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Let's calculate the mode for the given frequency distribution using both the formula method and the grouping table method.
### 1. Formula Method
The formula for the mode in a grouped frequency distribution is:
[tex]\[ \text{Mode} = l + \left( \frac{f_1 - f_0}{(2f_1 - f_0 - f_2)} \right) \times h \][/tex]
where:
- [tex]\( l \)[/tex] is the lower boundary of the modal class
- [tex]\( f_1 \)[/tex] is the frequency of the modal class
- [tex]\( f_0 \)[/tex] is the frequency of the class preceding the modal class
- [tex]\( f_2 \)[/tex] is the frequency of the class succeeding the modal class
- [tex]\( h \)[/tex] is the class width
Step-by-Step Solution:
1. Identify the modal class:
- The modal class is the class with the highest frequency.
- Here, the highest frequency is [tex]\( 15 \)[/tex], which corresponds to the class interval [tex]\( 29-32 \)[/tex].
2. Determine the values needed:
- The lower boundary ([tex]\( l \)[/tex]) of the modal class [tex]\( 29-32 \)[/tex] is [tex]\( 29 \)[/tex].
- Frequency of the modal class ([tex]\( f_1 \)[/tex]) is [tex]\( 15 \)[/tex].
- Frequency of the class preceding the modal class ([tex]\( f_0 \)[/tex]) is [tex]\( 14 \)[/tex] (class interval [tex]\( 25-28 \)[/tex]).
- Frequency of the class succeeding the modal class ([tex]\( f_2 \)[/tex]) is [tex]\( 11 \)[/tex] (class interval [tex]\( 33-36 \)[/tex]).
- The class width ([tex]\( h \)[/tex]) is calculated as [tex]\( 32 - 29 + 1 = 4 \)[/tex].
3. Plug the values into the formula:
[tex]\[ \text{Mode} = 29 + \left( \frac{15 - 14}{(2 \times 15 - 14 - 11)} \right) \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + \left( \frac{1}{(30 - 14 - 11)} \right) \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + \left( \frac{1}{5} \right) \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + 0.2 \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + 0.8 \][/tex]
[tex]\[ \text{Mode} = 29.8 \][/tex]
Hence, the mode using the formula method is [tex]\( \boldsymbol{29.8} \)[/tex].
### 2. Grouping Table Method
The grouping table method involves creating a table that helps us identify the modal class more clearly.
Let's construct the grouping table:
| Class Interval | Frequency ([tex]\( f \)[/tex]) | Cumulative Frequency | [tex]\( f_i - f_{i-1} \)[/tex] | [tex]\( f_i - f_{i+1} \)[/tex] |
|--- | --- | --- | --- | --- |
| [tex]\( 1 - 4 \)[/tex] | 2 | 2 | | 2 - 5 |
| [tex]\( 5 - 8 \)[/tex] | 5 | 7 | 5 - 2 | 5 - 8 |
| [tex]\( 9 - 12 \)[/tex] | 8 | 15 | 8 - 5 | 8 - 9 |
| [tex]\( 13 - 16 \)[/tex] | 9 | 24 | 9 - 8 | 9 - 13 |
| [tex]\( 17 - 20 \)[/tex] | 13 | 37 | 13 - 9 | 13 - 14 |
| [tex]\( 21 - 24 \)[/tex] | 14 | 51 | 14 - 13 | 14 - 14 |
| [tex]\( 25 - 28 \)[/tex] | 14 | 65 | 14 - 14 | 14 - 15 |
| [tex]\( 29 - 32 \)[/tex] | 15 | 80 | 15 - 14 | 15 - 11 |
| [tex]\( 33 - 36 \)[/tex] | 11 | 91 | 11 - 15 | 11 - 13 |
| [tex]\( 37 - 40 \)[/tex] | 13 | 104 | 13 - 11 | |
Analysis:
- The highest [tex]\( f_i \)[/tex] is [tex]\( 15 \)[/tex] in the class interval [tex]\( 29 - 32 \)[/tex].
- Based on this highest frequency, let's check the column [tex]\( f_i - f_{i-1} \)[/tex]:
- [tex]\( 15 - 14 = 1 \)[/tex]
- In the column [tex]\( f_i - f_{i+1} \)[/tex]:
- [tex]\( 15 - 11 = 4 \)[/tex]
This confirms that the highest peak is indeed at the interval [tex]\( 29 - 32 \)[/tex], as it has the highest frequency with the significant difference in neighboring classes.
So, the mode using the grouping table method is again located at class interval [tex]\( 29-32 \)[/tex].
Thus, the mode calculated by both the formula method and the grouping table method is [tex]\( \boldsymbol{29.8} \)[/tex].
