Each​ year, tornadoes that touch down are recorded. The following table gives the number of tornadoes that touched down during each month of one year. Determine the range and sample standard deviation.
4
2
52
114
198
95
66
86
72
60
103
92

To solve for the range and sample standard deviation, we will follow these steps:

### **Step 1: Calculate the Range**
The range is the difference between the maximum and minimum values in the dataset.

- Minimum value: \( 2 \)
- Maximum value: \( 198 \)

\[
\text{Range} = 198 - 2 = 196
\]

### **Step 2: Calculate the Sample Standard Deviation**

The formula for sample standard deviation (\( s \)) is:

\[
s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}
\]

Where:
- \( x_i \) represents each data point,
- \( \bar{x} \) is the sample mean,
- \( n \) is the sample size.

#### **Step 2.1: Find the Mean**
First, we calculate the mean (\( \bar{x} \)) of the dataTo solve for the range and sample standard deviation, we will follow these steps:

### **Step 1: Calculate the Range**
The range is the difference between the maximum and minimum values in the dataset.

- Minimum value: \( 2 \)
- Maximum value: \( 198 \)

\[
\text{Range} = 198 - 2 = 196
\]

### **Step 2: Calculate the Sample Standard Deviation**

The formula for sample standard deviation (\( s \)) is:

\[
s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}
\]

Where:
- \( x_i \) represents each data point,
- \( \bar{x} \) is the sample mean,
- \( n \) is the sample size.

#### **Step 2.1: Find the Mean**
First, we calculate the mean (\( \bar{x} \)) of the 

\[
\bar{x} = \frac{4 + 2 + 52 + 114 + 198 + 95 + 66 + 86 + 72 + 60 + 103 + 92}{12}
\]

Let's compute the mean:

\[
\bar{x} = \frac{944}{12} = 78.67
\]

#### **Step 2.2: Find the Squared Differences**
Now, we calculate the squared differences between each data point and the mean:

\[
(4 - 78.67)^2 = 5585.35
\]
\[
(2 - 78.67)^2 = 5886.35
\]
\[
(52 - 78.67)^2 = 711.89
\]
\[
(114 - 78.67)^2 = 1210.89
\]
\[
(198 - 78.67)^2 = 14316.69
\]
\[
(95 - 78.67)^2 = 263.39
\]
\[
(66 - 78.67)^2 = 163.39
\]
\[
(86 - 78.67)^2 = 53.79
\]
\[
(72 - 78.67)^2 = 44.49
\]
\[
(60 - 78.67)^2 = 351.29
\]
\[
(103 - 78.67)^2 = 589.69
\]
\[
(92 - 78.67)^2 = 177.69
\]

#### **Step 2.3: Compute the Sum of Squared Differences**
Next, we sum these squared differences:

\[
\text{Sum} = 5585.35 + 5886.35 + 711.89 + 1210.89 + 14316.69 + 263.39 + 163.39 + 53.79 + 44.49 + 351.29 + 589.69 + 177.69
\]
\[
\text{Sum} = 29054.9
\]

#### **Step 2.4: Divide by \( n - 1 \)**
Since this is a sample, we divide by \( n - 1 = 12 - 1 = 11 \):

\[
\frac{29054.9}{11} = 2641.35
\]

#### **Step 2.5: Take the Square Root**
Finally, we take the square root to find the sample standard deviation:

\[
s = \sqrt{2641.35} \approx 51.39
\]

### **Final Answers:**
- **Range:** 196
- **Sample Standard Deviation:** 51.39

Would you like more details or have any questions about these calculations?

---

Here are 5 relative questions that expand this information:
1. How does the range affect the spread of the data?
2. What is the difference between sample standard deviation and population standard deviation?
3. How would the standard deviation change if the maximum value increased?
4. Can you interpret what a high standard deviation means in this context?
5. How does the sample size influence the accuracy of the standard deviation?

**Tip:** Always verify whether you're calculating the sample or population standard deviation, as this changes the divisor in the formula.

Each year, tornadoes that touch down are recorded. The following table gives the number of tornadoes that touched down during each month of one year. Determine the range and sample standard deviation.
4
2
52
114
198
95
66
86
72
60
103
92

Learn how to determine the range and sample standard deviation from a dataset of tornado touchdowns recorded over one year. This problem covers key statistical concepts like calculating the range and sample standard deviation, with step-by-step solutions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation