📊 STATISTICS

Class 10 Mathematics • Chapter 13

Data • Frequency • Mean • Median • Mode • Graphical Representation

THE GOAL • HANDWRITTEN EXAM NOTES
01. What is Statistics?
Statistics is the branch of mathematics concerned with the collection, organisation, presentation, analysis and interpretation of data.
🧠 Simple words mein: Statistics ka matlab hai data ko collect karke arrange karna, usko samajhna aur usse useful information nikalna.
✏️ Example
Marks of five students are:
12, 18, 15, 20, 15
This is a collection of numerical data.
02. Data
Data means a collection of facts, figures or observations.
Raw Data

Data in its original, unorganised form.

Example:
7, 3, 9, 4, 7, 2
Organised Data

Data arranged systematically for easy understanding.

Example:
2, 3, 4, 7, 7, 9
💡 Exam mein raw data ko arrange karke frequency table banana bahut common step hai.
03. Frequency
The number of times an observation occurs is called its frequency.
Value Frequency
2 3
4 2
5 4
✏️ Example
Data:
2, 3, 2, 5, 3, 2, 5, 5, 5
Frequency of 2 = 3
Frequency of 3 = 2
Frequency of 5 = 4
✓ Total frequency = 3 + 2 + 4 = 9
04. Grouped Frequency Distribution
When data is divided into class intervals and frequencies are given for each interval, it is called a grouped frequency distribution.
Class Interval Frequency (f)
0–10 5
10–20 8
20–30 12
30–40 7
🧠 Class interval ek range hoti hai. Example: 10–20 means observations belonging to that class.
05. Class Mark ⭐⭐⭐⭐⭐
The midpoint of a class interval is called its class mark.
Class Mark = Upper Limit + Lower Limit 2
✏️ Example
For class interval 20–30:
x = 20 + 30 2 = 25
✓ Class mark = 25
06. Mean of Ungrouped Data ⭐⭐⭐⭐⭐
Mean = Sum of observations Number of observations
✏️ Example
Find the mean of:
8, 10, 12, 14, 16
Mean = 8+10+12+14+16 5
= 60 5 = 12
✓ Mean = 12
07. Mean with Frequency
Mean = Σfx Σf
x f fx
2 3 6
4 2 8
6 5 30
Total 10 44
✏️ Example
Mean = 44 10 = 4.4
✓ Mean = 4.4
08. Mean of Grouped Data — Direct Method ⭐⭐⭐⭐⭐
For grouped data, first find the class mark x for every class, then calculate fx.
x = Upper Limit + Lower Limit 2
Mean = Σfx Σf
Class f x fx
0–10 2 5 10
10–20 3 15 45
20–30 5 25 125
Total 10 — 180
Mean = 180 10 = 18
✓ Mean = 18
09. Assumed Mean Method
When calculations using direct method become lengthy, an assumed mean can simplify the calculation.
d = x − A
Mean = A + Σfd Σf
🧠 A = assumed mean. Usually centre ke aas-paas ka convenient class mark A choose karna calculation ko easy banata hai.
✏️ Example
Suppose: A = 25 and
Σf = 20,   Σfd = −40
Then:
Mean = 25 + −40 20
= 25 − 2 = 23
✓ Mean = 23
10. Step-Deviation Method
u = x − A h
Mean = A + h Σfu Σf
Here:

