mode = 3 × Median − 2 × Mean

Or equivalently:
Mean − Mode = 3 × (Mean − Median)

Rearranged:
Median = (Mode + 2 × Mean) / 3
Mean = (3 × Median − Mode) / 2

💡 Why Does This Work?

In a perfectly symmetric (bell-shaped) distribution, Mean = Median = Mode. When the distribution becomes skewed, the three values separate in a predictable way. This formula captures that predictable separation.

⚠️ Important Caveats:
1. This is an approximation, not an exact formula.
2. Works well for moderately skewed distributions only.
3. For symmetric distributions: Mean = Median = Mode (the formula gives Mode = 3M − 2M = M).
4. For extremely skewed or multi-modal distributions, the formula may give inaccurate results.

Introduction to Statistics

Unit II · Lecture 12

🎬 Visual: How Skewness Affects the Three Averages

Click each type to see the relative positions of Mean, Median, and Mode:

Introduction to Statistics

Unit II · Lecture 12

📝 Empirical Formula — Worked Example

📝 Example 1: Find Mode from Mean & Median

For a moderately skewed distribution:
Mean = 40, Median = 35
Find the Mode using the empirical formula.

📝 Solution

Mode = 3 × Median − 2 × Mean
Mode = 3 × 35 − 2 × 40
Mode = 105 − 80
Mode = 25

Check: Mean (40) > Median (35) > Mode (25) → Right-skewed ✓

📝 Example 2: Find Median from Mean & Mode

For a moderately skewed distribution:
Mean = 50, Mode = 62
Find the Median.

📝 Solution

Using: Mean − Mode = 3(Mean − Median)
50 − 62 = 3(50 − Median)
−12 = 150 − 3 × Median
3 × Median = 162
Median = 54

Check: Mode (62) > Median (54) > Mean (50) → Left-skewed ✓

Introduction to Statistics

Unit II · Lecture 12

🎬 Visual: Empirical Formula Calculator

Enter any two values and the third will be calculated automatically:

Mean (x̄)

Median (M)

Mode (Z)

Introduction to Statistics

Unit II · Lecture 12

⚖️ Mean vs Median vs Mode — Complete Comparison

Feature	Mean (x̄)	Median (M)	Mode (Z)
Definition	Sum ÷ Count	Middle value	Most frequent value
Type	Mathematical average	Positional average	Positional average
Based on	All observations	Middle position only	Frequency only
Affected by outliers?	Yes — highly sensitive	No — robust	No — robust
Qualitative data?	No	Yes (ordinal only)	Yes (nominal too)
Open-end classes?	Cannot be calculated	Can be estimated	Can be estimated
Algebraic treatment?	Yes (variance, regression)	Limited	No
Unique?	Always unique	Always unique	May not exist or may have multiple
Graphical determination?	Not directly	From ogive	From histogram (highest bar)
Sampling stability	Most stable	Moderate	Least stable
Best for	Symmetric data	Skewed data	Most common value needed

Introduction to Statistics

Unit II · Lecture 12

🧭 When to Use Which Measure?

Use the MEAN when:

• Data is symmetric with no outliers
• You need further algebraic calculations
• Every observation matters
• Computing average marks, income, temperature

Use the MEDIAN when:

• Data is skewed (income, house prices)
• There are extreme outliers
• Distribution has open-end classes
• You want the centre of a ranked dataset

Use the MODE when:

• Data is qualitative/nominal
• You need the most typical or common value
• Finding peak demand (busiest hour, best-selling size)
• Data has no numerical meaning

💬 Practical Wisdom: In practice, the mean is used most often (in exams, research, business). The median is preferred for income/wealth data. The mode is used in market research and manufacturing. Understanding all three gives you a complete picture.

Introduction to Statistics

Unit II · Lecture 12

🎬 Visual: The Relationship Map

Introduction to Statistics

Unit II · Lectures 11–12

🔑 Complete Summary: Lectures 11–12

Median = middle value when data is ordered. For odd n: median at position (n+1)/2. For even n: average of the two middle values.
Median for grouped data: Median = L + [(N/2 − CF) / f] × h. The median class is where CF first reaches or exceeds N/2.
Median merits: Not affected by outliers, can handle open-end classes, can be found graphically from the ogive.
Mode = the most frequently occurring value. Can be unimodal, bimodal, trimodal, or no mode. Works for both qualitative and quantitative data.
Mode for grouped data: Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h. The modal class has the highest frequency.
Mode merits: Not affected by outliers, works for nominal data, easy to identify by inspection for ungrouped data.
Karl Pearson's empirical formula: Mode = 3 × Median − 2 × Mean (or equivalently Mean − Mode = 3(Mean − Median)). Works for moderately skewed distributions.
In a symmetric distribution: Mean = Median = Mode. In a right-skewed distribution: Mean > Median > Mode. In a left-skewed distribution: Mode > Median > Mean.
Mean: Best for symmetric data, uses all observations, algebraically tractable, but sensitive to outliers.
Median: Best for skewed data with outliers, robust, but ignores extreme values and is not algebraically tractable.
Mode: Best for finding the most common value, works for qualitative data, but may not be unique or may not exist.

Introduction to Statistics

Unit II · Lectures 11–12

🧠 Quick Quiz — Lectures 11–12

Q1: The median of 8, 12, 15, 20, 25, 30, 35 is:

A) 15

B) 20

C) 25

D) 17.9

Q2: The median of 10, 20, 30, 40 is:

A) 20

B) 30

C) 25

D) 22

Q3: In a grouped frequency distribution, the median class is identified by:

A) The class with the highest frequency

B) The class where CF first reaches or exceeds N/2

C) The first class

D) The middle class in the table

Q4: The data {Red, Blue, Red, Green, Red, Blue} has mode:

A) Red

B) Blue

C) Green

D) No mode

Q5: If Mean = 50 and Median = 45, the Mode (using empirical formula) is approximately:

A) 40

B) 35

C) 55

D) 42

Q6: In a right-skewed distribution, the correct order is:

A) Mean > Median > Mode

B) Mode > Median > Mean

C) Mean = Median = Mode

D) Median > Mean > Mode

Q7: Which average can be found for qualitative (nominal) data?

A) Mean only

B) Median only

C) Mode

D) All three

Q8: If the modal class has f₁=20, the preceding class has f₀=12, and the succeeding class has f₂=15, with L=30 and h=10, the mode is:

A) 32

B) 38

C) 40

D) 35

Q9: The empirical formula Mean − Mode = 3(Mean − Median) was given by:

A) Bowley

B) Karl Pearson

C) Fisher

D) Gauss

Q10: The median is NOT affected by:

A) The number of observations

B) The order of observations

C) Extreme values (outliers)

D) The middle value

Introduction to Statistics

Unit II · Lectures 11–12

📝 Homework & Practice

📝 Practice Tasks

1. Find the median of: (a) 14, 28, 35, 42, 56, 63, 71 (b) 12, 18, 25, 30, 38, 44

2. Find the median from the following grouped data:

Class	0–10	10–20	20–30	30–40	40–50
f	6	10	14	12	8

3. Find the mode of: (a) 3, 5, 7, 5, 3, 5, 8, 5, 9 (b) Apple, Banana, Apple, Orange, Banana, Apple, Mango, Banana, Banana

4. Find the mode from the following grouped data:

Class	5–15	15–25	25–35	35–45	45–55
f	4	8	16	11	6

5. The mean of a moderately skewed distribution is 42 and the median is 38. Find the mode using the empirical formula. Is the distribution right-skewed or left-skewed?

6. The mode of a distribution is 55 and the median is 48. Find the mean using the empirical formula.

7. Explain why the median is a better measure of central tendency than the mean for income data. Give a numerical example.

8. For a symmetric distribution with mean = 30, what are the median and mode? Verify using the empirical formula.

9. BONUS: The following are the marks of 15 students: 65, 72, 58, 84, 90, 58, 72, 58, 68, 75, 82, 58, 63, 70, 78. Calculate the mean, median, and mode. Verify the empirical relationship. Comment on the skewness.

Introduction to Statistics

Unit II · Lectures 11–12

Thank You!

Questions? Let's discuss!

Upcoming in Unit II

Lecture 13–14: Partition Values — Quartiles, Deciles, Percentiles
Lecture 15: Unit II Review & Problem-Solving

Unit II · Lectures 11–12 · BDS113 Introduction to Statistics · B.Sc. Data Science, Semester II