📊 Chapter 5: Data Handling (Class 8) 📈
*(Chapter number correction: Assuming this corresponds to Chapter 5 on Data Handling in typical NCERT syllabus)*Hello Data Detectives! Humari daily life mein humein bahut saari information milti hai – cricket scores 🏏, temperature 🌡️, marks 💯, etc. Yeh information hi **Data** hai. Lekin sirf data hona kaafi nahi hai, use samajhne ke liye organize karna zaroori hai. Let’s learn about Data Handling! Hello Data Detectives! Hamari daily life mein humein bahut saari jaankari milti hai – cricket scores, तापमान, marks, etc. Yeh jaankari hi Data hai. Lekin sirf data hona kaafi nahi hai, use samajhne ke liye vyavasthit karna zaroori hai. Chalo Data Handling ke baare mein seekhein!
🔍 Looking for Information (Data Kya Hai?)
Data: A collection of facts or figures gathered to get some information. Usually collected in the context of a situation that we want to study.
Data: Kuch jaankari prapt karne ke liye ikattha kiye gaye tathyon ya aankdon ka sangrah. Aamtaur par kisi situation ke sandarbh mein ikattha kiya jaata hai jiska hum adhyayan karna chahte hain.Examples of Data:
- Marks obtained by students in a Math test.
- Number of different types of trees in a park 🌳.
- Runs scored by a batsman in last 10 matches 🏏.
- Temperature recorded in a city over a week 🌡️.
Data collected initially is called Raw Data. It needs to be organized to make sense.
Shuru mein ikattha kiya gaya data Raw Data kehlata hai. Ise samajhne yogya banane ke liye vyavasthit karne ki zaroorat hoti hai.📑 Organising Data (Data Ko Vyavasthit Karna)
Organizing data helps us understand it better and draw conclusions easily. One way is using a **Frequency Distribution Table**. Data ko vyavasthit karne se hum use behtar samajh paate hain aur aasani se nishkarsh nikaal sakte hain. Ek tareeka hai Frequency Distribution Table (Baarambaarta Vitaran Saarni) ka upyog karna.
- Frequency: The number of times a particular entry or observation occurs in the data. (Koi data value kitni baar aayi hai).
- Tally Marks (||||): A method to keep track of frequency while going through raw data. We mark a vertical line (|) for each occurrence, and the fifth mark is a diagonal slash (
||||) across the previous four. Makes counting easier. (Frequency count karne ka ek tareeka. Har baar ek (|), paanchveen baar ek cross (||||)). - Frequency Distribution Table: A table showing data values (or groups) and their corresponding frequencies (often with tally marks too). (Ek table jo data values aur unki frequency dikhati hai).
Example 1: Marks of 20 students in a test (out of 10) are: 7, 8, 5, 6, 8, 7, 9, 4, 5, 7, 8, 8, 6, 7, 9, 7, 5, 8, 7, 6. Organise this data using a frequency distribution table.
Solution:
| Marks | Tally Marks | Frequency (No. of Students) |
|---|---|---|
| 4 | | | 1 |
| 5 | ||| | 3 |
| 6 | ||| | 3 |
| 7 | 6 | |
| 8 | 5 | |
| 9 | || | 2 |
| Total | 20 |
📊 Grouping Data (Data Ka Samoohikaran)
When we have a large amount of data, listing individual values is difficult. So, we group the data into suitable intervals called **Classes** or **Class Intervals**. Jab hamare paas bahut sara data hota hai, toh har value ko likhna mushkil hota hai. Isliye, hum data ko uchit antralon mein samuhit karte hain jinhe Classes ya Class Intervals kehte hain.
- The distribution obtained is called a **Grouped Frequency Distribution**.
Key Terms for Grouped Data:
- Class Interval (Varg Antraal): A range into which data values are grouped (e.g., 0-10, 10-20, 20-30). (Ek range jisme data ko group kiya jaata hai).
- Lower Class Limit: The smallest data value that can go into the class interval. (In 10-20, 10 is the lower limit). (Class interval ki sabse chhoti value).
- Upper Class Limit: The largest data value that can go into the class interval. (In 10-20, 20 is the upper limit). (Class interval ki sabse badi value).
- Class Size (or Width): The difference between the upper and lower class limits. (For 10-20, width = 20 – 10 = 10). (Upper aur lower limit ke beech ka antar).
