🚀 Data Science Roadmap 2026

📘 Phase 2: Mathematics for Data Science

📖 Topic 7: Descriptive Statistics — Range, Percentiles, Quartiles, IQR & Five-Number Summary

Welcome back! 👋

In the previous lesson you covered Probability Distributions.

Now we’re moving to Descriptive Statistics — how we summarize data without predicting the population.

Today we’ll cover: Range, Percentiles, Quartiles, IQR, Five-number summary, Outlier detection

These are core for EDA.

🔹 1. What is Descriptive Statistics?

Summarizes key characteristics of a dataset.

Example: Salaries: 30000, 35000, 40000, 45000, 50000

Instead of checking each value, use: Min, Max, Mean, Median, Quartiles, Percentiles, Std Dev

🔹 2. Range

Formula:

Range = Maximum − Minimum

Example: 10, 20, 30, 40, 50 → Range = 50 − 10 = 40

Note: Very sensitive to outliers. 50 → 500 makes range jump to 490.

🔹 3. Percentiles ⭐

Value below which X% of observations fall.

50th Percentile = Median

25th Percentile = 25% at or below

90th Percentile = 90% at or below

🔹 4. Real-World Example

90th percentile score ≠ 90% marks. It means you did better than ∼90% of people.

🔹 5. Quartiles

Divide data into 4 equal parts:

Q1 = 25th percentile

Q2 = 50th percentile = Median

Q3 = 75th percentile

🔹 6. Visualizing Quartiles

0% ---- Q1 ---- Q2 ---- Q3 ---- 100%

25% 50% 75%

🔹 7. Interquartile Range (IQR) ⭐

Formula: IQR = Q3 − Q1

Example: Q1=20, Q3=60 → IQR = 40. Middle 50% spans 40 units.

🔹 8. Why IQR Matters

Less affected by outliers than Range.

Data: 10,20,30,40,50,1000 → Range=990 but IQR ignores the 1000.

🔹 9. Detecting Outliers Using IQR ⭐

Lower Bound = Q1 − 1.5 × IQR

Upper Bound = Q3 + 1.5 × IQR

Values outside = potential outliers

🔹 10. Outlier Example

Q1=20, Q3=60 → IQR=40

Lower = 20-60 = -40

Upper = 60+60 = 120

So < -40 or > 120 are outliers

🔹 11. Five-Number Summary ⭐

1. Minimum 2. Q1 3. Median 4. Q3 5. Maximum

Ex: 10, 20, 30, 40, 50

🔹 12. Box Plot

Visualizes the 5-number summary.

Box = Q1 to Q3. Line inside = Median. Whiskers = range without outliers.

🔹 13. Python Example

import numpy as np

data = [10, 20, 30, 40, 50, 60, 70]
q1 = np.percentile(data, 25)
median = np.percentile(data, 50)
q3 = np.percentile(data, 75)
iqr = q3 - q1
print("Q1:", q1, "Median:", median, "Q3:", q3, "IQR:", iqr)

🔹 14. Descriptive Statistics in Pandas

import pandas as pd

df = pd.DataFrame({"Salary": [30000, 35000, 40000, 45000, 50000]})
print(df["Salary"].describe())

describe() gives Count, Mean, Std, Min, 25%, 50%, 75%, Max

🔹 15. Real-World Example

Transactions: Q1=₹500, Median=₹1000, Q3=₹2000 → IQR=₹1500

Use IQR to flag fraud, bulk orders, errors, or VIP customers. Investigate before deleting.

🔹 16. Range vs IQR

Range: Easy but outlier-sensitive

IQR: Middle 50% only, robust to outliers

🔹 17. Percentile vs Percentage

Percentage = out of 100.

Ex: 80% marks

Percentile = relative position.

Ex: 90th percentile

🔹 18. Common Mistakes

❌ 90th percentile = 90% score

❌ Deleting all outliers blindly

❌ Thinking IQR covers all data

🎯 Practice Questions

1. Range of 10, 20, 30, 40, 50 = ?

2. Median = which percentile?

3. Q1=25, Q3=75 → IQR = ?

4. Upper outlier boundary formula?

5. 5 components of five-number summary?

🎯 Key Takeaways

✅ Range = Max - Min

✅ Q1=25th, Q2=50th=Median, Q3=75th

✅ IQR = Q3 - Q1

✅ 5-number summary = Min, Q1, Median, Q3, Max

✅ Percentile ≠ Percentage

👉 Double Tap ❤️ For More