For what value of x , the mode of the following data is 27? 25, 26, 27, 23,…

The mode of a set of values is whichever value occurs most often. When two or more values are tied for the highest frequency, the data has no single mode -- adding one more occurrence of one of the tied values breaks the tie and makes that value the unique mode of the updated data set.

1. List the eleven known values (excluding the blank): 25, 26, 27, 23, 27, 26, 24, 27, 26, 25, 25.

2. Count how many times each number occurs among these eleven values: 23 occurs once, 24 occurs once, 25 occurs three times, 26 occurs three times, and 27 occurs three times.

3. Before the blank is filled, 25, 26, and 27 are tied at three occurrences each, so the data has no unique mode yet.

4. For 27 to become the mode, the blank must add one more occurrence of 27, raising its count to four while 25 and 26 remain at three.

5. Since four occurrences is strictly more than three, x = 27 makes 27 the single most frequent value once it is substituted.

Substituting x = 27 back into the full list of twelve values and recounting confirms 27 appears four times while every other value appears at most three times, so 27 is indeed the mode. Trying x = 25 or x = 26 instead would raise that value's count to four and make it (not 27) the unique mode, confirming that x = 27 is the only value satisfying the given condition.