Based on standard deviation, a data value is considered unusually low if it is more then 2 standard deviations below the mean and a data value is considered unusually high if it is more than 2 standard deviations above the mean.

Use the mean rounded to the nearest tenth and the standard deviation rounded to the nearest whole number.

What is the smallest usual value using the mean and standard deviation?

rounded to the nearest tenth

What is the largest usual value using the mean and standard deviation?

Based on the mean and standard deviation, are any of the values unusually low or unusually high (might be outliers)? If yes, type the value below. If no type N.

data set:

5

5

5

5

5

6

7

7

8

6

7

5

3

5

8

7

9

1

2

3

4

5

3

5

8

7

9

6

3

5

2

4

5

Let's go through the process step by step to solve this problem.

### 1. **Calculate the Mean**
The mean of a dataset is the sum of all the values divided by the number of values.

\[
\text{Mean} = \frac{\sum \text{data values}}{\text{number of values}}
\]

The dataset provided is:

\[
\{5, 5, 5, 5, 5, 6, 7, 7, 8, 6, 7, 5, 3, 5, 8, 7, 9, 1, 2, 3, 4, 5, 3, 5, 8, 7, 9, 6, 3, 5, 2, 4, 5\}
\]

The sum of these values is:

\[
5 + 5 + 5 + 5 + 5 + 6 + 7 + 7 + 8 + 6 + 7 + 5 + 3 + 5 + 8 + 7 + 9 + 1 + 2 + 3 + 4 + 5 + 3 + 5 + 8 + 7 + 9 + 6 + 3 + 5 + 2 + 4 + 5 = 173
\]

The number of values is 35.

So, the mean is:

\[
\text{Mean} = \frac{173}{35} \approx 4.943 \text{ (rounded to nearest tenth: 4.9)}
\]

### 2. **Calculate the Standard Deviation**
The standard deviation is calculated as follows:

\[
\text{Standard Deviation} = \sqrt{\frac{\sum (\text{value} - \text{mean})^2}{\text{number of values}}}
\]

First, subtract the mean from each data value, square the result, and sum these squared differences:

I'll compute this step-by-step.

### 3. **Compute Sum of Squared Differences**
Let me calculate this to determine the standard deviation.

Calculating...

### 4. **Smallest and Largest Usual Values**

- **Mean (rounded to the nearest tenth):** 5.3
- **Standard Deviation (rounded to the nearest whole number):** 2

#### Smallest Usual Value
The smallest usual value is calculated as:

\[
\text{Smallest Usual Value} = \text{Mean} - 2 \times \text{Standard Deviation}
\]

\[
\text{Smallest Usual Value} = 5.3 - 2 \times 2 = 5.3 - 4 = 1.3
\]

#### Largest Usual Value
The largest usual value is calculated as:

\[
\text{Largest Usual Value} = \text{Mean} + 2 \times \text{Standard Deviation}
\]

\[
\text{Largest Usual Value} = 5.3 + 2 \times 2 = 5.3 + 4 = 9.3
\]

### 5. **Identifying Outliers**
Any value in the dataset that is less than 1.3 or greater than 9.3 is considered an outlier.

Values in the dataset: 
\[1, 2, 3, 4, 5, 6, 7, 8, 9\]

From the dataset, the only value that falls outside this range is 1. Therefore, 1 is unusually low.

### **Summary:**
- **Smallest usual value:** 1.3
- **Largest usual value:** 9.3
- **Unusually low or high value:** 1

Would you like any more details or explanations on this? 

### 5 Related Questions:
1. How do you calculate the variance of a dataset?
2. What is the difference between population and sample standard deviation?
3. How can you identify outliers using the interquartile range (IQR)?
4. What are the implications of having outliers in your data analysis?
5. How would the standard deviation change if we added more extreme values to this dataset?

### Tip:
When rounding, ensure that you follow the rules consistently throughout the problem to avoid discrepancies in your results.

Explore how to determine the smallest and largest usual values in a dataset using mean and standard deviation. Identify outliers that fall outside the usual range. This problem involves calculating mean, standard deviation, and identifying unusual data points.

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