Answer :
- Calculate the total number of data points: $3 + 5 + 8 + 4 + 2 = 22$.
- Determine the median position: average of the 11th and 12th values.
- Calculate cumulative frequencies: 2, 6, 14, 19, 22.
- Identify the interval containing the 11th and 12th values: $\boxed{70-79}$.
### Explanation
1. Understand the problem
We are given a frequency table showing grades, intervals, and frequencies. Our goal is to find the interval in which the median is located. The median is the middle value of a dataset.
2. Calculate the total number of data points
First, we need to find the total number of data points. We sum the frequencies: $3 + 5 + 8 + 4 + 2 = 22$. So there are 22 data points in total.
3. Determine the position of the median
Since there are 22 data points, the median is the average of the 11th and 12th values when the data is sorted in ascending order.
4. Calculate cumulative frequencies
Next, we calculate the cumulative frequencies for each interval:
- 50-59: 2
- 60-69: $2 + 4 = 6$
- 70-79: $6 + 8 = 14$
- 80-89: $14 + 5 = 19$
- 90-100: $19 + 3 = 22$
5. Identify the interval containing the median
Now we identify the interval that contains the 11th and 12th values based on the cumulative frequencies. The 11th and 12th values fall in the interval 70-79 because the cumulative frequency reaches 14 in this interval. This means that the 7th through 14th values are in the interval 70-79.
6. State the final answer
Therefore, the median is located in the interval 70-79.
### Examples
Understanding the median interval is useful in many real-world scenarios. For example, in education, it helps determine the grade range where the 'average' student's score lies. In economics, it can help identify the income bracket that represents the middle-income earners in a population. This information is valuable for making informed decisions and policies.
- Determine the median position: average of the 11th and 12th values.
- Calculate cumulative frequencies: 2, 6, 14, 19, 22.
- Identify the interval containing the 11th and 12th values: $\boxed{70-79}$.
### Explanation
1. Understand the problem
We are given a frequency table showing grades, intervals, and frequencies. Our goal is to find the interval in which the median is located. The median is the middle value of a dataset.
2. Calculate the total number of data points
First, we need to find the total number of data points. We sum the frequencies: $3 + 5 + 8 + 4 + 2 = 22$. So there are 22 data points in total.
3. Determine the position of the median
Since there are 22 data points, the median is the average of the 11th and 12th values when the data is sorted in ascending order.
4. Calculate cumulative frequencies
Next, we calculate the cumulative frequencies for each interval:
- 50-59: 2
- 60-69: $2 + 4 = 6$
- 70-79: $6 + 8 = 14$
- 80-89: $14 + 5 = 19$
- 90-100: $19 + 3 = 22$
5. Identify the interval containing the median
Now we identify the interval that contains the 11th and 12th values based on the cumulative frequencies. The 11th and 12th values fall in the interval 70-79 because the cumulative frequency reaches 14 in this interval. This means that the 7th through 14th values are in the interval 70-79.
6. State the final answer
Therefore, the median is located in the interval 70-79.
### Examples
Understanding the median interval is useful in many real-world scenarios. For example, in education, it helps determine the grade range where the 'average' student's score lies. In economics, it can help identify the income bracket that represents the middle-income earners in a population. This information is valuable for making informed decisions and policies.
