Answer :
- Calculate cumulative frequencies.
- Identify the median class and apply the median formula: $Median = 35 + \frac{64 - 51}{18} \times 5 = 38.61$.
- Identify the modal class and apply the mode formula: $Mode = 50 + \frac{22 - 6}{2(22) - 6 - 10} \times 5 = 52.86$.
- The median is $38.61$ and the mode is $52.86$, so the final answer is $\boxed{Median = 38.61, Mode = 52.86}$.
### Explanation
1. Problem Analysis
We are given a frequency distribution table and asked to find the median and mode. The class intervals and corresponding frequencies are provided.
2. Calculate Cumulative Frequencies
First, we need to calculate the cumulative frequencies to determine the median class. The cumulative frequencies are: 4, 12, 31, 51, 69, 90, 96, 118, 128. The total frequency, N, is 128.
3. Determine Median Class
The median is the value of the $\frac{N}{2}$-th item, which is $\frac{128}{2} = 64$. The median class is the class interval where the cumulative frequency is greater than or equal to 64. In this case, the median class is 35-40, since its cumulative frequency is 69, which is the first cumulative frequency greater than or equal to 64.
4. Calculate Median
Now, we apply the formula for calculating the median:
$Median = L + \frac{\frac{N}{2} - cf}{f} \times h$,
where L is the lower limit of the median class (35), N is the total frequency (128), cf is the cumulative frequency of the class preceding the median class (51), f is the frequency of the median class (18), and h is the class width (5).
$Median = 35 + \frac{64 - 51}{18} \times 5 = 35 + \frac{13}{18} \times 5 = 35 + \frac{65}{18} = 35 + 3.6111 = 38.6111$
5. Determine Modal Class
Next, we determine the modal class. The modal class is the class with the highest frequency. In this case, the modal class is 50-55, with a frequency of 22.
6. Calculate Mode
Now, we apply the formula for calculating the mode:
$Mode = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$,
where L is the lower limit of the modal class (50), $f_1$ is the frequency of the modal class (22), $f_0$ is the frequency of the class preceding the modal class (6), $f_2$ is the frequency of the class succeeding the modal class (10), and h is the class width (5).
$Mode = 50 + \frac{22 - 6}{2(22) - 6 - 10} \times 5 = 50 + \frac{16}{44 - 16} \times 5 = 50 + \frac{16}{28} \times 5 = 50 + \frac{80}{28} = 50 + 2.8571 = 52.8571$
7. Final Answer
Therefore, the median is approximately 38.61 and the mode is approximately 52.86.
### Examples
Understanding median and mode is very useful in analyzing data sets. For example, if you are analyzing the test scores of a class, the median score gives you the middle score, while the mode tells you the most common score. This helps in understanding the overall performance of the class and identifying areas where students may need more help. In business, understanding the mode of customer purchases can help optimize inventory.
- Identify the median class and apply the median formula: $Median = 35 + \frac{64 - 51}{18} \times 5 = 38.61$.
- Identify the modal class and apply the mode formula: $Mode = 50 + \frac{22 - 6}{2(22) - 6 - 10} \times 5 = 52.86$.
- The median is $38.61$ and the mode is $52.86$, so the final answer is $\boxed{Median = 38.61, Mode = 52.86}$.
### Explanation
1. Problem Analysis
We are given a frequency distribution table and asked to find the median and mode. The class intervals and corresponding frequencies are provided.
2. Calculate Cumulative Frequencies
First, we need to calculate the cumulative frequencies to determine the median class. The cumulative frequencies are: 4, 12, 31, 51, 69, 90, 96, 118, 128. The total frequency, N, is 128.
3. Determine Median Class
The median is the value of the $\frac{N}{2}$-th item, which is $\frac{128}{2} = 64$. The median class is the class interval where the cumulative frequency is greater than or equal to 64. In this case, the median class is 35-40, since its cumulative frequency is 69, which is the first cumulative frequency greater than or equal to 64.
4. Calculate Median
Now, we apply the formula for calculating the median:
$Median = L + \frac{\frac{N}{2} - cf}{f} \times h$,
where L is the lower limit of the median class (35), N is the total frequency (128), cf is the cumulative frequency of the class preceding the median class (51), f is the frequency of the median class (18), and h is the class width (5).
$Median = 35 + \frac{64 - 51}{18} \times 5 = 35 + \frac{13}{18} \times 5 = 35 + \frac{65}{18} = 35 + 3.6111 = 38.6111$
5. Determine Modal Class
Next, we determine the modal class. The modal class is the class with the highest frequency. In this case, the modal class is 50-55, with a frequency of 22.
6. Calculate Mode
Now, we apply the formula for calculating the mode:
$Mode = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$,
where L is the lower limit of the modal class (50), $f_1$ is the frequency of the modal class (22), $f_0$ is the frequency of the class preceding the modal class (6), $f_2$ is the frequency of the class succeeding the modal class (10), and h is the class width (5).
$Mode = 50 + \frac{22 - 6}{2(22) - 6 - 10} \times 5 = 50 + \frac{16}{44 - 16} \times 5 = 50 + \frac{16}{28} \times 5 = 50 + \frac{80}{28} = 50 + 2.8571 = 52.8571$
7. Final Answer
Therefore, the median is approximately 38.61 and the mode is approximately 52.86.
### Examples
Understanding median and mode is very useful in analyzing data sets. For example, if you are analyzing the test scores of a class, the median score gives you the middle score, while the mode tells you the most common score. This helps in understanding the overall performance of the class and identifying areas where students may need more help. In business, understanding the mode of customer purchases can help optimize inventory.