A = assumed mean
h = class size
u = step deviation
⚠️ Step-deviation method tab especially useful hota hai jab class intervals equal width ke hon.
11. Median — Basic Idea ⭐⭐⭐⭐⭐
Median is the middle value of an arranged data set. Data ko ascending ya descending order mein arrange karna zaroori hai.
✏️ Example
Data:
3, 7, 9, 12, 15
There are 5 observations. Middle observation = 3rd observation.
Median = 9
🧠 Median = beech wali value. But grouped data mein median class identify karke formula apply karna hota hai.
12. Median of Ungrouped Data
If number of observations is odd:
Median = n + 1 2 th observation
If n is even:
Median = ( n 2 )th observation + ( n 2 +1)th observation 2
✏️ Example — Even Data
2, 4, 6, 8 Here n = 4. Middle values are 2nd and 3rd:
4 and 6
Median = 4 + 6 2 = 5
✓ Median = 5
13. Cumulative Frequency ⭐⭐⭐⭐⭐
Cumulative frequency is the running total of frequencies.
Class f cf
0–10 4 4
10–20 6 10
20–30 8 18
30–40 5 23
🧠 cf banate waqt previous frequency ko next frequency mein add karte jao.
14. Median of Grouped Data ⭐⭐⭐⭐⭐
Median = l + ( N 2 − cf ) f × h
Where:

l = lower boundary of median class
N = total frequency
cf = cumulative frequency before median class
f = frequency of median class
h = class size
⚠️ Formula mein cf ka matlab median class se pehle wali cumulative frequency hai — median class ki cf nahi.
✏️ Example
Suppose:
N = 40
Median class:
20–30
and
l = 20,   cf = 14,   f = 12,   h = 10
Then:
Median = 20 + 20 − 14 12 × 10
= 20 + 5 = 25
✓ Median = 25
15. How to Find Median Class
First calculate:
N = Σf
Then calculate:
N/2
The class whose cumulative frequency is just greater than N/2 is the median class.
✏️ Example
Suppose:
N = 50
Then:
N/2 = 25
If cumulative frequencies are: 10, 18, 29, 40, 50 The first cf greater than 25 is 29. Therefore, the corresponding class is the median class.
16. Mode ⭐⭐⭐⭐⭐
Mode is the value that occurs most frequently.
✏️ Example
Data:
3, 5, 5, 7, 8, 5, 9
5 occurs three times.
✓ Mode = 5
🧠 Mode = jo value sabse zyada baar aaye.
17. Mode of Grouped Data ⭐⭐⭐⭐⭐
Mode = l + f₁ − f₀ 2f₁ − f₀ − f₂ × h
Where:

l = lower boundary of modal class
f₁ = frequency of modal class
f₀ = frequency of class preceding modal class
f₂ = frequency of class succeeding modal class
h = class size
✏️ Example
Suppose:
l = 20, f₁ = 15, f₀ = 8, f₂ = 10, h = 10
Then:
Mode = 20 + 15 − 8 30 − 8 − 10 × 10
= 20 + 7 12 × 10
= 20 + 35 6
≈ 25.83
✓ Mode ≈ 25.83
18. Modal Class
The class having the highest frequency is called the modal class.
Class Frequency
0–10 5
10–20 12
20–30 18
30–40 9
✓ Modal class = 20–30
19. Empirical Relationship ⭐⭐⭐⭐⭐
Mode = 3 Median − 2 Mean
or
3 Median = Mode + 2 Mean
✏️ Example
Mean = 20
Median = 22 Then:
Mode = 3(22) − 2(20)
= 66 − 40
= 26
✓ Mode = 26
⚠️ Is relation ko tabhi use karo jab question/data situation appropriate ho aur required relationship expected ho.
20. Mean, Median & Mode — Difference
Measure Basic Idea
Mean Average value
Median Middle value
Mode Most frequent value
🧠 Yaad rakhne ka easiest way:

Mean → Average
Median → Middle
Mode → Most repeated
21. Graphical Representation
Data can be represented visually using graphs. For cumulative frequency distributions, the important graph is the ogive.
Less-than Ogive

Uses upper class boundaries and corresponding less-than cumulative frequencies.
More-than Ogive

Uses lower class boundaries and corresponding more-than cumulative frequencies.
22. Less-than Ogive 📈
For a less-than cumulative frequency curve:

X-axis → upper class boundaries
Y-axis → less-than cumulative frequencies
Upper Class Boundary Cumulative Frequency Less-than Ogive
💡 Less-than ogive mein points generally upper boundaries ke against cumulative frequencies par plot hote hain.
23. More-than Ogive 📉
For a more-than cumulative frequency curve:

X-axis → lower class boundaries
Y-axis → more-than cumulative frequencies
Lower Class Boundary Cumulative Frequency More-than Ogive
24. Finding Median using Ogives
If less-than and more-than ogives are drawn on the same graph, their point of intersection gives the median when projected appropriately onto the x-axis.
🧠 Dono ogive ko same graph par draw karo.

