The numbers of trading cards owned by 10 middle-school students are given below.

(Note that these are already ordered from least to greatest.)
294, 402, 429, 437, 458, 486, 517, 533, 551, 573

Suppose that the number 573 from this list changes to 403. Answer the following.
(A) what happens to the median

(B) what happens to the mean

1 answer

Let's first analyze the provided data set and see the necessary calculations before and after the change.

Original Data Set: 294, 402, 429, 437, 458, 486, 517, 533, 551, 573

  1. Median:
  • The median is the middle value of an ordered data set. In a set with an even number of values, the median is the average of the two middle values.
  • Here, there are 10 data points. The two middle values are the 5th and 6th values (458 and 486).

\[ \text{Median} = \frac{458 + 486}{2} = \frac{944}{2} = 472 \]

  1. Mean:
  • The mean is calculated by adding all the numbers and dividing by the count of the numbers.

\[ \text{Sum of the original data} = 294 + 402 + 429 + 437 + 458 + 486 + 517 + 533 + 551 + 573 = 4580 \]

\[ \text{Mean} = \frac{4580}{10} = 458.0 \]

New Data Set (with 573 changed to 403): 294, 402, 403, 429, 437, 458, 486, 517, 533, 551

  1. New Median:
  • Now, we find the new median with the updated data set. The two middle values remain the same indices in a sorted list of 10 values (5th and 6th).

The 5th and 6th values are now 437 and 458.

\[ \text{New Median} = \frac{437 + 458}{2} = \frac{895}{2} = 447.5 \]

  1. New Mean:
  • We'll calculate the new mean with the updated data set.

\[ \text{Sum of the new data} = 294 + 402 + 403 + 429 + 437 + 458 + 486 + 517 + 533 + 551 = 4510 \]

\[ \text{New Mean} = \frac{4510}{10} = 451.0 \]

Now let's answer the questions:

(A) What happens to the median?

  • The original median was 472, and the new median is 447.5. So the median decreases.

(B) What happens to the mean?

  • The original mean was 458.0, and the new mean is 451.0. So the mean also decreases.