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.