Statistics - Class 9 Maths

📋 Statistics Kya Hai?

Statistics mathematics ki woh branch hai jo data (jaankari) collect karne, organize karne, present karne, aur analyse karne se related hai. Aaj ke digital zamane mein statistics bahut important hai — doctor patient ka data dekhte hain, government census karte hain, businesses sales data analyse karte hain.

🌾 Real Life Example:

Socho ek gaon mein school hai. Teacher jaanna chahti hain ki students ke marks kaisi distribution mein hain, kitne students average se upar hain, aur sabse common score kya hai. Ye sab statistics se pata chalta hai!

Do important terms:

Data: Facts, numbers, ya information jo collect ki jaati hai. Jaise: 40 students ke marks.
Statistics: Data ko samajhne ki science — patterns dhundna, summarize karna, conclusions nikalna.

1️⃣ Data Collection aur Frequency Table

Jab hum bahut saara data collect karte hain, toh pehle usse organize karna padta hai. Ek common method hai Frequency Distribution Table.

Raw Data: Data jaise collect kiya — koi order nahi. Jaise: 5, 3, 7, 5, 2, 5, 8, 3, 7.
Frequency: Har value kitni baar aayi. Jaise 5 teen baar aayi → frequency = 3.
Class Interval: Jab data range mein ho, toh groups banate hain. Jaise 0-10, 10-20.

📋 Example — Students ke marks (out of 10):

Raw Data: 6, 8, 5, 9, 7, 6, 8, 6, 5, 9, 7, 8, 6, 5, 8

Marks	Tally	Frequency (f)
5	III	3
6	IIII	4
7	II	2
8	IIII	4
9	II	2
Total		15

Ab dekho — sirf raw numbers se ye sab patterns clear nahi tha, par table se hum instantly dekh sakte hain ki 6 aur 8 sabse zyada baar aaye!

2️⃣ Graphical Representation (Graphs se Dikhao)

Numbers ko visually (graphs mein) dikhane se patterns aur easier samajh mein aate hain. Class 9 mein hum teen main types padhte hain:

📊 A. Bar Graph (Danda Graph)

Bar graph mein har category ke liye ek rectangular bar hoti hai. Bar ki height = frequency. Bars ke beech gap hota hai.

📌 Bar Graph vs Histogram:
Bar Graph: discrete/categorical data, bars ke beech gap hota hai.
Histogram: continuous data (class intervals), bars touch karte hain (koi gap nahi).

📈 B. Histogram (Class Intervals wala Graph)

Jab data continuous hota hai aur hum use groups (class intervals) mein rakhte hain, toh histogram banate hain. Marks 0-100 ko hum 0-20, 20-40, 40-60, 60-80, 80-100 mein divide kar sakte hain.

📉 C. Frequency Polygon

Frequency polygon banate hain histogram ke midpoints ko line se join karke. Ye ek line graph hota hai jo data ka overall shape dikhata hai.

Pehle histogram banao.
Har bar ke top midpoint par ek dot lagao.
In dots ko straight lines se connect karo.
Polygon ko x-axis se close karo (dono ends par).

📊 D. Stacked Bar Graph

Stacked bar graph mein ek bar ke andar kai categories dikhate hain ek ke upar ek. Ye do ya zyada groups ko compare karne ke liye use hota hai.

3️⃣ Measures of Central Tendency (Data Ka Center)

Ek dataset ka ek representative value nikaalte hain jisse hum poore data ko summarize kar sakein. Teen main measures hain:

📐 A. Mean (औसत / Ausat)

Mean = sabhi values ka sum ÷ total number of values. Iska symbol \(\bar{x}\) (x-bar) hota hai.

Mean (\(\bar{x}\)) = \(\frac{\sum x}{n} = \frac{\text{Sum of all values}}{\text{Total count}}\)

📝 Example:

5 students ke marks: 60, 75, 82, 68, 90. Mean nikalo.