- Class Mark (or Mid Point): (Upper Limit + Lower Limit) / 2. Represents the midpoint of the class. (For 10-20, Class Mark = (10+20)/2 = 15).
Convention: The common convention is that the upper limit of a class is not included in that class (e.g., in 10-20, the value 20 would go into the next class 20-30).
Aam niyam yeh hai ki ek class ki upper limit us class mein shamil *nahi* hoti hai (jaise, 10-20 mein, value 20 agle class 20-30 mein jayegi).Example 2: Weights (in kg) of 25 students: 32, 45, 51, 48, 60, 35, 38, 41, 44, 53, 58, 62, 36, 39, 42, 46, 50, 55, 59, 61, 31, 37, 49, 54, 47. Group this into classes 30-40, 40-50, etc.
Solution: Grouped Frequency Distribution Table
| Class Interval (Weight in kg) | Tally Marks | Frequency (No. of Students) |
|---|---|---|
| 30-40 | 7 | |
| 40-50 | 8 | |
| 50-60 | 6 | |
| 60-70 | |||| | 4 |
| Total | 25 |
(Note: A student weighing exactly 40kg would go into the 40-50 class, not 30-40).
Histogram (Aayat Chitra):
- A graphical representation of a grouped frequency distribution with continuous classes. (Grouped frequency distribution ka graph jisme classes lagatar hon).
- It consists of rectangles (bars) whose bases are the class intervals along the X-axis, and heights represent the frequencies along the Y-axis. (Aayat (bars) hote hain jinki chaudai class interval hoti hai aur unchai frequency).
- There are **no gaps** between consecutive bars (unlike a bar graph) because the classes are continuous. 📊 Lagatar bars ke beech koi gap nahi hota hai (bar graph ke vipreet).
- If the first class interval doesn’t start from 0, a jagged line (or kink) ~ is shown on the X-axis to indicate a break.
Example 3: Draw a histogram for the data in Example 2 (Student Weights).
How to Draw:
- Draw X-axis and Y-axis.
- Mark Class Intervals (30-40, 40-50, …) on X-axis with uniform width. (Use a kink ~ near origin if starting from 30).
- Mark Frequencies (e.g., 0, 2, 4, 6, 8, 10…) on Y-axis with uniform scale.
- For each class interval, draw a rectangle (bar) whose base is the interval and height is the corresponding frequency.
- Ensure bars touch each other (no gaps).
- Label axes (X-axis: Weight in kg, Y-axis: Number of Students) and give a title.
पाई Circle Graph or Pie Chart (Vritt Aalekh ya Pie Chart)
Pie Chart: A way of representing data as sectors of a circle. The whole circle represents the total, and the size of each sector is proportional to the information (or part) it represents.
Pie Chart: Data ko ek vritt ke sectors (triyakhand) ke roop mein represent karne ka tareeka. Poora vritt total ko represent karta hai, aur har sector ka size uske dwara represent ki gayi jaankari ke anupaatik hota hai.- Shows the relationship between a whole and its parts.
- The sum of angles of all sectors at the center is 360°.
Steps to Draw a Pie Chart:
- Calculate the fraction (or percentage) that each category represents out of the total. Har category total ka kitna hissa (fraction ya percentage) hai, calculate karo.
- Calculate the central angle for each sector.
Central Angle = (Fraction of category) × 360°
ORCentral Angle = (Value of componentTotal Value) × 360°
Har sector ke liye kendriy kon (central angle) calculate karo.
- Draw a circle of a suitable radius.
- Draw one radius as the starting line.
- Use a protractor 📐 to draw sectors one by one according to the calculated central angles. Protractor ka istemal karke calculate kiye gaye central angles ke anusar ek-ek karke sectors banao.
- Label each sector clearly.
Example 1: On a particular day, the sales (in Rupees) of different items of a baker’s shop are given below. Draw a pie chart.
Ordinary Bread: 320, Fruit Bread: 80, Cakes & Pastries: 160, Biscuits: 120, Others: 40. Total = 720.
Solution:
| Item | Sales (₹) | Fraction | Central Angle (Fraction × 360°) |
|---|---|---|---|
| Ordinary Bread | 320 | 320720 = 49 | 49 × 360° = 160° |
| Fruit Bread | 80 | 80720 = 19 | 19 × 360° = 40° |
| Cakes & Pastries | 160 | 160720 = 29 | 29 × 360° = 80° |
| Biscuits | 120 | 120720 = 16 | 16 × 360° = 60° |
| Others | 40 | 40720 = 118 | 118 × 360° = 20° |
| Total | 720 | 1 | 360° |
Now, draw a circle and use a protractor to mark sectors with these angles (160°, 40°, 80°, 60°, 20°). Label each sector.
