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If the mode of the following data is 6, then find the value of k. 2,4,6,7,5,6,10,6,7,2k+1,9,7,13
3
5
2
5
5
Count the frequency of each number. For 6 to be the mode, it must appear more than any other number (specifically 7, which appears 3 times). Thus, the unknown term 2k+1 must not create a new mode, and 6 must be the most frequent.
Data set: 2, 4, 6, 7, 5, 6, 10, 6, 7, 2k + 1, 9, 7, 13; Mode = 6
Mode=value with the highest frequency
Count the frequency of each number. For 6 to be the mode, it must appear more than any other number (specifically 7, which appears 3 times). Thus, the unknown term 2k+1 must not create a new mode, and 6 must be the most frequent.
Assuming 2k + 1 must be 6. In reality, 6 already appears 3 times, so 2k + 1 can be any value that does not result in 7 appearing 4 times, but typically such problems imply the variable term completes the mode frequency.
Analyze the frequencies
List the frequency of each number in the set: 2: 1, 4: 1, 6: 3, 7: 3, 5: 1, 10: 1, 9: 1, 13: 1, (2k + 1): 1.
6→3,7→3,others→1
Determine the condition for mode
The mode is given as 6. Currently, both 6 and 7 appear 3 times. For 6 to be the unique mode, 2k + 1 must be equal to 6, so that the frequency of 6 becomes 4.
2k+1=6
Solve for k
Subtract 1 from both sides and then divide by 2: 2k=6−1=5, so k=25=2.5.
k=2.5
D is correct because setting 2k + 1 = 6 makes 6 the most frequent value (occurring 4 times), resulting in k = 2.5.
Always verify if the calculated value makes the mode count unique; if 2k+1 were to equal 7, the data would become bimodal (6 and 7).
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