Sum = 60 + 75 + 82 + 68 + 90 = 375
n = 5
Mean = \(\frac{375}{5} = \mathbf{75}\) ✅

Weighted Mean (Vajan wala Ausat):

Jab har value ka weight (importance) alag hota hai, toh weighted mean use karte hain. Jaise exams mein alag subjects ka alag weightage hota hai.

Weighted Mean = \(\frac{\sum (w_i \times x_i)}{\sum w_i}\) (jahan w = weight)

📝 Weighted Mean Example:

Maths (weight=3): score 80, Science (weight=2): score 70, Hindi (weight=1): score 90.

Weighted Mean = \(\frac{3 \times 80 + 2 \times 70 + 1 \times 90}{3+2+1} = \frac{240+140+90}{6} = \frac{470}{6} \approx \mathbf{78.3}\)

📐 B. Median (Beech ki Value)

Median woh value hai jo data ko exactly aadha-aadha divide karti hai — aadhe values usse badi aur aadhe choti. Sabse pehle data ko ascending order mein arrange karte hain.

Median:
Odd n: Middle term = \(\left(\frac{n+1}{2}\right)^{th}\) term
Even n: Average of \(\left(\frac{n}{2}\right)^{th}\) and \(\left(\frac{n}{2}+1\right)^{th}\) terms

📝 Example (Even n):

Data: 12, 5, 8, 15, 3, 10. Median nikalo.

Step 1 — Ascending order: 3, 5, 8, 10, 12, 15 (n=6)
Step 2 — Middle terms: \(\frac{n}{2} = 3^{rd}\) term = 8, \(\frac{n}{2}+1 = 4^{th}\) term = 10
Median = \(\frac{8+10}{2} = \mathbf{9}\)

📐 C. Mode (Sabse Zyada Aane Wali Value)

Mode woh value hai jo data mein sabse zyada baar aati hai. Ek dataset mein ek, do (bimodal), ya koi mode nahi bhi ho sakta.

📝 Example:

Data: 4, 7, 3, 7, 9, 4, 7, 2. Mode = 7 (3 baar aaya — sabse zyada!) ✅

Data: 2, 4, 6, 8. Mode = koi nahi (sab ek baar). Bimodal: 1, 2, 2, 3, 5, 5, 6 → Mode = 2 aur 5.

🧠 Mean, Median, Mode — Kab Kya Use Karein?

Measure	Best For	Example
Mean	Symmetric data, koi extreme values nahi	Class ka average score
Median	Skewed data ya extreme values hon	Sheher ki median income (beggars + billionaires dono hon)
Mode	Most common item jaanna ho	Shoe shop mein sabse zyada bikne wala size

🎯 Practice Questions

❓ Q1:

7 students ke marks: 55, 72, 61, 84, 72, 90, 68. Mean, Median, aur Mode nikalo.

Answer:
Sum = 502 → Mean = 502/7 ≈ 71.7
Sorted: 55, 61, 68, 72, 72, 84, 90 → n=7, 4th term → Median = 72
72 do baar aayi → Mode = 72

❓ Q2:

Data ka mode 15 hai aur mean 12 hai. Kya median 15 ho sakti hai? Kyun ya kyun nahi?

Answer: Nahi necessarily. Mode ≠ Median hamesha. Mode sabse frequent value hai (15), aur median middle value hai. Median 12 aur 15 ke beech kuch bhi ho sakti hai depending on data.

❓ Q3 (Challenge):

Frequency table mein marks 10, 20, 30, 40, 50 ki frequencies 3, 4, 2, 5, 1 hain. Mean nikalo.

Answer:
\(\sum fx = 10\times3 + 20\times4 + 30\times2 + 40\times5 + 50\times1 = 30+80+60+200+50 = 420\)
\(\sum f = 3+4+2+5+1 = 15\)
Mean = \(420 \div 15 = \mathbf{28}\) ✅

📊 Statistics