How to find median and mode and range in the table

Okay, let's find the median, mode, and range using the grouped frequency table.

Here is our grouped frequency table:

$$\begin{array}{|c|c|} \hline \textbf{Score Interval} & \textbf{Frequency} \\ \hline 10 - 19 & 3 \\ 20 - 29 & 3 \\ 30 - 39 & 5 \\ 40 - 49 & 3 \\ 50 - 59 & 2 \\ \hline \textbf{Total} & \textbf{16} \\ \hline \end{array}$$

1. How to find the Mode from a grouped frequency table:
For grouped data, we find the modal class. This is the class interval with the highest frequency.

• Look at the "Frequency" column. The highest frequency is 5.
• The class interval corresponding to this frequency is 30 - 39. So, the modal class is $\boxed{30 - 39}$. This means the most common scores fall within this range.

2. How to find the Median from a grouped frequency table:
For grouped data, we find the median class. This is the class interval where the middle value lies.

• First, find the position of the median. Since there are 16 total scores (N=16), the median is between the $\frac{N}{2}$-th and $(\frac{N}{2} + 1)$-th values, which are the 8th and 9th values.
• Now, we find which class interval contains these values by adding up frequencies (cumulative frequency):
  • 10 - 19: 3 scores (scores 1, 2, 3)
  • 20 - 29: 3 scores. Cumulative: $3 + 3 = 6$ scores (scores 4, 5, 6)
  • 30 - 39: 5 scores. Cumulative: $6 + 5 = 11$ scores (scores 7, 8, 9, 10, 11)
• Since the 8th and 9th scores fall within the 30 - 39 interval, this is the median class. So, the median class is $\boxed{30 - 39}$.

3. How to find the Range from a grouped frequency table:
The range is the difference between the upper limit of the highest class and the lower limit of the lowest class.

• Highest class interval: 50 - 59. Its upper limit is 59.
• Lowest class interval: 10 - 19. Its lower limit is 10.

$$\text{Range} = \text{Upper limit of highest class} - \text{Lower limit of lowest class} = 59 - 10 = 49$$

The range is $\boxed{49}$.