Data Set Analysis: Mean, Median, and Mode

The question asks to identify the incorrect statement regarding the relationships between the mean, median, and mode for the given data set.

Data Set Preparation

First, list the data set and sort it in ascending order:

Data: 1, 2, 3, 5, 2, 3, 4, 6, 6, 8, 3, 4, 5

Sorted Data: 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 8

The total number of data points, $n$, is 13.

Calculations

Mean Calculation:

The mean is the sum of all values divided by the number of values.

Sum = $1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 6 + 6 + 8 = 52$

Mean = $\frac{\text{Sum}}{n} = \frac{52}{13} = 4$

Median Calculation:

The median is the middle value in a sorted data set. For $n=13$ (an odd number), the median is the $\left(\frac{n+1}{2}\right)^{th}$ value.

Median Position = $\frac{13+1}{2} = \frac{14}{2} = 7^{th}$ value

In the sorted data (1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 8), the 7th value is 4.

Median = 4

Mode Calculation:

The mode is the value that appears most frequently in the data set.

• 1 appears 1 time
• 2 appears 2 times
• 3 appears 3 times
• 4 appears 2 times
• 5 appears 2 times
• 6 appears 2 times
• 8 appears 1 time

The value 3 occurs most frequently (3 times).

Mode = 3

Option Evaluation

We have calculated: Mean = 4, Median = 4, Mode = 3.

Now, let's evaluate each option:

• Option 1: Median > Mode
  Is $4 > 3$? Yes, this statement is TRUE.
• Option 2: Mean = Median
  Is $4 = 4$? Yes, this statement is TRUE.
• Option 3: Mode < Mean
  Is $3 < 4$? Yes, this statement is TRUE.
• Option 4: Mean = Mode
  Is $4 = 3$? No, this statement is FALSE.
• Option 5 is empty.

The question asks for the INCORRECT option. Based on our analysis, the statement Mean = Mode is incorrect.