Mean Calculation for the Given Data

To find the difference between the mean and the median of the given data, we first need to calculate both values. The provided data set is: 21, 11, 27, 8, 5, 12, 7, 23, 3, 14, 9, 19.

The mean (average) of a data set is calculated by summing all the values and then dividing by the total number of values.

• Sum of Data: We add all the numbers in the data set:
• $$\Sigma x = 21 + 11 + 27 + 8 + 5 + 12 + 7 + 23 + 3 + 14 + 9 + 19 = 159$$
• Number of Data Points (n): Count how many numbers are in the data set:
• $$n = 12$$
• Mean Formula: The formula for the mean ($\bar{x}$) is:
• $$\text{Mean} (\bar{x}) = \frac{\text{Sum of all values}}{\text{Number of values}}$$
• Mean Calculation:
• $$\bar{x} = \frac{159}{12} = 13.25$$

So, the mean of the given data is 13.25.

Median Determination for the Data

The median is the middle value in a data set that is ordered from least to greatest. If there is an even number of data points, the median is the average of the two middle values.

Here are the steps to find the median:

• Order the Data: First, arrange the given data set in ascending order:
• 3, 5, 7, 8, 9, 11, 12, 14, 19, 21, 23, 27
• Identify Middle Values: Since there are 12 data points (an even number), the median will be the average of the 6^th and 7^th values in the ordered list.
• The 6^th value is 11.
• The 7^th value is 12.
• Median Calculation: The median is the average of these two middle values:
• $$\text{Median} = \frac{11 + 12}{2} = \frac{23}{2} = 11.5$$

Thus, the median of the given data is 11.5.

Difference Between Mean and Median

Finally, to find the difference between the mean and median, we subtract the median from the mean.

• Mean: 13.25
• Median: 11.5
• Difference Calculation:
• $$\text{Difference} = \text{Mean} - \text{Median}$$
• $$\text{Difference} = 13.25 - 11.5 = 1.75$$

The difference between the mean and median of the given data is 1.75.

This matches option 2.