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Number of Pairs of Shoes Number of 6th Graders 2 5 3 6 4 1 5 1 6 4 7 7 8 2 10 1 Michael wants to know how many pairs of shoes 6th graders typically own. He surveys the 6th graders at his summer camp. His results are given in the table. After Michael creates the table, he realizes that he forgot to include himself. Michael owns 4 pairs of shoes. Calculate the median of the original dataset and the median of the new dataset with the data point for Michael’s 4 pairs of shoes added. (1 point)

1 answer

To calculate the median of the original dataset and then the new dataset after adding Michael's pair of shoes, we will follow these steps:

1. Original Dataset

The original data collected by Michael is as follows:

  • 2 pairs: 5 students
  • 3 pairs: 6 students
  • 4 pairs: 1 student
  • 5 pairs: 1 student
  • 6 pairs: 4 students
  • 7 pairs: 7 students
  • 8 pairs: 2 students
  • 10 pairs: 1 student

Let's list out the number of pairs of shoes owned by each student:

  • From "2 pairs: 5 students", we have: 2, 2, 2, 2, 2
  • From "3 pairs: 6 students", we have: 3, 3, 3, 3, 3, 3
  • From "4 pairs: 1 student", we have: 4
  • From "5 pairs: 1 student", we have: 5
  • From "6 pairs: 4 students", we have: 6, 6, 6, 6
  • From "7 pairs: 7 students", we have: 7, 7, 7, 7, 7, 7, 7
  • From "8 pairs: 2 students", we have: 8, 8
  • From "10 pairs: 1 student", we have: 10

Now, let's list out all the numbers: \[ 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 10 \]

2. Finding the Median of the Original Dataset

There are 27 (5 + 6 + 1 + 1 + 4 + 7 + 2 + 1) students. Since 27 is odd, the median will be the middle number, which is the 14th number in the ordered list.

Counting through the listed pairs, we find the 14th number:

  1. \(2, 2, 2, 2, 2\) (5 values)
  2. \(3, 3, 3, 3, 3, 3\) (6 values) = 11 values total
  3. \(4\) (1 value) = 12 values total
  4. \(5\) (1 value) = 13 values total
  5. \(6\) (4 values) = 17 values total

The 14th number is \(6\).

3. New Dataset (Including Michael)

We add Michael's 4 pairs of shoes: \[ 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 10 \]

Now, there are 28 students. Since 28 is even, the median is the average of the two middle numbers (14th and 15th).

Counting through the new ordered list:

  1. The 14th number is \(5\).
  2. The 15th number is \(6\).

4. Calculate the Median of the New Dataset

\[ \text{Median} = \frac{5 + 6}{2} = \frac{11}{2} = 5.5 \]

Final Answers:

  • Median of the original dataset: 6
  • Median of the new dataset (with Michael): 5.5