1. Range Calculation

The range of a data set is calculated by subtracting the minimum value from the maximum value.

• Maximum value: 25 grams
• Minimum value: 15 grams

$\text{Range} = \text{Maximum value} - \text{Minimum value} = 25 - 15 = 10 \text{ grams}$

So, the range is 10 grams.

2. Standard Deviation Calculation

To calculate the standard deviation, we use the following steps:

Step 1: Find the mean (average)

$\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}$

Sum of all data points: $23 + 21 + 16 + 20 + 24 + 20 + 19 + 22 + 24 + 22 + 16 + 20 + 20 + 24 + 21 + 22 + 25 + 15 + 16 + 25 = 425 \text{ grams}$

Number of data points: 20

$\text{Mean} = \frac{425}{20} = 21.25 \text{ grams}$

Step 2: Calculate the squared differences from the mean

$\text{Squared difference} = (x_i - \text{Mean})^2$ where $x_i$ represents each data point.

We calculate this for each data point:

$(23 - 21.25)^2 = 3.0625$ $(21 - 21.25)^2 = 0.0625$ $(16 - 21.25)^2 = 27.5625$ $\ldots$ (Calculations continue for all data points)

Step 3: Find the average of the squared differences

$\text{Variance} = \frac{\text{Sum of squared differences}}{\text{Number of data points}}$

Step 4: Take the square root of the variance to find the standard deviation

Let me calculate these values for you.### Results:

1. Mean: 20.75 grams
2. Variance: 9.19 grams² (rounded to two decimal places)
3. Standard Deviation: 3.0 grams (rounded to the nearest tenth)

Final Answers:

• Range: 10 grams
• Standard Deviation: 3.0 grams

Would you like to know more details or have any questions?

Related Questions:

1. What is the variance if you used the sample formula instead of the population formula?
2. How would the standard deviation change if the highest value was 30 grams instead of 25 grams?
3. What does a low standard deviation indicate about the data set?
4. How would adding more data points at the mean affect the standard deviation?
5. What is the impact of outliers on the range and standard deviation?

Tip:

When calculating the standard deviation for a sample rather than an entire population, remember to divide by $n - 1$ instead of $n$ when finding the variance.