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!