The image shows a frequency distribution table with the following dataThe image shows a frequency distribution table with the following

The question asks for the "upper quartile" (third quartile or $Q_3$) of the data.

Steps to find the upper quartile:

1. Determine the total number of observations: $N = 3 + 4 + 2 + 4 + 2 = 15$

2. Find the position of the third quartile: The formula for $Q_3$ is: $Q_3 = \frac{3(N+1)}{4}$ Substituting $N = 15$: $Q_3 = \frac{3(15+1)}{4} = \frac{48}{4} = 12$ The 12th observation corresponds to $Q_3$.

3. Locate the 12th observation in the cumulative frequency:

   • Values of 6 appear 3 times.
   • Values of 7 appear 4 times. Cumulative frequency after 7 is $3 + 4 = 7$.
   • Values of 8 appear 2 times. Cumulative frequency after 8 is $7 + 2 = 9$.
   • Values of 9 appear 4 times. Cumulative frequency after 9 is $9 + 4 = 13$.

   The 12th observation is between the 9th and 13th observations, so $Q_3 = 9$.

Answer: $Q_3 = 9$ (option d)

Would you like more details on any of the steps?

Here are some related questions you might explore:

1. What is the first quartile $Q_1$ for this dataset?
2. How do we calculate the median $Q_2$ from the frequency table?
3. What is the cumulative frequency for this dataset?
4. How do you calculate the interquartile range (IQR)?
5. How does the mode of a frequency distribution differ from the quartiles?

Tip: The cumulative frequency helps determine which value a particular quartile corresponds to by summing frequencies sequentially.