To determine the relationship between the mean and the median of the data and interpret what it reveals about the shape of the data, we will perform the following steps:
1. Calculate the Mean:
- The mean (average) is calculated by summing all the values and then dividing by the number of values.
2. Calculate the Median:
- The median is the middle value in a data set that is ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle numbers.
3. Compare the Mean and Median:
- By comparing the mean to the median, we can infer whether the data distribution is symmetrical, skewed to the left, or skewed to the right.
Given the data:
[tex]\[
10, 5, 8, 10, 12, 6, 8, 10, 15, 6, 12, 18
\][/tex]
### Step-by-Step Solution
1. Calculate the Mean:
[tex]\[
\text{Mean} = \frac{10 + 5 + 8 + 10 + 12 + 6 + 8 + 10 + 15 + 6 + 12 + 18}{12} = \frac{120}{12} = 10.0
\][/tex]
2. Calculate the Median:
- First, order the data from least to greatest:
[tex]\[
5, 6, 6, 8, 8, 10, 10, 10, 12, 12, 15, 18
\][/tex]
- Since there are 12 data points (an even number), the median is the average of the 6th and 7th values in the ordered list:
- The 6th value is 10.
- The 7th value is 10.
[tex]\[
\text{Median} = \frac{10 + 10}{2} = 10.0
\][/tex]
3. Compare the Mean and Median:
- The mean is 10.0 and the median is also 10.0.
Since the mean is equal to the median, it indicates that the data distribution is symmetrical.
Conclusion:
The relationship between the mean and the median reveals that the data is symmetrical. Thus, the correct interpretation is:
[tex]\[
\textbf{The mean is equal to the median, so the data is symmetrical.}
\][/tex]
