The following are the amounts of total fat (in grams) in different kinds of sweet treats available at the local donut shop.

17	24	24	20	15	18	15	15	18	16
25	16	16	21	23	22	22	22	17	20
What is the range for this data set?
grams

What is the standard deviation for this data set? Round your answer to the nearest tenth, if necessary.
grams

Let's calculate the range and the standard deviation for the given data set.

### Data Set:
\( 17, 24, 24, 20, 15, 18, 15, 15, 18, 16, 25, 16, 16, 21, 23, 22, 22, 22, 17, 20 \)

### Step 1: Find the Range
The range is calculated by subtracting the smallest value in the data set from the largest value.

- **Smallest value**: \( 15 \)
- **Largest value**: \( 25 \)

\[
\text{Range} = 25 - 15 = 10 \text{ grams}
\]

### Step 2: Find the Standard Deviation

1. **Find the Mean**:
   \[
   \text{Mean} (\mu) = \frac{\sum x_i}{n}
   \]
   Where \( \sum x_i \) is the sum of all data points, and \( n \) is the number of data points.
   \[
   \text{Mean} = \frac{17 + 24 + 24 + 20 + 15 + 18 + 15 + 15 + 18 + 16 + 25 + 16 + 16 + 21 + 23 + 22 + 22 + 22 + 17 + 20}{20} = \frac{366}{20} = 18.3
   \]

2. **Calculate the Variance**:
   \[
   \text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{n}
   \]
   We calculate the squared differences between each data point and the mean, then take the average of those squared differences.

3. **Calculate the Standard Deviation**:
   \[
   \text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}}
   \]

I'll compute the variance and standard deviation now.

\[
\sigma^2 = \frac{(17-18.3)^2 + (24-18.3)^2 + \dots + (20-18.3)^2}{20}
\]
\[
\sigma^2 \approx 9.41
\]
\[
\sigma \approx \sqrt{9.41} \approx 3.1 \text{ grams}
\]

### Final Answers:
- **Range**: \( 10 \) grams
- **Standard Deviation**: \( 3.1 \) grams

Would you like any more details or have any other questions?

Here are some related questions you might find interesting:

1. What is the median of the data set?
2. How would the range change if the largest value increased by 2 grams?
3. What would be the new standard deviation if an additional value of 30 grams was added?
4. How does the presence of multiple values close to the mean affect the standard deviation?
5. Can you describe the significance of the standard deviation in understanding the data set?

**Tip:** The standard deviation helps to understand how much the values in a data set deviate from the mean on average.

Learn how to calculate the range and standard deviation of total fat content in various sweet treats from a local donut shop. This problem covers descriptive statistics concepts including range and standard deviation, with a detailed step-by-step solution. Suitable for students and learners interested in data analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation