Statistics

Q1: Arrange the following data in ascending order:

i. 8, 3, 11, 15, 1, 7, 16, 25, 9, 4

Step 1: Write all the given numbers clearly.
8, 3, 11, 15, 1, 7, 16, 25, 9, 4
Step 2: To arrange in ascending order, start from the smallest and go to the largest number.
Ascending order means — smallest to greatest (1 → 2 → 3 → …).
Step 3: Compare all numbers and arrange them step-by-step.
Smallest number is 1.
Next greater numbers are 3, 4, 7, 8, 9, 11, 15, 16, 25.
Answer: 1, 3, 4, 7, 8, 9, 11, 15, 16, 25

ii. 5.7, 9.2, 3.1, 4.6, 1.8, 10.4, 2.5, 0.9, 1.2

Step 1: Write the given decimal numbers clearly.
5.7, 9.2, 3.1, 4.6, 1.8, 10.4, 2.5, 0.9, 1.2
Step 2: Arrange in ascending order — start with the smallest decimal and move to the largest.
Step 3: Compare each decimal value carefully.
Smallest = 0.9
Then 1.2 < 1.8 < 2.5 < 3.1 < 4.6 < 5.7 < 9.2 < 10.4
Answer: 0.9, 1.2, 1.8, 2.5, 3.1, 4.6, 5.7, 9.2, 10.4

Q2: Arrange the following data in descending order:

i. 15, 3, 0, 11, 7, 2, 23, 14, 29, 5, 1, 17

Step 1: Write all the given numbers clearly.
15, 3, 0, 11, 7, 2, 23, 14, 29, 5, 1, 17
Step 2: To arrange in descending order, start from the greatest and go to the smallest number.
Descending order means — greatest to smallest (29 → 0).
Step 3: Compare all numbers and arrange them step-by-step.
Greatest number is 29.
Then next smaller numbers are 23, 17, 15, 14, 11, 7, 5, 3, 2, 1, 0.
Answer: 29, 23, 17, 15, 14, 11, 7, 5, 3, 2, 1, 0

ii. 2.3, 4.6, 1.2, 0.4, 3.5, 9.7, 6.1, 4.8, 0.8, 1.0

Step 1: Write the given decimal numbers clearly.
2.3, 4.6, 1.2, 0.4, 3.5, 9.7, 6.1, 4.8, 0.8, 1.0
Step 2: To arrange in descending order, start from the largest and move toward the smallest.
Descending order means — greatest to least (9.7 → 0.4).
Step 3: Compare each decimal value carefully.
Greatest = 9.7
Then 6.1 > 4.8 > 4.6 > 3.5 > 2.3 > 1.2 > 1.0 > 0.8 > 0.4
Answer: 9.7, 6.1, 4.8, 4.6, 3.5, 2.3, 1.2, 1.0, 0.8, 0.4

Q3: Construct a frequency table for each of the following data:

i. 3, 2, 5, 4, 1, 3, 2, 2, 5, 3, 1, 2, 1, 1, 2, 2, 3, 4, 5, 3, 1, 2, 3

Step 1: Write down all distinct observations (without repetition).
Distinct values are: 1, 2, 3, 4, 5
Step 2: Count how many times each value occurs in the data.

Answer: Frequency distribution shows that 2 and 3 occur most frequently (6 times each).

ii. 7.8, 6.5, 6, 7, 7, 9, 8, 10, 7, 6, 7, 8, 8, 9, 10, 5, 7, 8, 7, 6

Step 1: List all distinct values.
Distinct values: 5, 6, 6.5, 7, 7.8, 8, 9, 10
Step 2: Count the frequency for each value.

Answer: The most frequent value (mode) is 7, occurring 6 times.

iii. 152, 165, 172, 144, 135, 156, 175, 140, 132, 150, 153, 147

Step 1: Arrange data in ascending order.
132, 135, 140, 144, 147, 150, 152, 153, 156, 165, 172, 175
Step 2: Since no number repeats, each has frequency 1.

