se the magnitudes​ (Richter scale) of the 120120 earthquakes listed in the accompanying data table. Use technology to find the​ range, variance, and standard deviation. If another​ value, 8.008.00​, is added to those listed in the data​ set, do the measures of variation change​ much?
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Part 1
Without the extra data​ value, the range is 3.530
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 2
Without the extra data​ value, the standard deviation is 0.652.
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 3
Without the extra data​ value, the variance is 0.425.
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 4
With the extra data​ value, the range is enter your response here.
​(Type an integer or decimal rounded to three decimal places as​ needed.)

Let's address each part systematically, especially focusing on how the additional data value affects the range, variance, and standard deviation.

### Part 1: Range without the extra data value
- **Range** is the difference between the maximum and minimum values in the dataset. Without the extra data value, the range has already been given as **3.530**.

### Part 2: Standard Deviation without the extra data value
- **Standard deviation** measures the dispersion of the dataset values from the mean. It is given as **0.652** without the additional data.

### Part 3: Variance without the extra data value
- **Variance** is the square of the standard deviation, given as **0.425** without the additional data.

### Part 4: Adding the extra data value (8.00)

Now, if the extra value (8.00) is added to the dataset, we need to see how the range, variance, and standard deviation change.

1. **Range with the extra data value:**
   If the original maximum value in the dataset is less than 8.00, adding 8.00 will increase the range. However, if the maximum value was already greater than or equal to 8.00, the range will remain the same. 

   Without knowing the original maximum value, if we assume that adding 8.00 increases the maximum, the new range would be:
   \[
   \text{New range} = 8.00 - (\text{Minimum value})
   \]

   Since the original range is 3.530, the maximum value before adding 8.00 would have been approximately \(3.530 + \text{minimum value}\). If adding 8.00 changes this maximum value, the new range will be:

   \[
   \text{New range} = 8.00 - \text{minimum value}
   \]

### Does the measures of variation change much?
- **Standard deviation** and **variance** may slightly increase due to the addition of an outlier value (8.00), which might push the data further from the mean. However, whether the change is significant depends on the distribution of the remaining values.

Would you like to go over how this change affects the variance or standard deviation specifically?

### Follow-up questions:
1. How is the standard deviation calculated in a dataset?
2. What happens to the variance when an outlier is introduced to a dataset?
3. Could you explain why adding an extreme value like 8.00 might or might not affect the range?
4. What steps can we take to calculate the variance with the new data point added?
5. How does the shape of the distribution affect the standard deviation?

**Tip:** Outliers often have the most noticeable impact on range and variance. The more extreme the outlier, the greater the potential shift.

Use the magnitudes (Richter scale) of the 120 earthquakes listed in the accompanying data table. Use technology to find the range, variance, and standard deviation. If another value, 8.00, is added to those listed in the data set, do the measures of variation change much?

This page covers how to calculate the range, variance, and standard deviation of a dataset with 120 earthquakes. It also explores how the measures of variation change when an additional value (8.00) is added. Learn how outliers affect these statistical measures. Suitable for high school students in Grades 10-12.

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