Calculate the mean and variance of the first n natural numbers.

Step 1: Define the first n natural numbers

The set of the first $n$ natural numbers is simply $1, 2, 3,$ up to $n$.

Step 2: Calculate the mean (average)

The mean of a set of numbers is the sum of all the numbers divided by the count of numbers. For the first $n$ natural numbers, this is the sum $1+2+\dots+n$ divided by $n$.

Step 3: Use the formula for the sum of first n natural numbers

The sum of the first $n$ natural numbers is a well-known arithmetic series, and its formula is $n(n+1)/2$.

Step 4: Substitute the sum into the mean formula

Substitute the sum from the previous step into the mean calculation. The $n$ in the numerator and denominator cancels out, simplifying the expression for the mean.

Step 5: Define the variance formula

The variance measures how far each number in the set is from the mean. It is the average of the squared differences from the mean.

Step 6: Expand the variance formula

An alternative and often simpler formula for variance is the mean of the squares minus the square of the mean. This form is derived by expanding the squared term in the sum.

Step 7: Use the formula for the sum of squares of first n natural numbers

The sum of the squares of the first $n$ natural numbers is another standard formula that will be used in our variance calculation.

Step 8: Substitute values into the variance formula

Substitute the formula for the sum of squares and the previously calculated mean into the expanded variance formula.

Step 9: Simplify the first term

The $n$ in the numerator and denominator of the first term cancels out, simplifying it to $(n+1)(2n+1)/6$.

Step 10: Continue simplifying the variance expression

Now we need to subtract the square of the mean from the sum of squares. To do this, we find a common denominator, which is 12.

Step 11: Find a common denominator and combine terms

Multiply the first term by $2/2$ and the second term by $3/3$ to get a common denominator of 12. Then factor out the common term $(n+1)$.

Step 12: Simplify the expression inside the brackets

Expand the terms inside the brackets and combine like terms. The expression simplifies to $n-1$.

Step 13: Final variance formula

Multiply $(n+1)$ by $(n-1)$ to get $n^2-1$. This gives the final simplified formula for the variance of the first $n$ natural numbers.

More from this chapter

• For a statistical data x_1, x_2,, x_10 of 10 values, a student obtained the mean as 5.5 and Σ_i=1^(10) x_i² = 371. He later found that he had noted two values i
• The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observ
• Let the mean and variance of five observations x_1=1, x_2=3, x_3=a, x_4=7 and x_5=b, with a>b, be 5 and 10 respectively. Then the variance of the observations n
• The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.
• If each of the observation x_1, x_2,..., x_n is increased by ' a ', where a is a negative or positive number, show that the variance remains unchanged.
• The variance of the numbers 8, 21, 34, 47,, 320 is:
• The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.
• Find the mean deviation about the median for the following data: x_i: 3, 6, 9, 12, 13, 15, 21, 22 f_i: 3, 4, 5, 2, 4, 5, 4, 3.

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