Let's analyze the given data and determine the relationship between the mean and the median, which will reveal the shape of the data.
1. Listing the Data:
The budgets for lunch recorded by 12 teachers at the school are:
[tex]\[10, 5, 8, 10, 12, 6, 8, 10, 15, 6, 12, 18\][/tex]
2. Calculating the Mean:
To find the mean, we sum up all the budgets and then divide by the number of budgets. The calculation is as follows:
[tex]\[ \text{Mean} = \frac{10 + 5 + 8 + 10 + 12 + 6 + 8 + 10 + 15 + 6 + 12 + 18}{12} = \frac{120}{12} = 10.0 \][/tex]
3. Calculating the Median:
To find the median, we first need to sort the data in ascending order:
[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 will be the average of the 6th and 7th values in the ordered list. The 6th and 7th values are both 10:
[tex]\[ \text{Median} = \frac{10 + 10}{2} = 10.0 \][/tex]
4. Comparing Mean and Median:
- The mean of the data is [tex]\(10.0\)[/tex].
- The median of the data is [tex]\(10.0\)[/tex].
Since the mean and median are equal, this indicates that the data is symmetrical.
Conclusion:
The relationship between the mean and the median reveals the shape of the data. Specifically, when the mean is equal to the median, it suggests that the data is symmetrical.
Therefore, the correct conclusion is:
[tex]\[ \text{The mean is equal to the median, so the data is symmetrical.} \][/tex]
