Find the range, the standard deviation, and the variance for the given samples. Round non-integer results to the nearest tenth.
94, 59, 62, 58, 56, 46, 86, 52, 48

Let's calculate the **range**, **standard deviation**, and **variance** for the given sample:

### Given Data:
94, 59, 62, 58, 56, 46, 86, 52, 48

---

### 1. **Range**:
The range is the difference between the maximum and minimum values in the data set.
- Maximum value = 94
- Minimum value = 46

\[
\text{Range} = 94 - 46 = 48
\]

### 2. **Variance** and **Standard Deviation**:
First, let's find the variance and then the standard deviation (which is the square root of the variance).

#### Step 1: Find the Mean (\(\mu\))
\[
\mu = \frac{94 + 59 + 62 + 58 + 56 + 46 + 86 + 52 + 48}{9} = \frac{561}{9} = 62.33
\]

#### Step 2: Find the Squared Differences from the Mean
For each number, we subtract the mean and square the result.

- \((94 - 62.33)^2 = (31.67)^2 = 1002.6\)
- \((59 - 62.33)^2 = (-3.33)^2 = 11.1\)
- \((62 - 62.33)^2 = (-0.33)^2 = 0.1\)
- \((58 - 62.33)^2 = (-4.33)^2 = 18.7\)
- \((56 - 62.33)^2 = (-6.33)^2 = 40.1\)
- \((46 - 62.33)^2 = (-16.33)^2 = 266.7\)
- \((86 - 62.33)^2 = (23.67)^2 = 560.3\)
- \((52 - 62.33)^2 = (-10.33)^2 = 106.7\)
- \((48 - 62.33)^2 = (-14.33)^2 = 205.4\)

#### Step 3: Find the Variance (\(\sigma^2\))
Variance is the average of these squared differences:
\[
\sigma^2 = \frac{1002.6 + 11.1 + 0.1 + 18.7 + 40.1 + 266.7 + 560.3 + 106.7 + 205.4}{9} = \frac{2211.7}{9} = 245.7
\]

#### Step 4: Find the Standard Deviation (\(\sigma\))
The standard deviation is the square root of the variance:
\[
\sigma = \sqrt{245.7} = 15.7
\]

---

### Final Results:
- **Range** = 48
- **Variance** = 245.7
- **Standard Deviation** = 15.7

---

Would you like further details on any of the calculations or have any questions? Here are five related questions to practice:

1. How would the range change if the sample included an additional value of 100?
2. What effect does an outlier (a value much higher or lower than the others) have on the standard deviation?
3. Calculate the mean for a new data set: 34, 76, 54, 89, 67, 23, 90.
4. If you added a value of 80 to this sample, how would it affect the mean and standard deviation?
5. Explain the difference between population variance and sample variance.

**Tip**: Standard deviation is sensitive to extreme values; a few high or low numbers can significantly affect it.

Learn how to calculate the range, variance, and standard deviation for the data set: 94, 59, 62, 58, 56, 46, 86, 52, 48. This problem demonstrates key statistical concepts and includes a step-by-step solution. Perfect for students in Grades 9-12.

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