Question

Here are the prices for
biweekly lawn mowing
services for 13 local
companies:
15, 15, 20, 30, 30, 35, 40,
40, 40, 40, 50, 50, 50
Which measure of center—
mean or median—best
represents the shape of the
distribution?
Median; the shape
of the distribution is
skewed to the right.
Median; the shape
of the distribution is
skewed to the left.
Mean; the shape of
the distribution is
skewed to the left.
Mean; the shape of
the distribution is
close to
symmetrical.

Answer 1

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.**