Let's address each part of the question step by step.
### Part a: Understanding the Formula Component \( f \)
The formula for the first quartile \( Q_1 \) in a continuous series is:
[tex]\[ Q_1 = L + \left( \frac{N}{4} - cf \right) \times \frac{i}{f} \][/tex]
In this formula:
- \( L \) is the lower boundary of the first quartile class,
- \( N \) is the total number of observations,
- \( cf \) is the cumulative frequency of the class preceding the first quartile class,
- \( f \) is the frequency of the first quartile class,
- \( i \) is the class interval.
So, \( f \) denotes the frequency of the first quartile class.
### Part b: Frequency Distribution Table
Given data:
[tex]\[ 53, 27, 63, 53, 46, 58, 78, 21, 39, 51, 32, 43, 52, 59, 33, 45, 40, 32, 60, 63 \][/tex]
Step-by-step process to prepare a frequency distribution table:
1. Sort the data:
[tex]\[ 21, 27, 32, 32, 33, 39, 40, 43, 45, 46, 51, 52, 53, 53, 58, 59, 60, 63, 63, 78 \][/tex]
2. Determine class intervals:
- First class interval: 20-30
- Second class interval: 30-40
- Third class interval: 40-50
- Fourth class interval: 50-60
- Fifth class interval: 60-70
- Sixth class interval: 70-80
3. Count frequencies for each interval:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Class interval} & \text{Frequency} \\
\hline
20-30 & 2 \\
30-40 & 4 \\
40-50 & 4 \\
50-60 & 6 \\
60-70 & 3 \\
70-80 & 1 \\
\hline
\end{array}
\][/tex]
Thus, the frequency distribution table is:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Class interval} & \text{Frequency} \\
\hline
20-30 & 2 \\
30-40 & 4 \\
40-50 & 4 \\
50-60 & 6 \\
60-70 & 3 \\
70-80 & 1 \\
\hline
\end{array}
\][/tex]
### Part c: Finding the Mode
The mode is the value that appears most frequently in a data set.
Given the sorted data:
[tex]\[ 21, 27, 32, 32, 33, 39, 40, 43, 45, 46, 51, 52, 53, 53, 58, 59, 60, 63, 63, 78 \][/tex]
We count the occurrences of each value:
- 21 appears 1 time
- 27 appears 1 time
- 32 appears 2 times
- 33 appears 1 time
- 39 appears 1 time
- 40 appears 1 time
- 43 appears 1 time
- 45 appears 1 time
- 46 appears 1 time
- 51 appears 1 time
- 52 appears 1 time
- 53 appears 2 times
- 58 appears 1 time
- 59 appears 1 time
- 60 appears 1 time
- 63 appears 2 times
- 78 appears 1 time
The highest frequency is 2, and the values that appear with this frequency are \( 32, 53, \) and \( 63 \).
Therefore, the mode from the given data is:
[tex]\[ 32, 53, \text{ and } 63 \][/tex]
