Chapter 9: Statistics (Exercise 9.1)

Chapter 9: Statistics (Exercise 9.1)

Question 1:
Find the mean of the first 5 natural numbers.

Solution:
* The first 5 natural numbers are: 1, 2, 3, 4, 5.
* Sum of observations = 1 + 2 + 3 + 4 + 5 = 15
* Total number of observations = 5

Formula: Mean = (Sum of observations) / (Number of observations)
Mean = 15 / 5 = 3

Question 2:
Find the mean of the first 10 prime numbers.

Solution:
* The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
* Sum of observations = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129
* Total number of observations = 10

Mean = 129 / 10 = 12.9

Question 3:
If the mean of 6, 8, 9, x, 13 is 10, find the value of x.

Solution:
* Given observations: 6, 8, 9, x, 13 (Total = 5 observations)
* Given Mean = 10

Mean = (Sum of observations) / 5
10 = (6 + 8 + 9 + x + 13) / 5
10 = (36 + x) / 5
50 = 36 + x
x = 50 - 36 = 14

Question 4:
Find the mode of the following data: 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.

Solution:
* Frequency of observations:
- 14 occurs 4 times
- 18 occurs 3 times
- Other numbers occur 1 time each.

* Since 14 has the maximum frequency (occurs most frequently):
Mode = 14

Question 5:
Find the median of the data: 33, 86, 68, 32, 80, 48, 70.

Solution:
* Step 1: Arrange the data in ascending order:
32, 33, 48, 68, 70, 80, 86

* Step 2: Count the total number of observations (n):
n = 7 (which is an odd number)

* Formula for median (when n is odd):
Median = ½ (n + 1)^th observation
Median = ½ (7 + 1) = 4^th observation

* The 4^th observation in the ordered list is 68.
Median = 68

Chapter 9: Statistics (Exercise 9.1 - Remaining Questions)

Question 6:
The heights of 10 students (in cm) are given below:
150, 148, 152, 145, 150, 155, 150, 146, 149, 150
Find the mean, median, and mode of the heights.

Solution:
1. Calculating Mean:
Sum of heights = 150 + 148 + 152 + 145 + 150 + 155 + 150 + 146 + 149 + 150 = 1495 cm
Total students = 10
Mean = 1495 / 10 = 149.5 cm

2. Calculating Median:
Arranging data in ascending order:
145, 146, 148, 149, 150, 150, 150, 150, 152, 155
Since n = 10 (even), the median is the average of the 5^th and 6^th observations:
Median = (150 + 150) / 2 = 150 cm

3. Calculating Mode:
The height 150 cm occurs most frequently (4 times).
Mode = 150 cm

Question 7:
The weights (in kg) of 8 members of a club are: 60, 65, 70, 62, 75, 71, 64, 69.
If one person weighing 73 kg joins the group, find the new mean weight.

Solution:
* Sum of weights of original 8 members = 60 + 65 + 70 + 62 + 75 + 71 + 64 + 69 = 546 kg
* Weight of new member = 73 kg
* New total weight = 546 + 73 = 619 kg
* Total number of members = 8 + 1 = 9

New Mean = 619 / 9 = 68.78 kg

Question 8:
On the occasion of New Year's Day, a sweet stall prepared sweet packets. The price of each packet and the number of packets are given as follows:
* Price of packet (₹): 25, 50, 75, 100, 125, 150
* No. of packets: 20, 36, 32, 29, 22, 11
Find the mean, median, and mode of the data.

Solution:
Total packets N = 20 + 36 + 32 + 29 + 22 + 11 = 150

1. Calculating Mean:
Σ(f × x) = (25×20) + (50×36) + (75×32) + (100×29) + (125×22) + (150×11)
Σ(f × x) = 500 + 1800 + 2400 + 2900 + 2750 + 1650 = 12000
Mean = 12000 / 150 = ₹80

2. Calculating Median:
Cumulative Frequencies (CF):
* ₹25 → 20
* ₹50 → 56
* ₹75 → 88
* ₹100 → 117
* ₹125 → 139
* ₹150 → 150
N/2 = 150 / 2 = 75.
The cumulative frequency just greater than 75 is 88, which corresponds to ₹75.
Median = ₹75

3. Calculating Mode:
The highest frequency is 36, corresponding to the price ₹50.
Mode = ₹50

Question 9:
The mean (average) weight of three students is 40 kg. One of the students Ranga weighs 46 kg. The other two students, Rahim and Reshma, have the same weight. Find Rahim's weight.

Solution:
* Total weight of 3 students = Mean × 3 = 40 × 3 = 120 kg
* Weight of Ranga = 46 kg
* Let weight of Rahim = x kg and Reshma = x kg

Equation:
46 + x + x = 120
46 + 2x = 120
2x = 120 - 46
2x = 74
x = 37 kg

Therefore, Rahim's weight is 37 kg.

Question 10:
The mean of 10 observations was calculated as 15. It was later found that one observation 12 was misread as 21. Find the correct mean.

Solution:
* Incorrect Sum = Incorrect Mean × Number of observations = 15 × 10 = 150
* Correct Sum = Incorrect Sum - Misread Value + Correct Value
Correct Sum = 150 - 21 + 12 = 141

Correct Mean = Correct Sum / 10 = 141 / 10 = 14.1