The shoe sizes of 30 players selected for interschool competitions are given below :
`{:("Shoe size"(x_(i)),7,10,9,8,5,11,6),("Number of players"(f_(i)),4,5,8,3,5,2,3):}`
Find the median of this frequency distribution.

Answer

To find the median of the given frequency distribution, we will follow these steps:

### Step 1: Organize the Data
First, we will organize the shoe sizes and their corresponding frequencies into a tabular form.

| Shoe Size (x_i) | Frequency (f_i) |
|------------------|-----------------|
| 5 | 4 |
| 6 | 5 |
| 7 | 8 |
| 8 | 3 |
| 9 | 5 |
| 10 | 2 |
| 11 | 3 |

### Step 2: Calculate the Total Frequency (N)
Next, we need to calculate the total number of players (N) by summing up the frequencies.

\[ N = 4 + 5 + 8 + 3 + 5 + 2 + 3 = 30 \]

### Step 3: Calculate the Cumulative Frequency
Now, we will calculate the cumulative frequency (CF) for each shoe size.

| Shoe Size (x_i) | Frequency (f_i) | Cumulative Frequency (CF) |
|------------------|-----------------|---------------------------|
| 5 | 4 | 4 |
| 6 | 5 | 4 + 5 = 9 |
| 7 | 8 | 9 + 8 = 17 |
| 8 | 3 | 17 + 3 = 20 |
| 9 | 5 | 20 + 5 = 25 |
| 10 | 2 | 25 + 2 = 27 |
| 11 | 3 | 27 + 3 = 30 |

### Step 4: Find the Median Position
The formula for finding the median position in a frequency distribution is:

\[ \text{Median Position} = \frac{N + 1}{2} \]

Substituting the value of N:

\[ \text{Median Position} = \frac{30 + 1}{2} = \frac{31}{2} = 15.5 \]

### Step 5: Locate the Median
Now, we will determine which cumulative frequency covers the median position of 15.5.

From the cumulative frequency table, we see:
- CF for shoe size 7 is 17 (covers 15.5)
- CF for shoe size 6 is 9 (does not cover 15.5)

Thus, the median shoe size corresponds to the shoe size where the cumulative frequency first exceeds 15.5, which is 7.

### Conclusion
The median shoe size of the players is **7**.

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