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📘 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

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Essential Excel Functions for Data Analysts 🚀

1️⃣ Basic Functions

SUM() – Adds a range of numbers. =SUM(A1:A10)

AVERAGE() – Calculates the average. =AVERAGE(A1:A10)

MIN() / MAX() – Finds the smallest/largest value. =MIN(A1:A10)


2️⃣ Logical Functions

IF() – Conditional logic. =IF(A1>50, "Pass", "Fail")

IFS() – Multiple conditions. =IFS(A1>90, "A", A1>80, "B", TRUE, "C")

AND() / OR() – Checks multiple conditions. =AND(A1>50, B1<100)


3️⃣ Text Functions

LEFT() / RIGHT() / MID() – Extract text from a string.

=LEFT(A1, 3) (First 3 characters)

=MID(A1, 3, 2) (2 characters from the 3rd position)


LEN() – Counts characters. =LEN(A1)

TRIM() – Removes extra spaces. =TRIM(A1)

UPPER() / LOWER() / PROPER() – Changes text case.


4️⃣ Lookup Functions

VLOOKUP() – Searches for a value in a column.

=VLOOKUP(1001, A2:B10, 2, FALSE)


HLOOKUP() – Searches in a row.

XLOOKUP() – Advanced lookup replacing VLOOKUP.

=XLOOKUP(1001, A2:A10, B2:B10, "Not Found")



5️⃣ Date & Time Functions

TODAY() – Returns the current date.

NOW() – Returns the current date and time.

YEAR(), MONTH(), DAY() – Extracts parts of a date.

DATEDIF() – Calculates the difference between two dates.


6️⃣ Data Cleaning Functions

REMOVE DUPLICATES – Found in the "Data" tab.

CLEAN() – Removes non-printable characters.

SUBSTITUTE() – Replaces text within a string.

=SUBSTITUTE(A1, "old", "new")



7️⃣ Advanced Functions

INDEX() & MATCH() – More flexible alternative to VLOOKUP.

TEXTJOIN() – Joins text with a delimiter.

UNIQUE() – Returns unique values from a range.

FILTER() – Filters data dynamically.

=FILTER(A2:B10, B2:B10>50)



8️⃣ Pivot Tables & Power Query

PIVOT TABLES – Summarizes data dynamically.

GETPIVOTDATA() – Extracts data from a Pivot Table.

POWER QUERY – Automates data cleaning & transformation.


You can find Free Excel Resources here: https://t.me/excel_data

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