Answer: The required frequency distribution table for the given data is shown above.

iv. 13, 25, 19, 16, 8, 30, 27, 6, 0, 34, 40, 11, 4, 17

Step 1: Arrange data in ascending order.
0, 4, 6, 8, 11, 13, 16, 17, 19, 25, 27, 30, 34, 40
Step 2: Each number appears only once, so frequency = 1.

Answer: The required frequency distribution table for the given data is shown above.

Q4: Fill in the blanks:

i. Lower limit of the class interval 14–22 is _________.

Step 1: The first number in a class interval represents the lower limit.
So, in the class interval 14–22,
Lower limit = 14
Answer: 14

ii. Upper limit of the class interval 16–25 is _________.

Step 1: The second number in a class interval represents the upper limit.
So, in the class interval 16–25,
Upper limit = 25
Answer: 25

iii. Class-size of the class interval 20–30 is _________.

Step 1: Formula for class-size:
Class-size = Upper limit – Lower limit
Step 2: Substitute the values:
= 30 – 20
= 10
Answer: 10

iv. If the upper and lower limits of a class interval be 18 and 11 respectively, then the class-interval is _________.

Step 1: Class interval is written as:
Lower limit – Upper limit
Step 2: Substituting given values:
= 11 – 18
Answer: 11–18

v. If the lower and upper limits of a class interval be 11.5 and 21.5 respectively, then the class-interval is __________.

Step 1: Class interval = Lower limit – Upper limit
Step 2: Substitute given values:
= 11.5 – 21.5
Answer: 11.5–21.5

Q5: The marks out of 50, obtained by 30 students of a class in an examination are given below:
40, 12, 46, 37, 17, 27, 30, 6, 2, 23, 19, 39, 25, 5, 33, 19, 21, 12, 41, 17, 12, 19, 17, 8, 10, 1, 9, 21, 13, 48
Arrange them in ascending order and present it as a grouped data, by taking class-intervals 0-10, 10-20, 20-30, 30-40, 40-50.

Step 1: Arrange the data in ascending order

Procedure: Sort the list from smallest to largest.
Ascending order (sorted values): 1, 2, 5, 6, 8, 9, 10, 12, 12, 12, 13, 17, 17, 17, 19, 19, 19, 21, 21, 23, 25, 27, 30, 33, 37, 39, 40, 41, 46, 48
Check: Total observations = 30 (matches given).

Step 2: Grouped frequency table (class-intervals as given)

Convention used: Each class interval includes both end-points (e.g. 0–10 contains 10).
Count each class: 0–10 : 1, 2, 5, 6, 8, 9  →  6 observations.
10–20: 10, 12, 12, 12, 13, 17, 17, 17, 19, 19, 19  →  11 observations.
20–30: 21, 21, 23, 25, 27  →  5 observations.
30–40: 30, 33, 37, 39  →  4 observations.
40–50: 40, 41, 46, 48  →  4 observations.
Sanity check: 6 + 11 + 5 + 4 + 4 = 30 (all students accounted for).

Step 3: Tally & Frequency table (grouped)

Answer: The grouped frequency table with tally marks is shown above.

Q6: The fine imposed (in rupees) on 40 pupils for being absent is given below:
18, 8, 12, 6, 8, 16, 12, 5, 23, 2, 16, 23, 2, 10, 20, 12, 9, 7, 0, 5, 3, 6, 27, 21, 19, 15, 20, 24, 2, 16, 14, 8, 23, 24, 10, 14, 3, 8, 11, 2
Present the data in the form of frequency distribution table by taking class intervals 0-5, 5-10, 10-15, 15-20, 20-25.

Step 1: Arrange the data in ascending order
0, 2, 2, 2, 2, 3, 3, 5, 5, 6, 6, 7, 8, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 14, 14, 15, 16, 16, 16, 18, 19, 20, 20, 21, 23, 23, 23, 24, 24, 27
Step 2: Form class intervals and count frequency
Class intervals: 0–5, 5–10, 10–15, 15–20, 20–25
Now count how many observations fall into each class:
0–5 : 0, 2, 2, 2, 2, 3, 3 → 7 observations
5–10 : 5, 5, 6, 6, 7, 8, 8, 8, 8, 9 → 10 observations
10–15 : 10, 10, 11, 12, 12, 12, 14, 14 → 8 observations
15–20 : 15, 16, 16, 16, 18, 19 → 6 observations
20–25 : 20, 20, 21, 23, 23, 23, 24, 24 → 8 observations
25-30 : 27 → 1 observations
Check: 8 + 9 + 7 + 6 + 8 = 38 (2 pupils have fines beyond 25, i.e., 27 → fits outside the last class)
So, we add another class interval: 25–30, which includes 27.
Now total = 8 + 9 + 7 + 6 + 8 + 2 = 40 ✅
Step 3: Prepare the frequency distribution table (with tally marks)

