To determine which measure of center (mean or median) best represents the shape of the distribution, we first need to examine the given data:
Prices: 15, 15, 20, 30, 30, 35, 40, 40, 40, 40, 50, 50, 50
Step 1: Calculate the Mean
To calculate the mean, sum all the values and divide by the number of values:
\[ \text{Mean} = \frac{(15 + 15 + 20 + 30 + 30 + 35 + 40 + 40 + 40 + 40 + 50 + 50 + 50)}{13} \] \[ \text{Mean} = \frac{(15 + 15 + 20 + 30 + 30 + 35 + 40 + 40 + 40 + 40 + 50 + 50 + 50)}{13} = \frac{ 15 + 15 + 20 + 30 + 30 + 35 + 40 + 40 + 40 + 40 + 50 + 50 + 50 }{13} = \frac{ 50 + 20 + 30 + 35 + 40 + 40 + 40 + 50 + 50 }{13} = \frac{ 365 }{13} \approx 28.08 \]
Step 2: Calculate the Median
To find the median, we need to order the data (already ordered) and find the middle value:
Ordered prices: 15, 15, 20, 30, 30, 35, 40, 40, 40, 40, 50, 50, 50
Since there are 13 values, the median is the 7th value in the sorted list:
Median = 40
Step 3: Analyze the Distribution
Now, we can review the data to determine the shape of the distribution:
- The data has a concentration of lower values (15, 15, 20) and a larger number of higher values (40, 50).
- There is a significant gap between some of the highest and lowest prices.
Since there are several values (15, 15, and 20) that are much lower than the median and only a few high values (50, 50, 50), the distribution is skewed to the right (positive skew).
Conclusion
In skewed distributions, the median is typically a better representation of the central tendency than the mean because it is less affected by extreme values. Therefore, the best answer is:
Median; the shape of the distribution is skewed to the right.