In a neighborhood donut shop, one type of donut has 470 calories, five types of donuts have 480 calories, four types of donuts have 350 calories, six types of donuts have 500 calories, and seven types of donuts have 520 calories.

Find the range.

 calories

Find the standard deviation. Round your answer to the nearest tenth, if necessary.

 calories

To find the **range** and **standard deviation** of the calories in the different types of donuts, we need to first summarize the data and then perform the necessary calculations.

### Step 1: Summarize the Data
We have the following calories and counts of donut types:

- 1 type with 470 calories
- 5 types with 480 calories
- 4 types with 350 calories
- 6 types with 500 calories
- 7 types with 520 calories

### Step 2: List All the Calorie Values
First, list each donut's calories based on the given counts:

- \( 470 \)
- \( 480, 480, 480, 480, 480 \) (five times)
- \( 350, 350, 350, 350 \) (four times)
- \( 500, 500, 500, 500, 500, 500 \) (six times)
- \( 520, 520, 520, 520, 520, 520, 520 \) (seven times)

Now the full list of calories is:
\[
\{470, 480, 480, 480, 480, 480, 350, 350, 350, 350, 500, 500, 500, 500, 500, 500, 520, 520, 520, 520, 520, 520, 520\}
\]

### Step 3: Find the Range
The range is the difference between the maximum and minimum values in the list.

- Minimum value = 350
- Maximum value = 520

**Range** = Maximum - Minimum = \( 520 - 350 = 170 \) calories.

### Step 4: Calculate the Standard Deviation
To calculate the standard deviation, we will use the formula for the standard deviation of a sample:

\[
\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N}}
\]

where:
- \( x_i \) are the individual calorie values
- \( \bar{x} \) is the mean (average) of the calories
- \( N \) is the total number of data points

1. **Calculate the mean \(\bar{x}\)**:
   
   \[
   \bar{x} = \frac{\text{Sum of all values}}{\text{Number of values}}
   \]

2. **Find each deviation from the mean, square it, and sum them up**.
   
3. **Divide by the total number of data points and take the square root**.

Let's calculate this step-by-step.### Results:
1. **Mean (Average) Calories**: \( \approx 474.3 \) calories
2. **Standard Deviation**: \( 59.2 \) calories (rounded to the nearest tenth)

### Summary:
- **Range**: 170 calories
- **Standard Deviation**: 59.2 calories

Would you like to see the detailed steps of these calculations or have any other questions? Here are some related questions:

1. How is the mean affected if a higher-calorie donut type is introduced?
2. What would happen to the range if the lowest-calorie donut type is removed?
3. How does the standard deviation change if all donut types have equal calories?
4. What is the impact on standard deviation if the number of donut types doubles?
5. How would the calculations change if calories are measured in kilojoules instead?

**Tip**: Understanding the distribution of values (like calories) helps in identifying variations, which is key in quality control and dietary assessments!

In a neighborhood donut shop, one type of donut has 470 calories, five types of donuts have 480 calories, four types of donuts have 350 calories, six types of donuts have 500 calories, and seven types of donuts have 520 calories. Find the range. Find the standard deviation. Round your answer to the nearest tenth, if necessary.

Learn how to find the range and standard deviation of calorie counts in different types of donuts from a neighborhood shop. This problem involves calculating the range (170 calories) and standard deviation (59.2 calories) based on data for various donut types. Suitable for Grades 8-10 students.

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