Answer :
Let's analyze the given frequency distribution of the number of letters in the surnames of some students in the school:
| Number of letters | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|-------------------|---|---|---|---|---|---|----|
| Number of students| 7 | 3 | 2 | 8 | 5 | 3 | 1 |
### (a) Determining Statistical Measures
#### (i) Mode
The mode of a data set is the value that appears most frequently. In this case, we look for the number of letters that corresponds to the highest frequency (number of students).
- Number of letters: 4, 5, 6, 7, 8, 9, 10
- Number of students: 7, 3, 2, 8, 5, 3, 1
The highest frequency is 8 (for 7 letters), hence:
Mode = 7
#### (ii) Median
To find the median, we need to arrange the data in ascending order and find the middle value. If the number of data points is odd, the median is the middle number. If even, it is the average of the two middle numbers.
Construct the list of letters sorted by frequency:
- 4 appears 7 times
- 5 appears 3 times
- 6 appears 2 times
- 7 appears 8 times
- 8 appears 5 times
- 9 appears 3 times
- 10 appears 1 time
Arranging these, the data becomes:
[tex]\[4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10\][/tex]
This data set has 29 values. Since 29 is an odd number, the median is the 15th value:
Median = 7
#### (iii) Mean
The mean (average) is calculated by dividing the total number of letters by the total number of students.
Total students:
[tex]\[7 + 3 + 2 + 8 + 5 + 3 + 1 = 29\][/tex]
Total letters:
[tex]\[ (4 \times 7) + (5 \times 3) + (6 \times 2) + (7 \times 8) + (8 \times 5) + (9 \times 3) + (10 \times 1) \][/tex]
[tex]\[ = 28 + 15 + 12 + 56 + 40 + 27 + 10 \][/tex]
[tex]\[ = 188\][/tex]
Now, compute the mean:
[tex]\[ \text{Mean} = \frac{188}{29} \approx 6.482758620689655 \][/tex]
Mean = 6.48 (rounded to two decimal places)
### (b) Probability for More Than 7 Letters
To find the probability that a randomly selected student has a surname with more than 7 letters (i.e., 8, 9, or 10 letters):
Number of students with 8, 9, or 10 letters:
[tex]\[5 (for 8) + 3 (for 9) + 1 (for 10) = 9\][/tex]
Total number of students:
[tex]\[29\][/tex]
Probability:
[tex]\[ \text{Probability} = \frac{9}{29} \approx 0.3103448275862069 \][/tex]
Probability = 0.31 (fraction)
### (c) Bar Chart
To draw a bar chart, you would plot the number of letters on the x-axis and the number of students on the y-axis. The heights of the bars would correspond to the frequencies given in the table.
For example:
- A bar for '4 letters' would have a height of 7.
- A bar for '5 letters' would have a height of 3.
- And so forth...
This visual representation helps in better understanding the distribution of the data.
Remember to label the axes and title your graph appropriately!
| Number of letters | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|-------------------|---|---|---|---|---|---|----|
| Number of students| 7 | 3 | 2 | 8 | 5 | 3 | 1 |
### (a) Determining Statistical Measures
#### (i) Mode
The mode of a data set is the value that appears most frequently. In this case, we look for the number of letters that corresponds to the highest frequency (number of students).
- Number of letters: 4, 5, 6, 7, 8, 9, 10
- Number of students: 7, 3, 2, 8, 5, 3, 1
The highest frequency is 8 (for 7 letters), hence:
Mode = 7
#### (ii) Median
To find the median, we need to arrange the data in ascending order and find the middle value. If the number of data points is odd, the median is the middle number. If even, it is the average of the two middle numbers.
Construct the list of letters sorted by frequency:
- 4 appears 7 times
- 5 appears 3 times
- 6 appears 2 times
- 7 appears 8 times
- 8 appears 5 times
- 9 appears 3 times
- 10 appears 1 time
Arranging these, the data becomes:
[tex]\[4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10\][/tex]
This data set has 29 values. Since 29 is an odd number, the median is the 15th value:
Median = 7
#### (iii) Mean
The mean (average) is calculated by dividing the total number of letters by the total number of students.
Total students:
[tex]\[7 + 3 + 2 + 8 + 5 + 3 + 1 = 29\][/tex]
Total letters:
[tex]\[ (4 \times 7) + (5 \times 3) + (6 \times 2) + (7 \times 8) + (8 \times 5) + (9 \times 3) + (10 \times 1) \][/tex]
[tex]\[ = 28 + 15 + 12 + 56 + 40 + 27 + 10 \][/tex]
[tex]\[ = 188\][/tex]
Now, compute the mean:
[tex]\[ \text{Mean} = \frac{188}{29} \approx 6.482758620689655 \][/tex]
Mean = 6.48 (rounded to two decimal places)
### (b) Probability for More Than 7 Letters
To find the probability that a randomly selected student has a surname with more than 7 letters (i.e., 8, 9, or 10 letters):
Number of students with 8, 9, or 10 letters:
[tex]\[5 (for 8) + 3 (for 9) + 1 (for 10) = 9\][/tex]
Total number of students:
[tex]\[29\][/tex]
Probability:
[tex]\[ \text{Probability} = \frac{9}{29} \approx 0.3103448275862069 \][/tex]
Probability = 0.31 (fraction)
### (c) Bar Chart
To draw a bar chart, you would plot the number of letters on the x-axis and the number of students on the y-axis. The heights of the bars would correspond to the frequencies given in the table.
For example:
- A bar for '4 letters' would have a height of 7.
- A bar for '5 letters' would have a height of 3.
- And so forth...
This visual representation helps in better understanding the distribution of the data.
Remember to label the axes and title your graph appropriately!
