To determine the median for the given frequency distribution, follow these steps:
1. List the Values and Frequencies:
The values and their corresponding frequencies are given in the table below:
```
Value Frequency
0 8
1 12
2 15
3 20
4 20
5 14
```
2. Calculate the Total Number of Observations:
Add up all the frequencies to get the total number of observations.
[tex]\[
8 + 12 + 15 + 20 + 20 + 14 = 89
\][/tex]
So, the total number of observations [tex]\(N\)[/tex] is 89.
3. Calculate Cumulative Frequencies:
To find the median, we need the cumulative frequencies for each value:
[tex]\[
\begin{aligned}
\text{Value} & \quad \text{Cumulative Frequency} \\
0 & \quad 8 \\
1 & \quad 8 + 12 = 20 \\
2 & \quad 20 + 15 = 35 \\
3 & \quad 35 + 20 = 55 \\
4 & \quad 55 + 20 = 75 \\
5 & \quad 75 + 14 = 89 \\
\end{aligned}
\][/tex]
So, the cumulative frequencies are: 8, 20, 35, 55, 75, and 89.
4. Determine the Median Position:
The position of the median in the ordered dataset is given by:
[tex]\[
\text{Median Position} = \frac{N + 1}{2} = \frac{89 + 1}{2} = 45
\][/tex]
5. Identify the Median Class:
Find the cumulative frequency that is equal to or just greater than the median position. From the cumulative frequencies:
- For value 3, the cumulative frequency is 55, which is the first cumulative frequency greater than or equal to 45.
Hence, the median value is 3.
Conclusion:
The median for the given frequency distribution is [tex]\(\boxed{3}\)[/tex].
