Let's break down how replacing the largest data value (50,000) with 1,000,000 will affect the mean and median.

Effect on the Mean

• The mean is the average of all the data points. It is calculated by summing all the values and dividing by the number of data points.

  Original Mean Calculation: $\text{Mean} = \frac{\sum \text{Data Values}}{n}$

  When you replace the largest value (50,000) with a much larger value (1,000,000), the sum of all the data points increases dramatically. Since the mean depends directly on the sum of the values, replacing a large value with an even larger one will increase the mean significantly.

  How Much of an Effect: The mean will increase by approximately the difference between 1,000,000 and 50,000, which is: $1,000,000 - 50,000 = 950,000$ So, the mean will increase by 950,000.

Effect on the Median

• The median is the middle value in the dataset when the values are ordered from least to greatest. If there is an odd number of data points, the median is the exact middle value. If there is an even number of data points, the median is the average of the two middle values.

  Since the median depends on the position of the values rather than their magnitude, replacing the largest value (50,000) with 1,000,000 will not affect the middle values unless the data set is very close to having 50,000 as the median.

  However, in most cases (especially with large datasets), the largest value does not significantly impact the median unless the dataset is small or the distribution is skewed in such a way that the median is influenced by extreme values.

  How Much of an Effect: The median will likely remain unchanged, as it is based on the relative ordering of values rather than the absolute magnitude of the largest one. However, if the largest data point was near the median position (e.g., in the middle of the dataset), the median could shift. But for a dataset of 50 values, it's highly unlikely for the median to change by this large of an amount unless the original dataset was tightly clustered around the largest value.

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Summary of Effects:

• Mean: Will increase significantly by approximately 950,000.
• Median: Likely remains unchanged unless the data is heavily skewed or the number of data points is small and concentrated.

Would you like further clarification or more examples to see how these effects play out in specific datasets?

Here are some related questions you might find interesting:

1. How does the mean compare to the median when a dataset has extreme outliers?
2. What other measures of central tendency are there, and how do they behave with outliers?
3. How would the effect on the mean change if a smaller value (e.g., 10,000) was replaced with 1,000,000 instead?
4. What is the effect of replacing values on the standard deviation of the dataset?
5. How can you identify outliers in a dataset using both the mean and the median?

Tip: When working with data sets that contain extreme outliers, the median is often a better measure of central tendency because it is less sensitive to large changes in individual data points.