The problem shown in the image asks about the effect of eliminating the highest value, $10$, from a data set on both the mean and the median. The data set provided is:

$\{6, 3, 3, 3, 3, 5, 6, 6, 8, 10\}$

Let's analyze both the mean and median before and after removing the highest value, $10$.

1. Mean Calculation

• Original data set: $\{6, 3, 3, 3, 3, 5, 6, 6, 8, 10\}$
• Sum of values: $6 + 3 + 3 + 3 + 3 + 5 + 6 + 6 + 8 + 10 = 53$
• Number of values (n): 10
• Mean: $\frac{53}{10} = 5.3$

Now, let's remove the highest value, $10$.

• New data set: $\{6, 3, 3, 3, 3, 5, 6, 6, 8\}$
• Sum of new values: $6 + 3 + 3 + 3 + 3 + 5 + 6 + 6 + 8 = 43$
• Number of values (n): 9
• New mean: $\frac{43}{9} \approx 4.78$

2. Median Calculation

• Original data set (sorted): $\{3, 3, 3, 3, 5, 6, 6, 6, 8, 10\}$
• Since there are 10 values, the median is the average of the 5th and 6th values: $\text{Median} = \frac{5 + 6}{2} = 5.5$

Now, after removing $10$:

• New data set (sorted): $\{3, 3, 3, 3, 5, 6, 6, 6, 8\}$
• There are now 9 values, so the median is the 5th value in the list: $\text{Median} = 5$

3. Conclusion

• Effect on the mean: The mean decreases from 5.3 to approximately 4.78 after removing $10$.
• Effect on the median: The median decreases from 5.5 to 5.

Would you like a more detailed breakdown or explanation of these calculations?

5 Related Questions:

1. How would the mean and median change if the lowest value were removed instead?
2. What is the effect of removing the second highest value on the mean and median?
3. How does the range of the data set change when the highest value is removed?
4. What would happen to the mean if we removed multiple values, not just the highest one?
5. Can we predict the change in median without recalculating the entire data set?

Tip:

When a value far from the mean (like an outlier) is removed, it can significantly shift both the mean and the median, but more so for the mean.