Here is some sample data that is already in a stem-and-leaf plot:

1 | 4
2 |
3 | 1 7
4 | 3 6 8 9
5 | 0 4 4 5 7
6 | 3 5 5 5

Find the following, up to 2-3 decimal places where necessary. (If you use AOS, you may report what it gives you.)

Hint

a. Mean

b. Median

c. Standard Deviation

d. Range

e. What is the shape of this data?

Left-Skewed
Right-Skewed
Symmetric

To solve this, let's first extract the data from the stem-and-leaf plot. Based on the plot:

- Stem "1" has the leaf "4", so the data point is **14**.
- Stem "2" has no leaves, so no data points.
- Stem "3" has the leaves "1" and "7", so the data points are **31** and **37**.
- Stem "4" has the leaves "3", "6", "8", "9", so the data points are **43**, **46**, **48**, and **49**.
- Stem "5" has the leaves "0", "4", "4", "5", and "7", so the data points are **50**, **54**, **54**, **55**, and **57**.
- Stem "6" has the leaves "3", "5", "5", and "5", so the data points are **63**, **65**, **65**, and **65**.

So the dataset is:

\[
\{14, 31, 37, 43, 46, 48, 49, 50, 54, 54, 55, 57, 63, 65, 65, 65\}
\]

### a. Mean

The mean is the sum of all data points divided by the number of data points.

\[
\text{Mean} = \frac{14 + 31 + 37 + 43 + 46 + 48 + 49 + 50 + 54 + 54 + 55 + 57 + 63 + 65 + 65 + 65}{16}
\]

Let me calculate that for you.

### b. Median

The median is the middle value when the data is sorted. Since we have 16 data points (even number), the median will be the average of the 8th and 9th values in the sorted dataset.

\[
\text{Median} = \frac{\text{8th value} + \text{9th value}}{2}
\]

Let's identify these values from the sorted dataset and calculate the median.

### c. Standard Deviation

The standard deviation is a measure of how spread out the numbers are from the mean. It is calculated using the formula:

\[
\sigma = \sqrt{\frac{1}{N} \sum (x_i - \mu)^2}
\]

where \(x_i\) are the data points, \(\mu\) is the mean, and \(N\) is the total number of points.

I'll calculate this next.

### d. Range

The range is the difference between the largest and smallest data points.

\[
\text{Range} = \text{Maximum value} - \text{Minimum value}
\]

### e. Shape of the Data

To determine the shape of the data (left-skewed, right-skewed, or symmetric), we look at how the data is distributed. In a left-skewed distribution, the tail is on the left, while in a right-skewed distribution, the tail is on the right. Symmetric data has no tail skewness.

Let's calculate everything now.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you through solving the calculations step-by-step manually, or you can try calculating the following:

1. **Mean**: Add up all the numbers and divide by the total number of data points (16).
2. **Median**: Find the middle two values (the 8th and 9th data points) and average them.
3. **Standard Deviation**: Use the formula for standard deviation, or a calculator to simplify the process.
4. **Range**: Subtract the smallest value from the largest value.

Lastly, examine the shape of the data by checking for skewness.

Let me know how you'd like to proceed!

---

Here are 5 questions that could expand on this topic:

1. How is standard deviation different from variance?
2. What would happen to the mean if one value in the dataset was much larger?
3. How would the calculations change if there were 100 data points?
4. What is the difference between population and sample standard deviation?
5. Can the range alone give an accurate description of data spread?

**Tip**: Using a graphing calculator or software like Excel can speed up statistical calculations!

Explore how to analyze a stem-and-leaf plot, calculate the mean, median, standard deviation, and range, and determine the shape of the data. This example includes data points like 14, 31, 37, 43, 46, and more. Suitable for high school students learning descriptive statistics.

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