Consider the dataset [3, 5, 7, 8, 9, 11, 13, 15, 17, 19]. What is the first quartile (Q1) of this data set?
Answer
Step 3: Since the position of Q1 is not an integer, we need to interpolate to find the actual value of Q1. We take the first integer part of the Q1 position (2) and the next integer (3), and find the values at these positions in the dataset (5 and 7, respectively). Then, we interpolate using the formula \(Q1 = value_{lower} + (Q1_{position} - lower_{position}) * (value_{upper} - value_{lower})\). Substituting the known values, we get \(Q1 = 5 + (2.75 - 2) * (7 - 5) = 6.5\).
Step 1 ：Step 1: First, we need to sort the dataset in ascending order. In this case, the dataset is already sorted: \(3, 5, 7, 8, 9, 11, 13, 15, 17, 19\).
Step 2 ：Step 2: Next, we calculate the position of the first quartile (Q1) using the formula \(Q1 = \frac{1}{4}(n + 1)\), where n is the number of data points. Substituting n = 10 (the number of data points in our dataset), we get \(Q1 = \frac{1}{4}(10 + 1) = 2.75\).
Step 3 ：Step 3: Since the position of Q1 is not an integer, we need to interpolate to find the actual value of Q1. We take the first integer part of the Q1 position (2) and the next integer (3), and find the values at these positions in the dataset (5 and 7, respectively). Then, we interpolate using the formula \(Q1 = value_{lower} + (Q1_{position} - lower_{position}) * (value_{upper} - value_{lower})\). Substituting the known values, we get \(Q1 = 5 + (2.75 - 2) * (7 - 5) = 6.5\).
