To solve the given problem, let's start by interpreting the stem-and-leaf plot to identify the data values. After listing all the data, we'll compute the required statistics:

Step 1: List the Data from the Stem-and-Leaf Plot

The stem-and-leaf plot represents the following numbers:

• 1 | 6: 16
• 2 | : (No values)
• 3 | 7 7: 37, 37
• 4 | 0 1 1 3: 40, 41, 41, 43
• 5 | 0 0 3 9 9: 50, 50, 53, 59, 59
• 6 | 0 3 6 8: 60, 63, 66, 68

So, the data set is:
16, 37, 37, 40, 41, 41, 43, 50, 50, 53, 59, 59, 60, 63, 66, 68

Step 2: Calculate the Mean

The mean (average) is calculated by dividing the sum of all data points by the total number of data points.

$\text{Mean} = \frac{\sum \text{Data}}{n}$

Let's calculate the mean.

Step 3: Calculate the Median

To find the median, we need to locate the middle value(s) of the sorted data. Since there are 16 data points (an even number), the median will be the average of the 8th and 9th values in the sorted data.

Step 4: Calculate the Standard Deviation

The standard deviation measures the spread of the data around the mean. For a sample, it is calculated as:

$\sigma = \sqrt{\frac{\sum (x_i - \text{Mean})^2}{n}}$

Where:

• $x_i$ are the data points,
• Mean is the average,
• $n$ is the number of data points.

Step 5: Calculate the Range

The range is the difference between the maximum and minimum values of the data set:

$\text{Range} = \text{Maximum Value} - \text{Minimum Value}$

Step 6: Determine the Shape of the Data

To determine the shape, we analyze the distribution of data.

Let's compute all of these values step by step.Here are the calculated values based on the given dataHere are the calculated values based on the given

a. Mean: 48.938
b. Median: 50.0
c. Standard Deviation: 13.245
d. Range: 52

Step 7: Determine the Shape of the Data

To determine the shape of the data, we can look at the distribution:

• The data is somewhat symmetric around the median (50), with values spreading almost equally on both sides.
• There is no clear skew to the right or left.

e. Shape of the Data: Symmetric.

Would you like any further details or have additional questions?

Additional Questions for Further Exploration:

1. How would the values change if an outlier were added to the data set?
2. What is the interquartile range (IQR) of this data?
3. How do you interpret the standard deviation in the context of this data?
4. What would the effect on the mean and median be if the smallest value were removed?
5. How would a different shape (e.g., right-skewed) impact the mean and median?

Tip: Knowing the shape of data helps in choosing the right statistical methods for analysis.