Table of Contents
Detailed solutions for all exercises from Chapter 8 – “Statsistics” in Class 9 Mathematics. This chapter focuses on the concept of statistics, including the representation of data, mean, median, and mode, and how to interpret various types of data.
Introduction Statistics
Statistics is the branch of mathematics that deals with the collection, analysis, interpretation, and presentation of masses of numerical data. It is important for understanding trends and patterns in various real-life scenarios. In this blog post, we will dive into all the exercises from Chapter 8, providing step-by-step solutions to ensure a complete understanding of the topics involved.
Exercise 8.1: Representation of Data
Q1. The marks obtained by 20 students in a mathematics test are given below:
Marks: 10, 20, 20, 30, 30, 30, 40, 40, 50, 50, 50, 50, 60, 60, 60, 60, 70, 70, 80, 80
- Construct a frequency distribution table for the given data.
Solution:
A frequency distribution table organizes the data into classes and provides the frequency of each class (i.e., how many times each mark occurs).
Step 1: Identify the classes
- Here, we have marks ranging from 10 to 80. We will group the marks into intervals (classes). For simplicity, we can use intervals of 10 marks.
Step 2: Create the table
| Marks | Frequency (f) |
|---|---|
| 10-19 | 1 |
| 20-29 | 3 |
| 30-39 | 4 |
| 40-49 | 4 |
| 50-59 | 4 |
| 60-69 | 4 |
| 70-79 | 2 |
| 80-89 | 2 |
Answer: This is the frequency distribution table for the given marks.
Exercise 8.2: Mean
Q1. Find the mean of the following data:
Marks: 10, 20, 30, 40, 50
Number of students: 5, 7, 6, 8, 4
Solution:
To find the mean of a data set, we use the following formula:Mean=∑f×x∑f\text{Mean} = \frac{\sum f \times x}{\sum f}Mean=∑f∑f×x
Where:
- fff is the frequency of each value.
- xxx is the value (data points).
- ∑\sum∑ represents the sum of the terms.
Step 1: Multiply each data point by its frequency
| Marks (x) | Frequency (f) | f×xf \times xf×x |
|---|---|---|
| 10 | 5 | 50 |
| 20 | 7 | 140 |
| 30 | 6 | 180 |
| 40 | 8 | 320 |
| 50 | 4 | 200 |
Step 2: Sum up the values∑f×x=50+140+180+320+200=890\sum f \times x = 50 + 140 + 180 + 320 + 200 = 890∑f×x=50+140+180+320+200=890 ∑f=5+7+6+8+4=30\sum f = 5 + 7 + 6 + 8 + 4 = 30∑f=5+7+6+8+4=30
Step 3: Calculate the meanMean=89030=29.67\text{Mean} = \frac{890}{30} = 29.67Mean=30890=29.67
Answer: The mean of the given data is 29.67.
Exercise 8.3: Mode
Q1. Find the mode of the following data:
Marks: 5, 8, 10, 15, 5, 8, 8, 15, 10, 8, 5
Solution:
The mode is the value that appears most frequently in a data set.
Step 1: Count the frequency of each data point
| Marks | Frequency |
|---|---|
| 5 | 3 |
| 8 | 4 |
| 10 | 2 |
| 15 | 2 |
Step 2: Identify the most frequent value
From the table, we can see that 8 appears 4 times, which is the highest frequency.
Answer: The mode of the given data is 8.
Exercise 8.4: Median
Q1. Find the median of the following data:
Marks: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Solution:
The median is the middle value when the data is arranged in ascending order. If the data set has an even number of observations, the median is the average of the two middle values.
Step 1: Arrange the data in ascending order (already done)
Data: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Step 2: Find the middle values
Since there are 10 data points (even number), the median is the average of the 5th and 6th values.
The 5th value is 50, and the 6th value is 60.
Step 3: Calculate the medianMedian=50+602=1102=55\text{Median} = \frac{50 + 60}{2} = \frac{110}{2} = 55Median=250+60=2110=55
Answer: The median of the given data is 55.
Exercise 8.5: Cumulative Frequency
Q1. The following table shows the marks obtained by 30 students in a class:
| Marks | Number of Students |
|---|---|
| 0-10 | 3 |
| 10-20 | 5 |
| 20-30 | 7 |
| 30-40 | 6 |
| 40-50 | 9 |
| 50-60 | 0 |
Calculate the cumulative frequency for the data.
Solution:
To calculate the cumulative frequency, we add up the frequencies as we move from top to bottom.
Step 1: Calculate cumulative frequencies
| Marks | Number of Students | Cumulative Frequency |
|---|---|---|
| 0-10 | 3 | 3 |
| 10-20 | 5 | 3 + 5 = 8 |
| 20-30 | 7 | 8 + 7 = 15 |
| 30-40 | 6 | 15 + 6 = 21 |
| 40-50 | 9 | 21 + 9 = 30 |
| 50-60 | 0 | 30 + 0 = 30 |
Answer: The cumulative frequency table is as follows:
| Marks | Number of Students | Cumulative Frequency |
|---|---|---|
| 0-10 | 3 | 3 |
| 10-20 | 5 | 8 |
| 20-30 | 7 | 15 |
| 30-40 | 6 | 21 |
| 40-50 | 9 | 30 |
| 50-60 | 0 | 30 |
Additional Questions for Chapter 8: Statistics
1. Find the mean of the following data:
Data (marks obtained by 10 students): 10, 15, 20, 25, 30, 35, 40, 45, 50, 55
Solution: To find the mean, use the formula:Mean=∑f×x∑f\text{Mean} = \frac{\sum f \times x}{\sum f}Mean=∑f∑f×x
Where fff is the frequency and xxx is the data point.
2. The following table shows the ages of a group of 15 people:
| Age (in years) | Number of People (f) |
|---|---|
| 10-20 | 3 |
| 20-30 | 5 |
| 30-40 | 4 |
| 40-50 | 2 |
| 50-60 | 1 |
Calculate the median age of the group.
Solution:
To calculate the median, find the cumulative frequency and use the median formula for grouped data.
3. The following data shows the number of books read by 25 students:
Data: 3, 5, 7, 8, 9, 9, 7, 6, 4, 8, 6, 3, 4, 6, 9, 5, 7, 4, 8, 5, 6, 7, 8, 6, 9
Find the mode of the data.
Solution:
The mode is the value that appears most frequently in the data. First, identify the frequency of each value and choose the most frequent one.
4. A teacher records the following number of students present in her class on each day for a week:
Data (students present each day): 20, 25, 22, 23, 24, 25, 22
Find the mean, median, and mode of the number of students present.
Solution:
- Mean: Add all the values and divide by the number of days (7).
- Median: Arrange the data in ascending order and find the middle value.
- Mode: Identify the value that appears most frequently.
5. The following data shows the height (in cm) of 20 students:
Data (Heights of students in cm): 150, 160, 145, 155, 160, 165, 150, 155, 155, 160, 170, 175, 180, 165, 160, 150, 160, 150, 175, 160
Construct a frequency distribution table and calculate the mean height.
Solution:
- Step 1: Group the data into intervals.
- Step 2: Calculate the frequency for each interval.
- Step 3: Use the formula for mean after constructing the table.
6. A group of 30 students in a class were surveyed to find their favorite colors. The results are as follows:
| Color | Number of Students |
|---|---|
| Red | 7 |
| Blue | 10 |
| Green | 5 |
| Yellow | 4 |
| Pink | 2 |
| Purple | 2 |
What is the mode of the favorite color among the students?
Solution:
The mode is the color with the highest frequency, which is Blue (10 students).
7. The following data represents the number of goals scored by a football team in 10 matches:
Data (Goals scored in matches): 3, 5, 2, 6, 4, 3, 5, 2, 7, 6
Calculate the median number of goals scored.
Solution:
- Arrange the data in ascending order and find the middle value, since there are 10 data points (even number, so average of the two middle values).
8. The following data shows the weekly earnings (in dollars) of 12 workers:
Data (Weekly earnings): 350, 400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 900
Find the cumulative frequency for the data.
Solution:
- Construct the cumulative frequency table by adding the frequencies as you move down the list.
9. The following table shows the number of hours spent on homework by 15 students in a week:
| Hours Spent (x) | Number of Students (f) |
|---|---|
| 0-2 | 4 |
| 2-4 | 5 |
| 4-6 | 3 |
| 6-8 | 2 |
| 8-10 | 1 |
Find the median hours spent on homework.
Solution:
- Construct a cumulative frequency table and then find the median from the cumulative frequency.
10. The following data shows the number of vehicles passing a traffic signal in a day for 7 days:
| Number of Vehicles (x) | Number of Days (f) |
|---|---|
| 50-100 | 2 |
| 100-150 | 3 |
| 150-200 | 1 |
| 200-250 | 1 |
Find the mean number of vehicles passing the signal in a day.
Solution:
- Use the midpoint of each class and apply the formula for mean:
Mean=∑f×x∑f\text{Mean} = \frac{\sum f \times x}{\sum f}Mean=∑f∑f×x
11. The following table shows the number of children in 15 families:
| Number of Children (x) | Number of Families (f) |
|---|---|
| 1-2 | 5 |
| 3-4 | 6 |
| 5-6 | 3 |
| 7-8 | 1 |
Find the mode of the number of children per family.
Solution:
- Identify the class with the highest frequency, which gives the mode.
12. A class test was conducted in which 100 students appeared. The scores of the students are as follows:
Scores: 50, 55, 60, 45, 50, 55, 60, 50, 45, 60, 50, 45, 55, 60, 50, 45, 60, 50, 55, 60, … (and so on)
Find the mean, median, and mode of the scores.
Solution:
- Mean: Add all the scores and divide by the total number of students.
- Median: Find the middle score when arranged in order.
- Mode: Find the most frequent score.
13. The following table shows the marks obtained by 20 students in a class:
| Marks (x) | Number of Students (f) |
|---|---|
| 0-10 | 2 |
| 10-20 | 4 |
| 20-30 | 6 |
| 30-40 | 5 |
| 40-50 | 3 |
Find the cumulative frequency and the class interval with the highest cumulative frequency.
Solution:
- Calculate the cumulative frequency and identify the class with the highest cumulative frequency.
14. The following table shows the ages of 20 people:
| Age (Years) | Number of People (f) |
|---|---|
| 10-20 | 2 |
| 20-30 | 6 |
| 30-40 | 4 |
| 40-50 | 5 |
| 50-60 | 3 |
Find the median age of the group.
Solution:
- Find the cumulative frequency and use the median formula for grouped data.
15. The following data represents the weight of 15 people:
Weights (in kg): 45, 50, 55, 60, 60, 65, 70, 70, 75, 75, 80, 80, 85, 90, 95
Find the mode of the weights.
Solution:
- Identify the value that occurs most frequently.
16. The following table shows the number of children in 25 families:
| Number of Children | Number of Families (f) |
|---|---|
| 1 | 5 |
| 2 | 7 |
| 3 | 6 |
| 4 | 4 |
| 5 | 3 |
Find the mean number of children per family.
Solution:
- Use the formula for mean after calculating the frequencies.
These questions cover a variety of problems related to mean, median, mode, and cumulative frequency, helping you get a firm grip on the important concepts of Statistics. Practicing them will boost your confidence and improve your problem-solving skills for your upcoming exams!
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