Verification: Total frequency = 7 + 10 + 8 + 6 + 8 + 1 = 40 ✅
Answer: The frequency distribution table with class intervals 0–5, 5–10, 10–15, 15–20, 20–25, and 25–30 is shown above. Total pupils = 40.

Q7: The weights (in kg) of 30 students are given below:
48, 55, 30, 43, 41, 36, 48, 37, 34, 38, 46, 42, 38, 52, 37, 34, 40, 42, 50, 37, 35, 46, 42, 45, 43, 37, 39, 46, 48, 46
Form the grouped frequency table by taking class-intervals 30-35, 35-40, 40-45, 45-50, 50-55.

Step 1: Arrange the data in ascending order
30, 34, 34, 35, 36, 37, 37, 37, 37, 38, 38, 39, 40, 41, 42, 42, 42, 43, 43, 45, 46, 46, 46, 46, 48, 48, 48, 50, 52, 55
Step 2: Form class intervals and count frequency
Class intervals: 30–35, 35–40, 40–45, 45–50, 50–55
Now count the number of observations in each class:
30–35 : 30, 34, 34 → 3
35–40 : 35, 36, 37, 37, 37, 37, 38, 38, 39 → 9
40–45 : 40, 41, 42, 42, 42, 43, 43 → 7
45–50 : 45, 46, 46, 46, 46, 48, 48, 48 → 8
50–55 : 50, 52, 55 → 3
Verification: 3 + 9 + 7 + 8 + 3 = 30 ✅
Step 3: Prepare the grouped frequency distribution table (with tally marks)

Check: Total frequency = 30 ✅
Answer: The grouped frequency distribution table for the weights of 30 students with class intervals 30–35, 35–40, 40–45, 45–50, and 50–55 is shown above.

Q8: The daily income (in rupees) of 30 workers is given below:
83, 96, 91, 80, 97, 111, 90, 83, 112, 120, 103, 77, 82, 90, 70, 68, 89, 85, 78, 105, 94, 97, 90, 89, 102, 113, 90, 106, 70, 84
Construct a frequency table using groups 65-75, 75-85, 85-95, 95-105, 105-115, 115-125.

Step 1: Arrange the data in ascending order
68, 70, 70, 77, 78, 80, 82, 83, 83, 84, 85, 89, 89, 90, 90, 90, 90, 91, 94, 96, 97, 97, 102, 103, 105, 106, 111, 112, 113, 120
Step 2: Form class intervals and count frequency
Class intervals: 65–75, 75–85, 85–95, 95–105, 105–115, 115–125
Now count the number of observations in each class:
65–75 : 68, 70, 70 → 3
75–85 : 77, 78, 80, 82, 83, 83, 84 → 7
85–95 : 85, 89, 89, 90, 90, 90, 90, 91, 94 → 9
95–105 : 96, 97, 97, 102, 103 → 5
105–115 : 105, 106, 111, 112, 113 → 5
115–125 : 120 → 1
Verification: 3 + 7 + 9 + 5 + 5 + 1 = 30 ✅
Step 3: Prepare the grouped frequency distribution table (with tally marks)

Check: Total frequency = 30 ✅
Answer: The grouped frequency distribution table for the daily income of 30 workers with class intervals 65–75, 75–85, 85–95, 95–105, 105–115, and 115–125 is shown above.

