Finding the Mean of Prime Numbers Between 1 and 30

To find the mean of a set of numbers, we need to sum all the numbers in the set and then divide by the count of numbers in the set.

The formula for the mean is:

\(\text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}\)

Step 1: Identify the Prime Numbers Between 1 and 30

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

We need to list all prime numbers that are greater than 1 and less than 30.

Let's list them:

• 2 (The only even prime number)
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• 29

These are the prime numbers between 1 and 30.

Step 2: Calculate the Sum of the Prime Numbers

Now, we add up all the prime numbers we identified:

\(\text{Sum} = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29\)

Let's perform the addition:

\(2 + 3 = 5\)
\(5 + 5 = 10\)
\(10 + 7 = 17\)
\(17 + 11 = 28\)
\(28 + 13 = 41\)
\(41 + 17 = 58\)
\(58 + 19 = 77\)
\(77 + 23 = 100\)
\(100 + 29 = 129\)

So, the sum of the prime numbers between 1 and 30 is 129.

\(\text{Sum} = 129\)

Step 3: Count the Number of Prime Numbers

Next, we count how many prime numbers we listed between 1 and 30.

The list is: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

There are 10 numbers in this list.

\(\text{Count} = 10\)

Step 4: Calculate the Mean

Finally, we use the mean formula with the sum and count we found:

\(\text{Mean} = \frac{\text{Sum}}{\text{Count}}\)

\(\text{Mean} = \frac{129}{10}\)

\(\text{Mean} = 12.9\)

The mean of the prime numbers between 1 and 30 is 12.9.

Using the table values:

\(\text{Mean} = \frac{129}{10} = 12.9\)

Revision Table: Mean of Prime Numbers

Additional Information: Properties of Prime Numbers and Mean

What is a Prime Number?

As mentioned, a prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. The number 1 is not considered prime. The first few prime numbers are 2, 3, 5, 7, 11, etc.

Why is 2 the Only Even Prime Number?

Any even number greater than 2 is divisible by 2 (and also by 1 and itself). Since it has a divisor other than 1 and itself (which is 2), it cannot be prime. The number 2 is only divisible by 1 and 2, making it prime.

Understanding the Mean

The mean, or average, is a measure of central tendency. It gives us a single value that represents the typical value in a set of numbers. It's calculated by summing all the values and dividing by the number of values.

Other Measures of Central Tendency

Besides the mean, other common measures of central tendency include:

• Median: The middle value in a dataset when the numbers are arranged in order.
• Mode: The number that appears most frequently in a dataset.

For the set of prime numbers between 1 and 30 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29):

• Median: Since there are 10 numbers (an even count), the median is the average of the 5th and 6th numbers (11 and 13). Median = \(\frac{11+13}{2} = \frac{24}{2} = 12\).
• Mode: All numbers appear only once, so there is no mode for this specific set.

This problem specifically asked for the mean, which we calculated as 12.9.