To find the third quartile (Q3) of the scores, follow these steps:

Step 1: Confirm the data is in ascending order.

This has already been done:

$58, 60, 61, 63, 66, 68, 69, 70, 71, 71, 72, 75, 76, 76, 76, 77, 77, 78, 83, 84, 84, 87, 87, 89, 91, 95, 96, 96, 97, 98$

Step 2: Determine the number of data points ($n$).

There are $n = 30$ data points.

Step 3: Find the position of Q3.

The third quartile (Q3) is the value at the 75th percentile. The position of Q3 can be found using the formula:

$Q3 = \frac{3(n + 1)}{4}$

Substituting the value of $n$:

$Q3 = \frac{3(30 + 1)}{4} = \frac{93}{4} = 23.25$

Step 4: Locate Q3.

Since $Q3$ is at position 23.25, it lies between the 23rd and 24th values in the data set. To find the exact value, we can interpolate between the 23rd and 24th values.

• The 23rd value is 87.
• The 24th value is 89.

The value of Q3 can be found by interpolating between these two values:

$Q3 = 87 + 0.25 \times (89 - 87) = 87 + 0.25 \times 2 = 87 + 0.5 = 87.5$

Final Answer:

The third quartile (Q3) of the scores is 87.5.

Would you like more details on the interpolation process or have any other questions?

Here are five related questions:

1. How do you calculate the interquartile range (IQR) using Q1 and Q3?
2. What is the significance of the third quartile in a data set?
3. How can quartiles help in identifying outliers in a dataset?
4. How would you find the quartiles if the data set had an odd number of observations?
5. What role do quartiles play in box plots?

Tip: The third quartile (Q3) represents the value below which 75% of the data falls. It is useful in understanding the spread of the upper half of the dataset.