STATISTICS

Ungrouped data mein, observations directly diye hote hain.

• Definition: Mean (या Arithmetic Mean) observations ka average hota hai.
• Formula: Agar \(x_1, x_2, ..., x_n\) n observations hain, toh

$$ \bar{x} = \frac{x_1 + x_2 + ... + x_n}{n} = \frac{\sum x_i}{n} $$

• Frequency ke saath: Agar observations \(x_1, x_2, ..., x_n\) ki respective frequencies \(f_1, f_2, ..., f_n\) hain, toh

$$ \bar{x} = \frac{f_1x_1 + f_2x_2 + ... + f_nx_n}{f_1 + f_2 + ... + f_n} = \frac{\sum f_ix_i}{\sum f_i} $$ Jahan \(\sum f_i = N\) total number of observations hai.

Yeh formula basic hai aur grouped data ke direct method ka base banta hai.

Grouped data mein, observations class intervals mein diye hote hain.

Steps to calculate Mean by Direct Method:

1. Class Mark (\(x_i\)) nikalna: Har class interval ka mid-point calculate karo.

$$ x_i = \frac{\text{Upper Limit} + \text{Lower Limit}}{2} $$

1. Product (\(f_ix_i\)) nikalna: Har class interval ki frequency (\(f_i\)) ko uske class mark (\(x_i\)) se multiply karo.
2. Summation (\(\sum f_ix_i\) aur \(\sum f_i\)): Sabhi \(f_ix_i\) values ka sum aur sabhi frequencies (\(f_i\)) ka sum nikalna.
3. Formula apply karna: Mean calculate karne ke liye formula use karo:

$$ \bar{x} = \frac{\sum f_ix_i}{\sum f_i} $$

Yeh method tab useful hai jab \(x_i\) aur \(f_i\) ki values badi hon, jisse \(f_ix_i\) calculations lengthy ho jaati hain.

Steps to calculate Mean by Assumed Mean Method:

1. Class Mark (\(x_i\)) nikalna: Har class interval ka mid-point calculate karo.
2. Assumed Mean (a) choose karna: \(x_i\) values mein se koi ek value ko assumed mean 'a' choose karo. Generally, middle wali value choose karte hain.
3. Deviation (\(d_i\)) nikalna: Har \(x_i\) se assumed mean 'a' ko subtract karo.

$$ d_i = x_i - a $$

1. Product (\(f_id_i\)) nikalna: Har frequency (\(f_i\)) ko uske corresponding deviation (\(d_i\)) se multiply karo.
2. Summation (\(\sum f_id_i\) aur \(\sum f_i\)): Sabhi \(f_id_i\) values ka sum aur sabhi frequencies (\(f_i\)) ka sum nikalna.
3. Formula apply karna: Mean calculate karne ke liye formula use karo:

$$ \bar{x} = a + \frac{\sum f_id_i}{\sum f_i} $$

Yeh method Assumed Mean method ka extension hai, aur bhi simple calculations provide karta hai, especially jab class intervals ki width (h) same ho.

Steps to calculate Mean by Step-Deviation Method:

1. Class Mark (\(x_i\)) nikalna: Har class interval ka mid-point calculate karo.
2. Assumed Mean (a) choose karna: \(x_i\) values mein se koi ek value ko assumed mean 'a' choose karo.
3. Class Size (h) nikalna: Har class interval ki width (upper limit - lower limit) nikalna. Is method ke liye, class size 'h' sabhi intervals ke liye same hona chahiye.
4. Step-deviation (\(u_i\)) nikalna: Har deviation (\(d_i = x_i - a\)) ko class size 'h' se divide karo.

$$ u_i = \frac{x_i - a}{h} $$

1. Product (\(f_iu_i\)) nikalna: Har frequency (\(f_i\)) ko uske corresponding step-deviation (\(u_i\)) se multiply karo.
2. Summation (\(\sum f_iu_i\) aur \(\sum f_i\)): Sabhi \(f_iu_i\) values ka sum aur sabhi frequencies (\(f_i\)) ka sum nikalna.
3. Formula apply karna: Mean calculate karne ke liye formula use karo:

$$ \bar{x} = a + \left( \frac{\sum f_iu_i}{\sum f_i} \right) \times h $$

Median central tendency ka woh measure hai jo data ko do equal halves mein divide karta hai. Grouped data mein, median ek specific class interval ke andar hota hai, jise Median Class kehte hain.

Steps to calculate Median of Grouped Data:

1. Cumulative Frequency (cf) Table banana: Har class interval ke liye cumulative frequency calculate karo. Cumulative frequency us class tak ki saari frequencies ka sum hoti hai.
2. Total Frequency (N) nikalna: Sabhi frequencies ka sum (\(\sum f_i = N\)) nikalna.
3. \(N/2\) calculate karna: \(N/2\) ki value find karo.
4. Median Class identify karna: Cumulative frequency table mein, woh class interval dhoondo jiski cumulative frequency \(N/2\) se just greater ho. Yeh class Median Class hogi.
5. Values identify karna: Median formula ke liye required values identify karo:

• \(l\): Median class ki lower limit.
• \(h\): Median class ka size (upper limit - lower limit).
• \(f\): Median class ki frequency.
• \(cf\): Median class se just pehle wali class ki cumulative frequency.