Example 2: If the total marks were 540, and a pie chart shows 90° for Maths, find the marks in Maths.
Solution:
Total Angle = 360°. Maths Angle = 90°.
Fraction for Maths = 90°360° = 14.
Marks in Maths = Fraction × Total Marks
= 14 × 540 = 135.
Answer: 135 marks.
Example 3: 36 people were asked about their favourite season. Results: Summer=9, Rainy=12, Winter=15. Find central angle for each.
Solution:
Total people = 36.
Summer Angle = (936) × 360° = 14 × 360° = 90°.
Rainy Angle = (1236) × 360° = 13 × 360° = 120°.
Winter Angle = (1536) × 360° = 512 × 360° = 5 × 30° = 150°.
Check: 90° + 120° + 150° = 360°. ✅
🎲 Chance and Probability (Sanyog Aur Prayikta)
Sometimes things happen by chance (like getting a ‘6’ when you roll a die 🎲). Probability measures the likelihood of such chance events.
Kabhi kabhi cheezein sanyog se hoti hain (jaise die roll karne par ‘6’ aana). Prayikta (Probability) aise sanyog wali ghatnaon ki sambhavna ko maapti hai.Experiment: An operation which can produce some well-defined outcomes.
Prayog (Experiment): Ek kriya jo kuch suparibhashit parinaam de sakti hai.Random Experiment: An experiment where all possible outcomes are known, but the exact outcome cannot be predicted in advance.
Yadrichhik Prayog (Random Experiment): Ek prayog jisme sabhi sambhav parinaam gyaat hon, lekin sahi parinaam ka pahle se anuman nahi lagaya ja sakta.Example: Tossing a coin 🪙, rolling a die.
Outcome: A possible result of an experiment.
Parinaam (Outcome): Ek prayog ka ek sambhav नतीजा.Example: Getting ‘Heads’ (H) or ‘Tails’ (T) when tossing a coin. Getting 1, 2, 3, 4, 5, or 6 when rolling a die.
Equally Likely Outcomes: Outcomes that have the same chance of occurring.
Sama Sambhaavi Parinaam: Ve parinaam jinke hone ki sambhavna barabar ho.Example: In a fair coin toss, H and T are equally likely. In a fair die roll, 1, 2, 3, 4, 5, 6 are equally likely.
Event: A specific outcome or a collection of outcomes from an experiment.
Ghatna (Event): Ek prayog ka ek vishisht parinaam ya parinaamon ka sangrah.Example: Getting an ‘even number’ (outcomes 2, 4, 6) when rolling a die is an event.
Probability of an Event (E):
P(E) = Number of outcomes favourable to ETotal number of possible outcomes
- Probability value always lies between 0 and 1 (inclusive). Probability ki value hamesha 0 aur 1 ke beech (ya unke barabar) hoti hai.
- P(Impossible Event) = 0. (Asambhav ghatna ki probability = 0).
- P(Sure Event) = 1. (Nishchit ghatna ki probability = 1).
Example 1: Find the probability of getting a ‘Head’ when a coin is tossed once.
Solution:
Total possible outcomes = {Head (H), Tail (T)} -> Total = 2.
Favourable outcome (Getting a Head) = {H} -> Number = 1.
Probability P(Head) = Favourable OutcomesTotal Outcomes = 12.
Answer: 1/2.
Example 2: A die is thrown once. Find the probability of getting (a) an even number (b) a number greater than 4.
Solution:
Total possible outcomes = {1, 2, 3, 4, 5, 6} -> Total = 6.
(a) Event E1: Getting an even number.
Favourable outcomes = {2, 4, 6} -> Number = 3.
P(E1) = 36 = 12.
(b) Event E2: Getting a number greater than 4.
Favourable outcomes = {5, 6} -> Number = 2.
P(E2) = 26 = 13.
Answer: (a) 1/2, (b) 1/3.
Example 3: A bag contains 3 red balls 🔴 and 2 blue balls 🔵. A ball is drawn randomly. Find probability of getting a blue ball.
