Let's analyze the data collected by Colin to understand the relationship between the mean and the median, which will allow us to infer the shape of the data distribution.
1. Collect the data:
[tex]\[
10, 5, 8, 10, 12, 6, 8, 10, 15, 6, 12, 18
\][/tex]
2. Calculate the mean:
The mean (average) is calculated by summing all the values and then dividing by the number of values.
[tex]\[
\text{Mean} = \frac{10 + 5 + 8 + 10 + 12 + 6 + 8 + 10 + 15 + 6 + 12 + 18}{12}
\][/tex]
[tex]\[
\text{Mean} = \frac{120}{12} = 10.0
\][/tex]
3. Calculate the median:
The median is the middle value of the dataset when it is ordered from smallest to largest. For an even number of observations, it is the average of the two middle numbers.
- First, we need to order the data:
[tex]\[
5, 6, 6, 8, 8, 10, 10, 10, 12, 12, 15, 18
\][/tex]
- The dataset has 12 values, so the median will be the average of the 6th and 7th values:
[tex]\[
\text{Median} = \frac{10 + 10}{2} = 10.0
\][/tex]
4. Determine the shape of the data:
We compare the mean and the median:
- If the mean < median, the data is skewed left.
- If the mean > median, the data is skewed right.
- If the mean = median, the data is symmetrical.
In this case:
[tex]\[
\text{Mean} = 10.0 \quad \text{and} \quad \text{Median} = 10.0
\][/tex]
Since the mean is equal to the median, this indicates that the data distribution is symmetrical.
Thus, the relationship between the mean and the median reveals that:
[tex]\[
\boxed{\text{The mean is equal to the median, so the data is symmetrical.}}
\][/tex]
