To find the median of a frequency distribution, follow these steps:
1. Construct the data set based on the given frequencies:
- Score 1 appears 3 times.
- Score 2 appears 3 times.
- Score 3 appears 3 times.
- Score 4 appears 3 times.
- Score 5 appears 7 times.
- Score 6 appears 5 times.
- Score 7 appears 7 times.
- Score 8 appears 5 times.
- Score 9 appears 3 times.
- Score 10 appears 3 times.
Combining these, we get the dataset:
[tex]\[ 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 6, 8, 9, 9, 9, 8, 10, 10, 10 \][/tex]
2. Arrange the data set in ascending order:
Arrange the numbers from smallest to largest:
[tex]\[
1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10
\][/tex]
3. Find the total number of data points:
Add up all the frequencies:
[tex]\[
3 + 3 + 3 + 3 + 7 + 5 + 7 + 5 + 3 + 3 = 42
\][/tex]
So, there are 42 data points.
4. Locate the position of the median:
The median is the middle value in an ordered list of data. For a dataset with an even number of observations (n = 42), the median will be the average of the 21st and 22nd values.
[tex]\[
\text{Median position} = \left( \frac{42}{2} \right) \text{th and } \left( \frac{42}{2} + 1 \right) \text{th observations}
\][/tex]
Hence, we need to find the 21st and 22nd values.
5. Determine the 21st and 22nd values in the data set:
Looking at the ordered dataset:
- The first 3 values are 1s.
- The next 3 values are 2s.
- The next 3 values are 3s.
- The next 3 values are 4s.
- The next 7 values are 5s.
- The next 5 values are 6s.
Therefore, the 21st and 22nd values are both 6.
6. Calculate the median:
Since the dataset has an even number of observations and both the 21st and 22nd values are 6, the median is:
[tex]\[
\text{Median} = 6
\][/tex]
The median for the data items in the given frequency distribution is [tex]\( \boxed{6.0} \)[/tex].
