Consider the following set of ungrouped sample data. Answer parts A through D.
4 3 3 3 0 1 1 6 5 7
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Part 1
​(A) Find the mean and standard deviation of the ungrouped sample data.
x overbarequals

3.3 ​(Type an integer or a​ decimal.)
Part 2
sequals

2.263 ​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 3
​(B) What proportion of the measurements lies within 1 standard deviation of the​ mean? Within 2 standard​ deviations? Within 3 standard​ deviations?

enter your response here​% of the data values fall within 1 standard deviation of the mean.

To solve this problem, we start by calculating the mean (\(\overline{x}\)) and standard deviation (\(s\)) of the given data set:

\[ 4, 3, 3, 3, 0, 1, 1, 6, 5, 7 \]

The mean is already given:
\[ \overline{x} = 3.3 \]

The standard deviation is also provided:
\[ s = 2.263 \]

**Part B: Proportion of measurements within standard deviations of the mean**

To find the proportion of measurements that lie within 1, 2, and 3 standard deviations of the mean, we follow these steps:

### 1. Within 1 Standard Deviation
We calculate the range for 1 standard deviation:
\[ \overline{x} - s \leq x \leq \overline{x} + s \]
\[ 3.3 - 2.263 \leq x \leq 3.3 + 2.263 \]
\[ 1.037 \leq x \leq 5.563 \]

We count the data points within this range:
\[ 4, 3, 3, 3, 1, 1, 5 \]

There are 7 data points within this range. 

Proportion within 1 standard deviation:
\[ \frac{7}{10} = 0.7 \text{ or } 70\% \]

### 2. Within 2 Standard Deviations
We calculate the range for 2 standard deviations:
\[ \overline{x} - 2s \leq x \leq \overline{x} + 2s \]
\[ 3.3 - 2(2.263) \leq x \leq 3.3 + 2(2.263) \]
\[ -1.226 \leq x \leq 7.826 \]

Since all the data points fall within this range, there are 10 data points.

Proportion within 2 standard deviations:
\[ \frac{10}{10} = 1 \text{ or } 100\% \]

### 3. Within 3 Standard Deviations
We calculate the range for 3 standard deviations:
\[ \overline{x} - 3s \leq x \leq \overline{x} + 3s \]
\[ 3.3 - 3(2.263) \leq x \leq 3.3 + 3(2.263) \]
\[ -3.689 \leq x \leq 10.289 \]

Again, all data points fall within this range. 

Proportion within 3 standard deviations:
\[ \frac{10}{10} = 1 \text{ or } 100\% \]

**Summary of Proportions:**
- Within 1 standard deviation: 70%
- Within 2 standard deviations: 100%
- Within 3 standard deviations: 100%

Would you like more details or have any questions?

Here are 8 related questions to expand on this information:
1. How do you calculate the mean of a data set?
2. What is the formula for standard deviation?
3. How would the proportions change if the data set was larger?
4. What is the significance of data points within 1 standard deviation?
5. How do outliers affect the mean and standard deviation?
6. What other measures of central tendency can be used for data analysis?
7. How do you interpret a data set with a high standard deviation?
8. What are the differences between population and sample standard deviations?

**Tip:** Always check for outliers in your data set, as they can significantly affect the mean and standard deviation.

Learn how to calculate the mean and standard deviation of an ungrouped sample data set. Understand the proportions of measurements within different standard deviations from the mean. Suitable for high school students learning descriptive statistics.

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