Q9: The heights of 45 students measured to the nearest centimetre are given below:
155, 166, 159, 161, 158, 154, 153, 159, 157, 152, 160, 157, 158.170, 152, 151, 162, 164, 171, 154, 165, 161, 150, 168, 162, 154, 160, 170, 163, 164, 165, 169, 165, 155, 157, 158, 160, 161, 156, 153, 150, 165, 167, 169, 172
Prepare a frequency table taking class intervals 150-155, 155-160, 160-165, 165-170, 170-175.

Step 1: Arrange the data in ascending order
150, 150, 151, 152, 152, 153, 153, 154, 154, 154, 155, 155, 156, 157, 157, 157, 158, 158, 158, 159, 159, 160, 160, 160, 161, 161, 161, 162, 162, 163, 164, 164, 165, 165, 165, 165, 166, 167, 168, 169, 169, 170, 170, 171, 172
Step 2: Form class intervals and count frequencies
Class intervals: 150–155, 155–160, 160–165, 165–170, 170–175
Now count the number of observations in each class:
150–155 : 150, 150, 151, 152, 152, 153, 153, 154, 154, 154 → 10
155–160 : 155, 155, 156, 157, 157, 157, 158, 158, 158, 159, 159 → 11
160–165 : 160, 160, 160, 161, 161, 161, 162, 162, 163, 164, 164 → 11
165–170 : 165, 165, 165, 165, 166, 167, 168, 169, 169 → 9
170–175 : 170, 170, 171, 172 → 4
Verification: 10 + 11 + 11 + 9 + 4 = 45 ✅
Step 3: Prepare the grouped frequency distribution table (with tally marks)

Check: Total frequency = 45 ✅
Answer: The grouped frequency distribution table for the heights of 45 students with class intervals 150–155, 155–160, 160–165, 165–170, and 170–175 is shown above.

Q10: Complete the table given below:

1. Find the Class Size

Formula: Class Size = Upper Limit – Lower Limit
For all classes:
= 30 – 25 = 5
= 35 – 30 = 5
= 40 – 35 = 5
= 45 – 40 = 5
Hence, Class Size = 5 for each interval.

2. Find the Class Mark (Midpoint)

Formula: Class Mark = (Lower Limit + Upper Limit) ÷ 2
For each class:
25–30 → (25 + 30) ÷ 2 = 27.5
30–35 → (30 + 35) ÷ 2 = 32.5
35–40 → (35 + 40) ÷ 2 = 37.5
40–45 → (40 + 45) ÷ 2 = 42.5
Thus, Class Marks = 27.5, 32.5, 37.5, 42.5

3. Find the Class Boundaries

Formula:For each class:
25–30 → 25 – 30
30–35 → 30 – 35
35–40 → 35 – 40
40–45 → 40 – 45
Thus, boundaries are obtained by subtracting 0.5 from lower limit and adding 0.5 to upper limit.

4. Complete the table

Answer: The completed table above shows that each class has a size of 5, class marks as 27.5, 32.5, 37.5, 42.5, and boundaries as (25–30), (30–35), (35–40), (40–45).

Q11: Complete the table given below:

1. Find the Class Size

Formula: Class Size = Upper Limit – Lower Limit
For all classes:
= 30 – 26 = 4
= 35 – 31 = 4
= 40 – 36 = 4
= 45 – 41 = 4
Hence, Class Size = 4 for each interval.

2. Find the Class Mark (Midpoint)

Formula: Class Mark = (Lower Limit + Upper Limit) ÷ 2
For each class:
26–30 → (26 + 30) ÷ 2 = 28
31–35 → (31 + 35) ÷ 2 = 33
36–40 → (36 + 40) ÷ 2 = 38
41–45 → (41 + 45) ÷ 2 = 43
Thus, Class Marks = 28, 33, 38, 43

3. Find the Class Boundaries

Formula: Class Boundaries = Class Limit ± 0.5
For each class:
26–30 → 25.5 – 30.5
31–35 → 30.5 – 35.5
36–40 → 35.5 – 40.5
41–45 → 40.5 – 45.5
Thus, subtract 0.5 from lower limit and add 0.5 to upper limit to get class boundaries.

4. Complete the table

Answer: The completed table shows class size = 4, class marks = 28, 33, 38, 43, and class boundaries as (25.5–30.5), (30.5–35.5), (35.5–40.5), (40.5–45.5).