Solution:
Total number of balls (Total possible outcomes) = 3 + 2 = 5.
Number of blue balls (Favourable outcomes) = 2.
Probability P(Blue Ball) = Number of Blue BallsTotal Number of Balls = 25.
Answer: 2/5.
❓ Sawal Jawab (Questions & Answers)
🤏 Very Short Answer Questions
1. What is raw data?
Data collected initially in its original form.
2. What does ‘frequency’ mean in data handling?
The number of times a particular observation occurs.
3. What do tally marks help us do?
Keep count of frequencies easily.
4. What is the range 20-30 called in grouped data?
Class interval.
5. What is the lower limit of the class interval 50-60?
50.
6. What is the upper limit of the class interval 10-20?
20.
7. What is the class size (width) of the interval 25-35?
35 – 25 = 10.
8. What graph uses bars of uniform width with no gaps for grouped continuous data?
Histogram.
9. What type of graph represents data as sectors of a circle?
Circle graph or Pie chart.
10. What is the sum of central angles in a pie chart?
360°.
11. What is an experiment?
An operation with well-defined outcomes.
12. What is an outcome?
A possible result of an experiment.
13. Give an example of a random experiment.
Tossing a coin, Rolling a die.
14. What is the probability of a sure event?
1.
15. What is the probability of an impossible event?
0.
16. What is the probability of getting ‘Tails’ when tossing a fair coin?
1/2.
17. What is the class mark of the interval 40-50?
(40+50)/2 = 45.
📝 Short Answer Questions
1. What is data? Why do we need to organize it?
- Data is a collection of facts/figures for information.
- Raw data is often difficult to understand or interpret directly.
- Organizing data (like in tables or graphs) helps to:
- Understand it easily.
- See patterns or trends.
- Draw meaningful conclusions quickly.
2. Make a frequency distribution table for: Red, Blue, Green, Red, Blue, Red, Yellow, Blue, Red, Green.
| Color | Tally Marks | Frequency |
|---|---|---|
| Red | |||| | 4 |
| Blue | ||| | 3 |
| Green | || | 2 |
| Yellow | | | 1 |
| Total | 10 |
3. What is a histogram? How is it different from a bar graph?
- Histogram: Graphical representation of grouped frequency distribution (continuous classes).
- Bars represent class intervals (base) and frequency (height).
- Difference from Bar Graph:
- Bars in histogram touch each other (no gaps, continuous data).
- Bars in bar graph have gaps between them (usually represent discrete categories).
- Base width of histogram bars represent class size; bar graph widths are usually uniform but arbitrary.
4. What is a pie chart used for? How is the central angle calculated?
- Used to show the relationship between a whole and its parts (comparing parts of a whole).
- Represents data as sectors of a circle पाई .
- Central Angle Calculation:
Central Angle = (Value of componentTotal Value) × 360°
5. List the outcomes when a single die is thrown.
- The possible outcomes are the numbers on the faces of the die.
- Outcomes: {1, 2, 3, 4, 5, 6}.
- Total number of outcomes = 6.
6. List the outcomes when two coins are tossed together.
- Let H=Head, T=Tail.
- Possible outcomes are combinations for Coin 1 and Coin 2.
- Outcomes: {HH, HT, TH, TT}.
- Total number of outcomes = 4.
7. A die is thrown. Find probability of getting: (a) a prime number (b) a number not greater than 5.
- Total outcomes = 6 ({1, 2, 3, 4, 5, 6}).
- (a) Prime numbers = {2, 3, 5}. Favourable = 3. P(Prime) = 3/6 = 1/2.
- (b) Numbers not greater than 5 = {1, 2, 3, 4, 5}. Favourable = 5. P(Not > 5) = 5/6.
8. From a standard deck of 52 playing cards, find probability of drawing a red king.
- Total outcomes = 52 cards.
- There are 2 red kings (King of Hearts ♥️, King of Diamonds ♦️).
- Favourable outcomes = 2.
- P(Red King) = 2 / 52 = 1 / 26.
9. What is the purpose of using Tally Marks?
- Tally marks provide a systematic way to count the frequency of each observation as you go through raw data.
- Marking a vertical line for each count and crossing the fifth count makes grouping in fives (
||||). - This reduces errors in counting and makes summing up frequencies easier, especially for large datasets.
10. In grouped data 10-20, 20-30 etc., where would you put the value 20?
- By convention, the upper limit is usually excluded from a class interval.