1. Formula apply karna: Median calculate karne ke liye formula use karo:

$$ \text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h $$

Mode central tendency ka woh measure hai jo data mein sabse zyada baar repeat hota hai (highest frequency). Grouped data mein, mode ek specific class interval ke andar hota hai, jise Modal Class kehte hain.

Steps to calculate Mode of Grouped Data:

1. Modal Class identify karna: Frequency distribution table mein, woh class interval dhoondo jiski frequency sabse zyada ho. Yeh class Modal Class hogi.
2. Values identify karna: Mode formula ke liye required values identify karo:

• \(l\): Modal class ki lower limit.
• \(h\): Modal class ka size (upper limit - lower limit).
• \(f_1\): Modal class ki frequency (highest frequency).
• \(f_0\): Modal class se just pehle wali class ki frequency.
• \(f_2\): Modal class se just baad wali class ki frequency.

1. Formula apply karna: Mode calculate karne ke liye formula use karo:

$$ \text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h $$

Empirical relationship in statistics jo Mean, Median, aur Mode ko relate karta hai:

$$ \text{3 Median} = \text{Mode} + \text{2 Mean} $$

• Yeh relationship symmetric distribution ke liye exact nahi hota, but moderately skewed distributions ke liye approximate hota hai.
• Is formula ka use tab hota hai jab teen measures mein se koi do diye hon aur teesra nikalna ho.

Example: Agar Mean = 50 aur Mode = 55 hai, toh Median kya hoga? \(3 \times \text{Median} = 55 + 2 \times 50\) \(3 \times \text{Median} = 55 + 100\) \(3 \times \text{Median} = 155\) \(\text{Median} = \frac{155}{3} = 51.67\)

Ogive cumulative frequency distribution ka graphical representation hai. Yeh do types ka hota hai:

1. Less Than Ogive

• Construction Steps:

1. Class intervals ko 'less than' type mein convert karo: Upper limits ko 'less than' values ke roop mein likho (e.g., 'less than 10', 'less than 20').
2. Cumulative frequencies calculate karo: Har 'less than' value ke corresponding cumulative frequency nikalna.
3. Points plot karo: X-axis par upper class limits lo aur Y-axis par corresponding cumulative frequencies lo. Points \((x_i, cf_i)\) plot karo.
4. Smooth curve ya line segments join karo: In points ko smooth curve ya line segments se join karo. Line segments se join karne par 'cumulative frequency polygon' banta hai, aur smooth curve se 'ogive'.

2. More Than Ogive

• Construction Steps:

1. Class intervals ko 'more than' type mein convert karo: Lower limits ko 'more than or equal to' values ke roop mein likho (e.g., 'more than or equal to 0', 'more than or equal to 10').
2. Cumulative frequencies calculate karo: Har 'more than' value ke corresponding cumulative frequency nikalna. Yeh total frequency (N) se start hokar decrease hoti jaati hai.
3. Points plot karo: X-axis par lower class limits lo aur Y-axis par corresponding cumulative frequencies lo. Points \((x_i, cf_i)\) plot karo.
4. Smooth curve ya line segments join karo: In points ko smooth curve ya line segments se join karo.

Ogive ka main use median nikalne mein hota hai.

Ogive ki help se grouped data ka median graphically find kiya ja sakta hai. Iske do main methods hain:

Method 1: Using one Ogive (Less Than or More Than)

1. Koi ek Ogive banao: Ya toh 'Less Than Ogive' ya 'More Than Ogive' graph paper par plot karo.
2. \(N/2\) mark karo: Y-axis par total frequency \(N\) ka half (\(N/2\)) mark karo.
3. Parallel line draw karo: Y-axis par \(N/2\) point se X-axis ke parallel ek horizontal line draw karo. Yeh line ogive ko ek point 'P' par cut karegi.
4. Perpendicular drop karo: Point 'P' se X-axis par ek perpendicular line drop karo. Jis point par yeh perpendicular X-axis ko cut karega, woh point median ki value hogi.

Method 2: Using both Ogives (Less Than and More Than)

1. Dono Ogives banao: 'Less Than Ogive' aur 'More Than Ogive' dono ko ek hi graph paper par plot karo.
2. Intersection Point dhoondo: Dono ogives ek dusre ko ek point 'P' par intersect karenge.
3. Perpendicular drop karo: Intersection point 'P' se X-axis par ek perpendicular line drop karo. Jis point par yeh perpendicular X-axis ko cut karega, woh point median ki value hogi.

Dono methods se same median value aati hai. Method 2 zyada accurate mana jaata hai kyunki yeh dono curves ke intersection par based hai.