Question

find the median of:
20 12 37 32 33 31 14 38
6 7 39 23 8
median = \boxed{}

Explanation:

Step1: Order the data set

First, we need to arrange the given numbers in ascending order. The numbers are: 20, 12, 37, 32, 33, 31, 14, 38, 6, 7, 39, 23, 8.
Arranging them from smallest to largest: 6, 7, 8, 12, 14, 20, 23, 31, 32, 33, 37, 38, 39. Wait, wait, let's count the number of elements. Wait, original data: first row: 20,12,37,32,33,31,14,38 (8 numbers), second row: 6,7,39,23,8 (5 numbers). So total numbers: 8 + 5 = 13 numbers. Wait, I must have miscounted earlier. Let's list all numbers: 20, 12, 37, 32, 33, 31, 14, 38, 6, 7, 39, 23, 8. Now let's sort them:
6, 7, 8, 12, 14, 20, 23, 31, 32, 33, 37, 38, 39. Wait, no, wait: 6,7,8,12,14,20,23,31,32,33,37,38,39? Wait, 6,7,8,12,14,20,23,31,32,33,37,38,39. Wait, let's check each number: 6,7,8,12,14,20,23,31,32,33,37,38,39. Wait, that's 13 numbers? Wait, original data: first row: 20,12,37,32,33,31,14,38 (8 numbers), second row: 6,7,39,23,8 (5 numbers). 8 + 5 = 13. So when sorted, the 13 numbers in ascending order: 6, 7, 8, 12, 14, 20, 23, 31, 32, 33, 37, 38, 39. Wait, no, wait: 6,7,8,12,14,20,23,31,32,33,37,38,39? Wait, 31,32,33,37,38,39. Wait, maybe I made a mistake. Let's list all numbers again:

Original numbers: 20, 12, 37, 32, 33, 31, 14, 38, 6, 7, 39, 23, 8.

Let's sort them:

Start with the smallest: 6, then 7, then 8, then 12, then 14, then 20, then 23, then 31, then 32, then 33, then 37, then 38, then 39. Wait, that's 13 numbers? Wait, 6 (1), 7 (2), 8 (3), 12 (4), 14 (5), 20 (6), 23 (7), 31 (8), 32 (9), 33 (10), 37 (11), 38 (12), 39 (13). Yes, 13 numbers.

Step2: Find the median position

For a data set with \( n \) elements, the median is the middle value when \( n \) is odd. The formula for the position of the median is \( \frac{n + 1}{2} \)th term. Here, \( n = 13 \), so the position is \( \frac{13 + 1}{2} = 7 \)th term.

Step3: Identify the median

Now, we look at the 7th term in the sorted data set. The sorted data set is: 6, 7, 8, 12, 14, 20, 23, 31, 32, 33, 37, 38, 39.

Counting the terms: 1:6, 2:7, 3:8, 4:12, 5:14, 6:20, 7:23. Wait, no, wait, wait, I think I made a mistake in sorting. Wait, original numbers: 20,12,37,32,33,31,14,38,6,7,39,23,8. Let's list all numbers again:

20, 12, 37, 32, 33, 31, 14, 38, 6, 7, 39, 23, 8.

Let's sort them correctly:

6, 7, 8, 12, 14, 20, 23, 31, 32, 33, 37, 38, 39. Wait, 23 is after 20, then 31, 32, 33, 37, 38, 39. So the 7th term is 23? Wait, no, wait: 1:6, 2:7, 3:8, 4:12, 5:14, 6:20, 7:23, 8:31, 9:32, 10:33, 11:37, 12:38, 13:39. Yes, so the 7th term is 23? Wait, but wait, maybe I miscounted the number of elements. Wait, first row: 8 numbers (20,12,37,32,33,31,14,38), second row: 5 numbers (6,7,39,23,8). 8 + 5 = 13. So n=13, which is odd. So median is the (13+1)/2 = 7th term. So in the sorted list, the 7th term is 23? Wait, no, wait, let's check again. Wait, 6 (1), 7 (2), 8 (3), 12 (4), 14 (5), 20 (6), 23 (7), 31 (8), 32 (9), 33 (10), 37 (11), 38 (12), 39 (13). Yes, so the 7th term is 23. Wait, but wait, maybe I made a mistake in sorting. Let's check each number:

Original numbers:

6,7,8,12,14,20,23,31,32,33,37,38,39. Let's check if all numbers are included:

From original data: 6,7,8,12,14,20,23,31,32,33,37,38,39. Yes, all 13 numbers are there. So the median is the 7th term, which is 23. Wait, but wait, maybe I made a mistake. Wait, let's count again. Wait, 6 (1), 7 (2), 8 (3), 12 (4), 14 (5), 20 (6), 23 (7), 31 (8), 32 (9), 33 (10), 37 (11), 38 (12), 39 (13). Yes, so the 7th term is 23. So the median is 23.

Wait, but wait, maybe I miscounted the number of elements. Wait, original data: first row: 20,12,37,32,33,31,14,38 (8 numbers), second row: 6,7,39,23,8 (5 numbers). 8 + 5 = 13. So n=13. So median is at position (13+1)/2 = 7. So the 7th term in sorted order is 23. So the median is 23.

Answer:

23