- Therefore, the value 20 would be included in the class interval **20-30**, not 10-20.
11. Find the class mark for the interval 15-25.
- Class Mark = (Upper Limit + Lower Limit) / 2
- Class Mark = (25 + 15) / 2 = 40 / 2 = 20.
📜 Long Answer Questions
1. The weekly wages (in Rs) of 30 workers in a factory are: 830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, 808, 812, 840, 885, 835, 835, 836, 878, 840, 868, 890, 806, 840. Using tally marks make a frequency table with intervals as 800–810, 810–820 and so on.
Frequency Distribution Table:
| Class Interval (Wages in Rs) | Tally Marks | Frequency (No. of Workers) |
|---|---|---|
| 800–810 | ||| | 3 |
| 810–820 | || | 2 |
| 820–830 | | | 1 |
| 830–840 | 9 | |
| 840–850 | 5 | |
| 850–860 | | | 1 |
| 860–870 | ||| | 3 |
| 870–880 | | | 1 |
| 880–890 | | | 1 |
| 890–900 | |||| | 4 |
| Total | 30 |
(Note: Workers earning exactly Rs 810 go into 810-820 class, Rs 820 into 820-830 etc.)
2. Draw a histogram for the frequency table made for the data in Q1.
Steps to draw histogram:
- Draw X-axis (Wages) and Y-axis (Number of Workers).
- Mark class intervals (800-810, 810-820… 890-900) on X-axis. Use a kink near origin as scale doesn’t start from 0.
- Choose a suitable scale for Y-axis (e.g., 1 unit = 1 worker). Mark frequencies (up to 9 or 10).
- Draw bars (rectangles) for each class interval with base = interval width, height = frequency. Bars must touch each other.
- Label axes and give a title “Weekly Wages of 30 Workers”.
3. The number of students in a hostel, speaking different languages is given below. Display the data in a pie chart:
Hindi: 40, English: 12, Marathi: 9, Tamil: 7, Bengali: 4. Total students = 72.
Calculate central angles:
| Language | Students | Fraction | Central Angle |
|---|---|---|---|
| Hindi | 40 | 40/72 = 5/9 | (5/9) × 360° = 200° |
| English | 12 | 12/72 = 1/6 | (1/6) × 360° = 60° |
| Marathi | 9 | 9/72 = 1/8 | (1/8) × 360° = 45° |
| Tamil | 7 | 7/72 | (7/72) × 360° = 7 × 5° = 35° |
| Bengali | 4 | 4/72 = 1/18 | (1/18) × 360° = 20° |
| Total | 72 | 1 | 360° |
Now, draw a circle and make sectors with angles 200°, 60°, 45°, 35°, 20° using a protractor. Label each sector.
4. A box contains numbers from 1 to 10 written on separate slips. One slip is chosen randomly. Find the probability of: (a) getting number 6 (b) getting a number less than 6 (c) getting a number greater than 6 (d) getting a 1-digit number.
Total possible outcomes = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Total = 10.
- (a) Getting number 6: Favourable = {6}. Number = 1. P(getting 6) = 1/10.
- (b) Getting number less than 6: Favourable = {1, 2, 3, 4, 5}. Number = 5. P(less than 6) = 5/10 = 1/2.
- (c) Getting number greater than 6: Favourable = {7, 8, 9, 10}. Number = 4. P(greater than 6) = 4/10 = 2/5.
- (d) Getting a 1-digit number: Favourable = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Number = 9. P(1-digit number) = 9/10.
5. What are the advantages of representing data graphically?
- Easy Understanding: Graphs make complex data easy to grasp quickly (‘a picture is worth a thousand words’).
- Visual Appeal: They are more engaging and interesting than tables of numbers.
- Comparison: Easy to compare different data sets or trends visually (e.g., heights of bars in a bar graph/histogram).
- Trend Identification: Patterns and trends in data can often be spotted easily in graphs (like line graphs).
- Quick Summary: Provides a quick overview of the data distribution or relationships (e.g., pie chart shows proportions).
- Effective Communication: Useful for presenting data effectively to others in reports or presentations.
6. Explain the terms: Raw Data, Frequency, Frequency Distribution Table.
- Raw Data: Data collected in its original, unorganized form. It’s just a collection of numbers or facts without any structure. Ex: List of marks obtained by students randomly written down. (Mool roop mein ikattha kiya gaya data, bina kisi vyavastha ke).
