The mean for Shop A is calculated as follows:
\[
\mu_A = \frac{\sum_{i=1}^N x_i}{N} = \frac{409}{8} = 51.12
\]
The sorted data for Shop A is:
\[
[47, 48, 50, 51, 52, 53, 53, 55]
\]
To find the median, we use the formula for the rank:
\[
\text{Rank} = Q \times (N + 1) = 0.5 \times (8 + 1) = 4.5
\]
The median is then calculated using the averaging formula:
\[
Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{51 + 52}{2} = 51.5
\]
The mean for Shop B is calculated as follows:
\[
\mu_B = \frac{\sum_{i=1}^N x_i}{N} = \frac{312}{8} = 39.0
\]
The sorted data for Shop B is:
\[
[34, 36, 38, 39, 40, 40, 41, 44]
\]
To find the median, we again use the formula for the rank:
\[
\text{Rank} = Q \times (N + 1) = 0.5 \times (8 + 1) = 4.5
\]
The median is calculated using the averaging formula:
\[
Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{39 + 40}{2} = 39.5
\]
For Shop A:
- Mean: \( \mu_A = 51.12 \)
- Median: \( Q_A = 51.5 \)
Since \( \mu_A \neq Q_A \), the data distribution for Shop A is not symmetric.
For Shop B:
- Mean: \( \mu_B = 39.0 \)
- Median: \( Q_B = 39.5 \)
Since \( \mu_B \neq Q_B \), the data distribution for Shop B is not symmetric.
Based on the calculations, the appropriate measures of central tendency are:
- Median for both coffee shops because the data distribution is not symmetric.
Thus, the answer is:
\(\boxed{\text{Median for both coffee shops because the data distribution is not symmetric.}}\)