Question

what is the arithmetic mean of the following numbers? 8, 10, 8, 5, 4, 7, 5, 10, 8

Explanation:

Step1: Sum the numbers

Add all the given numbers: \(8 + 10 + 8 + 5 + 4 + 7 + 5 + 10 + 8\). Let's calculate: \(8+10 = 18\), \(18+8 = 26\), \(26+5 = 31\), \(31+4 = 35\), \(35+7 = 42\), \(42+5 = 47\), \(47+10 = 57\), \(57+8 = 65\). Wait, no, wait: let's do it again carefully. \(8 + 10 = 18\), \(18 + 8 = 26\), \(26 + 5 = 31\), \(31 + 4 = 35\), \(35 + 7 = 42\), \(42 + 5 = 47\), \(47 + 10 = 57\), \(57 + 8 = 65\)? Wait, no, the numbers are 8,10,8,5,4,7,5,10,8. Let's count the count: 9 numbers. Let's sum again: 8+10=18, +8=26, +5=31, +4=35, +7=42, +5=47, +10=57, +8=65? Wait, 8+10+8+5+4+7+5+10+8. Let's group: (8+8+8) + (10+10) + (5+5) + 4 +7. 83=24, 102=20, 5*2=10, 4+7=11. Then total sum: 24+20=44, 44+10=54, 54+11=65. Yes, sum is 65.

Step2: Divide by count

Count of numbers is 9. So mean = sum / count = 65 / 9 ≈ 7.222... Wait, no, wait: 8+10+8+5+4+7+5+10+8. Let's add again: 8+10=18, +8=26, +5=31, +4=35, +7=42, +5=47, +10=57, +8=65. Yes, sum is 65. Number of terms: 9. So 65 divided by 9. Wait, 9*7=63, 65-63=2, so 7 + 2/9 ≈7.222... Wait, but maybe I made a mistake in sum. Let's check each number: 8,10,8,5,4,7,5,10,8. Let's list them:

8 (1), 10 (2), 8 (3), 5 (4), 4 (5), 7 (6), 5 (7), 10 (8), 8 (9). So positions 1-9.

Sum: 8+10=18; +8=26; +5=31; +4=35; +7=42; +5=47; +10=57; +8=65. Yes, sum is 65. Count is 9. So mean is 65/9 ≈7.222..., but wait, maybe I miscalculated. Wait, 83=24, 102=20, 5*2=10, 4+7=11. 24+20=44, 44+10=54, 54+11=65. Correct. So 65 divided by 9 is approximately 7.22, but let's check again. Wait, 8+10+8+5+4+7+5+10+8. Let's add 8+10=18, 8+5=13, 4+7=11, 5+10=15, 8. Then 18+13=31, +11=42, +15=57, +8=65. Yes. So mean is 65/9 ≈7.222..., but maybe the problem expects an exact fraction or decimal. Wait, 65 divided by 9 is 7 and 2/9, or approximately 7.22.

Wait, but maybe I made a mistake in the sum. Let's check with another approach. Let's list the numbers and their frequencies:

8: 3 times (positions 1,3,9)

10: 2 times (positions 2,8)

5: 2 times (positions 4,7)

4: 1 time (position 5)

7: 1 time (position 6)

So sum = (83) + (102) + (5*2) + 4 +7 = 24 + 20 + 10 + 4 +7 = 24+20=44, 44+10=54, 54+4=58, 58+7=65. Yes, that's correct. So count is 3+2+2+1+1=9. So mean is 65/9 ≈7.222...

Answer:

\(\frac{65}{9}\) (or approximately \(7.22\))