### 1. Formula Method
The formula for the mode in a grouped frequency distribution is:
[tex]\[ \text{Mode} = l + \left( \frac{f_1 - f_0}{(2f_1 - f_0 - f_2)} \right) \times h \][/tex]
where:
- [tex]\( l \)[/tex] is the lower boundary of the modal class
- [tex]\( f_1 \)[/tex] is the frequency of the modal class
- [tex]\( f_0 \)[/tex] is the frequency of the class preceding the modal class
- [tex]\( f_2 \)[/tex] is the frequency of the class succeeding the modal class
- [tex]\( h \)[/tex] is the class width
Step-by-Step Solution:
1. Identify the modal class:
- The modal class is the class with the highest frequency.
- Here, the highest frequency is [tex]\( 15 \)[/tex], which corresponds to the class interval [tex]\( 29-32 \)[/tex].
2. Determine the values needed:
- The lower boundary ([tex]\( l \)[/tex]) of the modal class [tex]\( 29-32 \)[/tex] is [tex]\( 29 \)[/tex].
- Frequency of the modal class ([tex]\( f_1 \)[/tex]) is [tex]\( 15 \)[/tex].
- Frequency of the class preceding the modal class ([tex]\( f_0 \)[/tex]) is [tex]\( 14 \)[/tex] (class interval [tex]\( 25-28 \)[/tex]).
- Frequency of the class succeeding the modal class ([tex]\( f_2 \)[/tex]) is [tex]\( 11 \)[/tex] (class interval [tex]\( 33-36 \)[/tex]).
- The class width ([tex]\( h \)[/tex]) is calculated as [tex]\( 32 - 29 + 1 = 4 \)[/tex].
3. Plug the values into the formula:
[tex]\[ \text{Mode} = 29 + \left( \frac{15 - 14}{(2 \times 15 - 14 - 11)} \right) \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + \left( \frac{1}{(30 - 14 - 11)} \right) \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + \left( \frac{1}{5} \right) \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + 0.2 \times 4 \][/tex]
[tex]\[ \text{Mode} = 29 + 0.8 \][/tex]
[tex]\[ \text{Mode} = 29.8 \][/tex]
Hence, the mode using the formula method is [tex]\( \boldsymbol{29.8} \)[/tex].
### 2. Grouping Table Method
The grouping table method involves creating a table that helps us identify the modal class more clearly.
Let's construct the grouping table:
| Class Interval | Frequency ([tex]\( f \)[/tex]) | Cumulative Frequency | [tex]\( f_i - f_{i-1} \)[/tex] | [tex]\( f_i - f_{i+1} \)[/tex] |
|--- | --- | --- | --- | --- |
| [tex]\( 1 - 4 \)[/tex] | 2 | 2 | | 2 - 5 |
| [tex]\( 5 - 8 \)[/tex] | 5 | 7 | 5 - 2 | 5 - 8 |
| [tex]\( 9 - 12 \)[/tex] | 8 | 15 | 8 - 5 | 8 - 9 |
| [tex]\( 13 - 16 \)[/tex] | 9 | 24 | 9 - 8 | 9 - 13 |
| [tex]\( 17 - 20 \)[/tex] | 13 | 37 | 13 - 9 | 13 - 14 |
| [tex]\( 21 - 24 \)[/tex] | 14 | 51 | 14 - 13 | 14 - 14 |
| [tex]\( 25 - 28 \)[/tex] | 14 | 65 | 14 - 14 | 14 - 15 |
| [tex]\( 29 - 32 \)[/tex] | 15 | 80 | 15 - 14 | 15 - 11 |
| [tex]\( 33 - 36 \)[/tex] | 11 | 91 | 11 - 15 | 11 - 13 |
| [tex]\( 37 - 40 \)[/tex] | 13 | 104 | 13 - 11 | |
Analysis:
- The highest [tex]\( f_i \)[/tex] is [tex]\( 15 \)[/tex] in the class interval [tex]\( 29 - 32 \)[/tex].
- Based on this highest frequency, let's check the column [tex]\( f_i - f_{i-1} \)[/tex]:
- [tex]\( 15 - 14 = 1 \)[/tex]
- In the column [tex]\( f_i - f_{i+1} \)[/tex]:
- [tex]\( 15 - 11 = 4 \)[/tex]
This confirms that the highest peak is indeed at the interval [tex]\( 29 - 32 \)[/tex], as it has the highest frequency with the significant difference in neighboring classes.
So, the mode using the grouping table method is again located at class interval [tex]\( 29-32 \)[/tex].
Thus, the mode calculated by both the formula method and the grouping table method is [tex]\( \boldsymbol{29.8} \)[/tex].
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