The image shows a list of numbers and asks how the median and mean change when the largest number (417) is replaced by 387. Here's a step-by-step breakdown of the solution.

The numbers given are: $260, 286, 310, 338, 342, 363, 365, 386, 403, 417$

Part (a): The Median

• The current median is the middle value of the ordered data.
• Since there are 10 numbers, the median is the average of the 5th and 6th numbers:
  $\text{Median} = \frac{342 + 363}{2} = 352.5$

Now, if 417 is replaced by 387, the new set is: $260, 286, 310, 338, 342, 363, 365, 386, 387, 403$

• The new median remains the average of the 5th and 6th numbers: $\text{New Median} = \frac{342 + 363}{2} = 352.5$

Thus, the median stays the same.

Part (b): The Mean

• The mean is the sum of all numbers divided by the total count.

• Original sum:
  $260 + 286 + 310 + 338 + 342 + 363 + 365 + 386 + 403 + 417 = 3470$
  Mean = $\frac{3470}{10} = 347.0$

• New sum after replacing 417 with 387:
  $260 + 286 + 310 + 338 + 342 + 363 + 365 + 386 + 403 + 387 = 3440$
  New mean = $\frac{3440}{10} = 344.0$

Thus, the mean decreases.

Final Answers:

• Median: Remains the same.
• Mean: Decreases.

Would you like further details or clarifications?

Here are 5 related questions you could consider:

1. How is the mode of a dataset affected by outliers?
2. How does changing multiple values affect the median?
3. What are the effects of skewness on the mean and median?
4. How does removing the highest and lowest value affect the mean and median?
5. How do you calculate the weighted mean in a dataset?

Tip: For an odd number of data points, the median is simply the middle value, but for an even number, it’s the average of the two central values.