Questions: For the data set 7, 15, 8, 9, 8, 4, 7, 26, 5, 11, 10, 14, 5, 6, 4, 13, 3, 6, 11, 7, 9, 12, Part 1 of 4 (a) Find the first and third quartiles.

Solution Steps

The given dataset is sorted to facilitate the calculation of quartiles. The sorted data is:

\[ [3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, 14, 15, 26] \]

To find the first quartile \( Q_1 \), we use the formula for the rank:

\[ \text{Rank} = Q \times (N + 1) = 0.25 \times (22 + 1) = 5.75 \]

Since the rank is not an integer, we take the average of the values at ranks 5 and 6:

\[ Q_1 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{5 + 6}{2} = 5.5 \]

To find the third quartile \( Q_3 \), we use the formula for the rank:

\[ \text{Rank} = Q \times (N + 1) = 0.75 \times (22 + 1) = 17.25 \]

Again, since the rank is not an integer, we take the average of the values at ranks 17 and 18:

\[ Q_3 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{11 + 12}{2} = 11.5 \]

Final Answer