- Frequency: The number of times a particular observation or data value occurs in a dataset. Ex: If ‘7 marks’ appeared 6 times, the frequency of ‘7’ is 6. (Koi value kitni baar aayi hai).
- Frequency Distribution Table: A table used to organize data by showing each distinct observation (or group/class interval) along with its corresponding frequency (and often tally marks). It summarizes the raw data neatly. (Ek table jo har observation/group aur uski frequency dikhati hai, data ko vyavasthit karti hai).
7. When is it more appropriate to use a Histogram compared to a Bar Graph?
- Histogram: Used to represent **grouped frequency distributions** where the data is **continuous** (or treated as continuous). The class intervals are usually adjacent, and there are **no gaps** between the bars. It shows the distribution of data over a range. Ex: Height of students grouped into intervals (140-145cm, 145-150cm etc.).
- Bar Graph: Used to represent **discrete data** or compare different **categories**. There are **gaps** between the bars, as each bar represents a distinct category. Ex: Favourite sports of students (Cricket, Football, Hockey etc.), marks in different subjects.
So, use Histogram for grouped, continuous data, and Bar Graph for discrete categories.
8. A survey was made to find the type of music certain young people liked. The adjoining pie chart shows the findings. Answer the following:
(i) If 20 people liked classical music, how many young people were surveyed?
(ii) Which type of music is liked by the maximum number?
(iii) If a company were to make 1000 CDs, how many of each type would they make?
*(Assume Pie Chart shows: Classical 10%, Semi Classical 20%, Light 40%, Folk 30%)*
- (i) Finding Total People:
- Classical music is 10% of total.
- Let Total people = T.
- 10% of T = 20 => (10/100) × T = 20.
- T = 20 × (100/10) = 20 × 10 = 200.
- Total young people surveyed = 200.
- (ii) Maximum Liked Music:
- The largest sector corresponds to Light music (40%).
- Maximum number liked Light music.
- (iii) CDs for 1000 Total:
- Classical = 10% of 1000 = (10/100) × 1000 = 100 CDs.
- Semi Classical = 20% of 1000 = (20/100) × 1000 = 200 CDs.
- Light = 40% of 1000 = (40/100) × 1000 = 400 CDs.
- Folk = 30% of 1000 = (30/100) × 1000 = 300 CDs.
9. What is probability? Explain with an example how to find the probability of an event.
Probability: It is a measure of the likelihood or chance that a particular event will occur during a random experiment. It is expressed as a number between 0 and 1.
Prayikta: Yeh kisi random prayog ke dauran kisi vishesh ghatna ke hone ki sambhavna ka maap hai. Ise 0 aur 1 ke beech ke number mein vyakt kiya jaata hai.How to Find Probability:
- Identify all possible outcomes of the experiment (Total Outcomes).
- Identify the specific outcomes that correspond to the event you are interested in (Favourable Outcomes).
- Calculate the probability using the formula:
P(Event) = (Number of Favourable OutcomesTotal Number of Possible Outcomes)
Example: Probability of getting an odd number when rolling a fair die 🎲.
- Total Possible Outcomes = {1, 2, 3, 4, 5, 6} -> Total = 6.
- Event = Getting an odd number.
- Favourable Outcomes = {1, 3, 5} -> Number = 3.
- Probability P(Odd) = 3 / 6 = 1 / 2.
10. Consider the data for weights of students in Example 2. Find the answers: (a) How many students weigh 50 kg or more? (b) Which class interval has the highest frequency? (c) What is the class size? (d) What is the lower limit of the class 50-60?
Data (from Example 2 Table): 30-40: 7 | 40-50: 8 | 50-60: 6 | 60-70: 4
- (a) Students weighing 50 kg or more: We need to sum frequencies of classes 50-60 and 60-70. Number = 6 + 4 = 10 students. (Intervals 50-60 aur 60-70 ki frequency jodo).
- (b) Highest Frequency Interval: The frequency is highest (8) for the class interval 40-50. (Sabse zyada frequency (8) 40-50 interval ki hai).
- (c) Class Size: The difference between upper and lower limits for any class. Eg: 40 – 30 = 10. Class size is 10 kg. (Kisi bhi interval ke upper aur lower limit ka antar).
- (d) Lower Limit of 50-60: The lower limit for the class 50-60 is 50.