Intersection point se x-axis par perpendicular drop karke median obtain kiya ja sakta hai.
📌 Graph question mein:

1. Correct cumulative frequency table banao.
2. Correct boundaries use karo.
3. Scale clearly mark karo.
4. Points accurately plot karo.
5. Smooth curve draw karo.
25. Missing Frequency ⭐⭐⭐⭐⭐
In some questions, one frequency is unknown. Let the missing frequency be x. Then use the given condition such as mean/median/mode to form an equation and solve x.
✏️ Example — Mean Based
Suppose total frequency is:
10 + x
and:
Σfx = 200 + 20x
If mean = 15:
15 = 200 + 20x 10 + x
Cross multiply:
150 + 15x = 200 + 20x
−50 = 5x
x = −10
⚠️ Negative frequency is impossible, so these assumed values would not represent a valid frequency question.
💡 Real exam question mein values aisi hongi ki frequency non-negative integer aaye. Calculation ke end mein frequency ko logically verify karna zaroori hai.
26. Choosing the Correct Formula
Question asks Use
Average of data Mean
Middle value Median
Most frequent value Mode
Grouped mean Σfx Σf
Grouped median l + (N/2 − cf) f h
Grouped mode l + f₁−f₀ 2f₁−f₀−f₂ h
27. Most Important Symbols
Symbol Meaning
x Observation / class mark
f Frequency
Σf = N Total frequency
cf Cumulative frequency
l Lower boundary of relevant class
h Class size
f₀ Frequency before modal class
f₁ Frequency of modal class
f₂ Frequency after modal class
A Assumed mean
28. Common Exam Mistakes ⚠️
❌ Data ko arrange kiye bina median find karna.

❌ Class mark incorrectly calculate karna.

❌ Σf ko total frequency na lena.

❌ Median formula mein wrong cf use karna.

❌ Modal class ko lowest frequency se select karna.

❌ f₀, f₁, f₂ ko wrong classes se lena.

❌ Ogive mein wrong class boundaries use karna.

❌ Graph ka scale mention na karna.

❌ Arithmetic calculation ke baad answer verify na karna.
29. One-Page Formula Revision ⭐⭐⭐⭐⭐
Mean = Σx n
Mean = Σfx Σf
Mean = A + Σfd Σf
Mean = A + h Σfu Σf
Median = l + (N/2 − cf) f h
Mode = l + f₁ − f₀ 2f₁ − f₀ − f₂ h
Mode = 3 Median − 2 Mean
30. Exam Attack Plan 🎯
STEP 1: Read the complete table.

STEP 2: Identify what is being asked: Mean / Median / Mode / Graph.

STEP 3: Calculate total frequency N.

STEP 4: Make class marks or cumulative frequencies as required.

STEP 5: Select the correct formula.

STEP 6: Substitute carefully.

STEP 7: Check the final answer.

STEP 8: For graph questions, check scale and plotting.
31. Final Revision Checklist 📝
☑ Raw data and frequency clear

☑ Class mark formula clear

☑ Direct mean method clear

☑ Assumed mean method clear

☑ Step-deviation method clear

☑ Median of ungrouped data clear

☑ Cumulative frequency clear

☑ Median class identification clear

☑ Grouped median formula clear

☑ Modal class identification clear

☑ Grouped mode formula clear

☑ Empirical relation clear

☑ Less-than ogive clear

☑ More-than ogive clear

☑ Graph se median identify karna clear