The median of a set of eight numbers is 4.5. Given that seven of the numbers are 7, 2, 13, 4, 8, 2 and 1, find the eight number and write down the mode of eight numbers.

Step 1: Understand the problem

The problem is asking for two things: the eighth number in a set where the median is known and seven of the numbers are given, and the mode of the set of eight numbers.

Step 2: Recall the definitions of median and mode

The median is the middle number in a sorted list of numbers. If the list has an even number of observations, the median is the average of the two middle numbers.

The mode is the number that appears most frequently in a set of numbers.

Step 3: Find the eighth number

We know that the median of the eight numbers is 4.5. Since eight is an even number, the median is the average of the fourth and fifth numbers when the numbers are arranged in ascending order.

First, let's arrange the seven given numbers in ascending order: 1, 2, 2, 4, 7, 8, 13.

We know that the median (4.5) is the average of the fourth and fifth numbers. Since the fourth number is 4, the fifth number must be 5 to make the average 4.5. Therefore, the eighth number is 5.

Step 4: Find the mode

The mode is the number that appears most frequently. In our set of eight numbers (1, 2, 2, 4, 5, 7, 8, 13), the number 2 appears twice and all other numbers appear once. Therefore, the mode of the eight numbers is 2.