Final answer:
The correct range of the given data set is 26, the IQR is 13.5, the median is 28.5, and there are no outliers. None of the provided answer options are correct. Calculations for these statistics are based on the ordered data values and quartile calculations.
Step-by-step explanation:
To correctly identify the range, IQR (Interquartile Range), median, and outliers for the given data set 20, 21, 25, 25, 26, 27, 30, 33, 37, 40, 42, 46, we need to calculate each of these statistical measures.
- Range: Subtract the smallest value in the data set from the largest value (46 - 20 = 26).
- To find the quartiles (which are required to calculate the IQR and to identify outliers), we need to divide the data into four equal parts after it has been sorted. With 12 values in our data set, the median (Q2) separates the data into two halves. The median of our data set is the average of the 6th and 7th values, which are 27 and 30. Therefore, median = (27 + 30) / 2 = 28.5. The lower quartile (Q1) is the median of the first half of the data, and the upper quartile (Q3) is the median of the second half of the data. Q1 is the average of the 3rd and 4th values, which are 25 and 25, so Q1 = (25 + 25) / 2 = 25. Q3 is the average of the 9th and 10th values, which are 37 and 40, so Q3 = (37 + 40) / 2 = 38.5.
- IQR: Calculate Q3 - Q1 = 38.5 - 25 = 13.5 (different from the options given).
- To identify potential outliers, we calculate the boundaries: Q1 - 1.5 * IQR and Q3 + 1.5 * IQR. This gives us 25 - 1.5 * 13.5 = 5.25 and 38.5 + 1.5 * 13.5 = 58.25. Any data values outside of this range would be considered outliers, and since all given data points are within these bounds, there are no outliers.
Based on our calculations, none of the provided options (a, b, c, d) are correct because the actual IQR is 13.5, and the median is 28.5.