Look at this set of 9 numbers:

Question

154944159578661987
Which value would change the most if the number 76 replaced the number 49 in the set?
mean,median,mode

Answer 1

To analyze how everything would change if the number 49 was replaced by the number 76 in the set of numbers, we first look at the original set: 154944159578661987.

1. Original Set:
   • 1
   • 5
   • 4
   • 9
   • 4
   • 4
   • 1
   • 5
   • 7
   • 8
   • 6
   • 6
   • 1
   • 9
   • 8
   • 7

Here the numbers can be extracted as:

• 1, 5, 4, 9, 4, 4, 1, 5, 7, 8, 6, 6, 1, 9, 8, 7.

2. Finding Mean, Median, Mode:
   • Mean: Sum of all values ÷ Total count of values.
   • Median: The middle value when sorted in order.
   • Mode: The value that appears most often.

Current Calculations:

1. Sum: \(1 + 5 + 4 + 9 + 4 + 4 + 1 + 5 + 7 + 8 + 6 + 6 + 1 + 9 + 8 + 7 = 81\)

2. Count: 16

3. Mean: \(\frac{81}{16} = 5.0625\)

4. Sorted Values:

   • 1, 1, 1, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9
   • Median: The average of the 8th and 9th values \(= \frac{5 + 6}{2} = 5.5\)

5. Mode:

   • The number that appears most often is 4 (appears 3 times) and 1 (appears 3 times).

Now if we replace one of the instances of 49 (4,9) in the original set with 76 (7,6), the set becomes:

New Set:

• 1, 5, 7, 6, 7, 4, 4, 1, 5, 7, 8, 6, 6, 1, 9, 8, 7.

New Calculations:

1. New Sum: \(1 + 5 + 7 + 6 + 7 + 4 + 4 + 1 + 5 + 7 + 8 + 6 + 6 + 1 + 9 + 8 + 7 = 88\)

2. New Mean: \(\frac{88}{16} = 5.5\)

3. Sorted Values:

   • 1, 1, 1, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9
   • New Median: The average of the 8th and 9th values \(= \frac{6 + 7}{2} = 6.5\)

4. New Mode:

   • The number that appears most often is now 7 (appears 3 times).

Summary of Changes:

• Mean: Changed from 5.0625 to 5.5 (0.4375 increase).
• Median: Changed from 5.5 to 6.5 (1 increase).
• Mode: The mode changed from 4 (and 1) to 7.

Conclusion:

The value that changes the most when replacing 49 with 76 in this set is mode, as it transited from two values to one distinct value which is 7.