Mean of First Six Odd Numbers: Step by Step Solution and Formula

The mean of the six odd numbers is 6.

That is the answer to the question “find the mean of six odd numbers” and it takes three lines to reach that result. The rest of this page shows those three lines, a faster method then why the mean equals 6 and not some random number. If you need to find the mean of the six odd numbers for a Class 9 worksheet the working below is what your teacher expects.

☰ Table of Contents

Step by Step Solution

The first six odd numbers are 1, 3, 5 7, 9 and 11. (In this question “odd numbers” means natural numbers so we start from 1 not from a negative number.)

Step 1. Add them up.

1 + 3 + 5. 7 + 9 + 11 = 36

Step 2. Count them. There are 6.

Step 3. Divide the sum by the count.

The mean = 36 ÷ 6 = 6

So the mean is 6. A habit before moving on: check that the mean sits between the smallest and largest numbers. 6 Is between 1 and 11 so it passes.

The Mean Formula

The mean = (sum of all the numbers) ÷ (how numbers there are)

Nothing more to it. Add, count, divide. The only place students slip is the counting. Six numbers means you divide by 6 not by 5 and not by the number.

A Quicker Way: Pair the Ends

The odd numbers go up by 2 each time so they form a progression. When numbers are evenly spaced like this the mean is the average of the first and the last:

The mean = ( + last) ÷ 2 = (1 + 11) ÷ 2 = 6

No full addition needed. It works because the numbers pair up around the middle: 1 and 11 make 12, 3 and 9 make 12, 5 and 7 make 12. Three pairs of 12 is 36 and 36 over 6 is 6.

Another Quick Way: Odd Numbers and Squares

Here is a pattern worth knowing. Add up the few odd numbers and watch the totals.

| How many odd numbers | Their sum |

|———————-|———–|

| 1 | 1 |

| 2 | 4 |

| 3 | 9 |

| 4 | 16 |

| 5 | 25 |

| 6 | 36 |

The sum of the odd numbers is n². For n = 6 that is 36. Divide by 6. You get the mean:

The mean = n² ÷ n = n

So the mean of the odd numbers is just n. The first six have a mean of 6. The first ten have a mean of 10. The first twenty have a mean of 20 because they add up to 400. You can check any of these by hand. It holds every time.

The Assumed Mean Method

Some syllabuses want this one, for bigger data sets but it works here too.

Pick a mean say a = 5. Find the deviation of each number from it:

1 − 5 = −4 3 − 5 = −2, 5 − 5 = 0 7 − 5 = 2 9 − 5 = 4 11 − 5 = 6

Add the deviations: −4 − 2. 0 +. 4 + 6 = 6.

The mean = a + (sum of deviations ÷ n) = 5. 6 ÷ 6 = 5 + 1 = 6

Same answer. Try a = 6. The deviations come out as −5 −3, −1, 1 3 5 which add up to 0. That is the clue that 6’s the mean because the deviations from the true mean always cancel to zero.

Why the Answer Is 6

The numbers 1, 3 5 7 9 and 11 are spread evenly.

There are three numbers below 6 and three numbers above 6. They are the same distances away: 5, 3 and 1 below and 1, 3 and 5 above.

The mean is the balance point of the list and in a symmetric list the balance point is the middle.

Here the middle falls between 5 and 7 which’s 6.

That is also the median so for this list the mean and median are equal.

Proof That the Sum Is n²

Write the sum of the n odd numbers twice, once forward and once backward.

S = 1 + 3 + 5….. 2N − 1)

S = (2n − 1) +…. 5 + 3 + 1

Add the two lines, column by column. Every column adds to 2n. There are n columns.

So 2S = n × 2n = 2n², which means S = n².

For n = 6 that gives 36 matching the sum.

Mean, Median and Mode of 1 3 5 7 9 11

The mean is 6.

The median is also 6 found by taking the two numbers, 5 and 7 and averaging them.

The mode does not exist, because every value appears once.

The range is 11 − 1 = 10.

A Word Problem

The houses on one side of a street are numbered 1, 3, 5, 7, 9 and 11.

What is the mean house number?

Add the numbers to get 36, divide by the 6 houses and the mean house number is 6.

There is no house with that number on this side of the street because 6 is even, which is a reminder that a mean does not have to be one of the values in the list.

Nearby Questions

These come up in the chapter and the pattern makes them quick.

Mean of the six even numbers: 2, 4 6 8 10 12.
Sum 42 divided by 6 gives 7.
Mean of the six natural numbers: 1 to 6.
Sum 21 divided by 6 gives 3.5.
Mean of the six odd prime numbers: 3 5, 7 11 13 17.
Sum 56 divided by 6 gives about 9.33.

That last one trips people up. Odd numbers and odd primes are lists. 9 Is odd but not prime. It is missing from the third list and the mean changes completely.

Common Mistakes

Starting from 3. Some people skip 1 because it is “not special.” It is the odd number and leaving it out changes the answer.
Including 0 or 2. Neither is odd.
Dividing by 5 or by 11 There are six numbers so the divisor is 6.
Adding wrongly. 1 + 3 + 5 + 7 + 9 + 11 is 36. A slip of one gives 35 or 37. Neither gives a whole number mean, which is a sign to recheck.
Mixing up numbers with odd primes as above.

Practice Questions

Find the mean of the four odd numbers.
Find the mean of the ten odd numbers.
Find the mean of the eight even numbers.
Use ( last) ÷ 2 to find the mean of 5, 7 9 11 13 15.
The mean of the odd numbers is 15. What is n?

Answers:

1) 1. 3 + 5 + 7 = 16 And 16 ÷ 4 = 4.

2) 10 Since the sum is 100.

3) 2. 4 +… + 16 = 72 And 72 ÷ 8 = 9.

4) (5 + 15) ÷ 2 = 10.

5) Because the mean equals n n = 15.

Add the six odd numbers to get 36, divide by 6 and the mean is 6.

The shortcut is (1 + 11) ÷ 2 and the pattern behind it is that the mean of the odd numbers is always n.

Frequently Asked Questions

How do you find the mean of six odd numbers?

To find the mean of the six odd numbers add 1, 3 5 7 9 and 11.

The total becomes 36.

Next divide 36 by 6.

The mean of the six odd numbers is 6.

What are the first six odd numbers?

The first six odd numbers are 1, 3, 5, 7, 9 and 11.

What is the sum of the six odd numbers?

The sum of the six odd numbers is 36.

36 Is the same as 6 squared.

What is the formula for the mean of the odd numbers?

The mean of the odd numbers equals n.

The sum of the odd numbers is n squared.

If you divide n squared by n the result is n.

Is the mean the same as the median ?

The mean is the same as the median.

Both are 6.

The numbers are evenly spaced and symmetric around 6.