To construct a set of six pieces of data with a mean, median, and midrange of 67, where no two pieces of data are the same, you can use the following values:
Smallest Value: 50
Second Smallest Value: 60
Middle Value (Median): 67
Second Largest Value: 74
Largest Value: 84
Midrange: [(50 + 84) / 2] = 67
In this set, all the values are distinct, and their mean, median, and midrange are all equal to 67.