RBSE Solutions Class 9 Maths Chapter 15 Statistics Exercise 15.4

 

Question 1. The following number of goals were scored by a football team in a series of 10 matches: 2, 3, 4, 5, 0, 1, 3, 3, 4, 3
Find the mean, median and mode of these scores.
Answer:
The number of goals scored by a football team in 10 matches are: 2, 3, 4, 5, 0, 1, 3, 3, 4, 3
(i) Mean
The mean is found by adding all the scores and dividing by the number of matches.
\( \overline{x} = \frac { \Sigma x }{ n } \)
\( = \frac { 2+3+4+5+0+1+3+3+4+3 }{ 10 } \)
\( = \frac { 28 }{ 10 } \)
\( = 2.8 \)
(ii) Median
First, arrange the data in ascending order: 0, 1, 2, 3, 3, 3, 3, 4, 4, 5.
There are \( n = 10 \) observations, which is an even number. For even numbers, the median is the average of the \( \frac { n }{ 2 } \)th and \( ( \frac { n }{ 2 } + 1 ) \)th terms.
\( \frac { n }{ 2 } \)th term \( = \frac { 10 }{ 2 } = 5 \)th term
\( ( \frac { n }{ 2 } + 1 ) \)th term \( = ( 5 + 1 ) = 6 \)th term
From the sorted data, the 5th term is 3 and the 6th term is 3.
Median \( = \frac { \text{5th term} + \text{6th term} }{ 2 } \)
\( = \frac { 3 + 3 }{ 2 } \)
\( = \frac { 6 }{ 2 } \)
\( = 3 \)
(iii) Mode
The mode is the number that appears most often in the data. In this dataset, the number 3 appears 4 times, which is more than any other number. Finding the mode helps understand the most common outcome.
Hence, Mode = 3
In simple words: The average number of goals is 2.8. The middle number of goals when sorted is 3. The number of goals that happened most often is 3.

🎯 Exam Tip: To find the median, always remember to sort the data first. For mode, count how many times each number appears.

 

Question 2. In a Mathematics test given by 15 students, the following marks (out of 100) are recorded: 41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60
Find the mean, median and mode of these data.
Answer:
The marks obtained by 15 students are: 41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60.
(i) Mean
The mean is calculated by summing all marks and dividing by the total number of students.
\( \Sigma x = 41+39+48+52+46+62+54+40+96+52+98+40+42+52+60 \)
\( = 862 \)
\( n = 15 \)
\( \overline{x} = \frac { \Sigma x }{ n } \)
\( = \frac { 862 }{ 15 } \)
\( = 57.466 \approx 57.47 \)
(ii) Median
First, arrange the marks in ascending order:
39, 40, 40, 41, 42, 46, 48, 52, 52, 52, 54, 60, 62, 96, 98.
There are \( n = 15 \) observations, which is an odd number. For an odd number of observations, the median is the middle term, which is the \( \frac { n+1 }{ 2 } \)th term.
Median \( = \frac { 15+1 }{ 2 } \)th term
\( = \frac { 16 }{ 2 } \)th term
\( = 8 \)th term
From the sorted data, the 8th term is 52. The median is the value that separates the higher half from the lower half of the data.
Hence, Median = 52.
(iii) Mode
The mode is the mark that appears most frequently. In the given data, 52 appears 3 times (4th, 5th, and 6th from the end), which is more than any other mark.
Hence, Mode = 52.
In simple words: The average mark is about 57.47. The middle mark when sorted is 52. The mark that students got most often is 52.

🎯 Exam Tip: When \(n\) is odd, the median is a single value in the data set. When \(n\) is even, it's the average of the two middle values.

 

Question 3. The following observations have been arranged in ascending order. If the median of the data is 63. Find the value of x.
29, 32, 48, 50, x, x + 2, 72, 78, 84, 95.
Answer:
The given observations are: 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95.
The number of observations is \( n = 10 \), which is an even number. The data is already arranged in ascending order.
For an even number of observations, the median is the average of the \( \frac { n }{ 2 } \)th and \( ( \frac { n }{ 2 } + 1 ) \)th terms.
\( \frac { n }{ 2 } \)th term \( = \frac { 10 }{ 2 } = 5 \)th term
\( ( \frac { n }{ 2 } + 1 ) \)th term \( = ( 5 + 1 ) = 6 \)th term
From the given data, the 5th term is \( x \) and the 6th term is \( x + 2 \).
We are given that the median is 63.
Median \( = \frac { \text{5th term} + \text{6th term} }{ 2 } \)
\( 63 = \frac { x + (x + 2) }{ 2 } \)
Now, multiply both sides by 2 to solve for x.
\( 63 \times 2 = x + x + 2 \)
\( 126 = 2x + 2 \)
Subtract 2 from both sides.
\( 126 - 2 = 2x \)
\( 124 = 2x \)
Divide by 2.
\( x = \frac { 124 }{ 2 } \)
\( x = 62 \)
In simple words: We know the middle value (median) and how the two middle numbers relate to 'x'. By using the median formula for an even set of numbers, we can calculate that 'x' is 62.

🎯 Exam Tip: When observations are already sorted and the median is given, set up the median formula carefully based on whether 'n' is even or odd, then solve the equation for the unknown variable.

 

Question 4. Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.
Answer:
The given observations are: 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.
To find the mode, we count how many times each observation appears:
14 appears 4 times.
25 appears 1 time.
28 appears 1 time.
18 appears 3 times.
17 appears 1 time.
23 appears 1 time.<
22 appears 1 time.
The observation that occurs most frequently is 14, which appears 4 times. This is the value with the highest frequency in the dataset.
Hence, Mode = 14.
In simple words: The number that shows up most often in the list is 14. So, 14 is the mode.

🎯 Exam Tip: To find the mode, it's helpful to first arrange the data or make a frequency table to easily spot the most frequent value.

 

Question 5. Find the mean salary of 60 workers of a factory from the following table:
Salary (in Rs) 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000
No. of Workers 16, 12, 10, 8, 6, 4, 3, 1
Answer:
To find the mean salary, we will use the formula for mean of grouped data: \( \overline{x} = \frac { \Sigma f_i x_i }{ \Sigma f_i } \). First, we need to calculate \( f_i x_i \) for each salary group.

Now, we apply the mean formula:
\( \overline{x} = \frac { \Sigma f_i x_i }{ \Sigma f_i } \)
\( = \frac { 305000 }{ 60 } \)
\( = 5083.33 \)
Therefore, the mean salary of 60 workers in the factory is Rs 5083.33. This calculation shows the average earnings per worker.
In simple words: To find the average salary, we multiply each salary by how many workers get it, add all those totals up, and then divide by the total number of workers. The average salary turns out to be Rs 5083.33.

🎯 Exam Tip: For problems involving frequency distributions, remember to calculate \( f_i x_i \) for each row and then sum these products before dividing by the total frequency.