Question

Find the mean of squares of the first 10 natural numbers.

Answer 1

Hint: We will use the formula of sum of the squares of first n natural numbers with n = 10 and then the formula of mean given as: sum of the observations / total number of observations to calculate the value of the mean of the squares of the first 10 natural numbers.

Complete step-by-step answer:
We are required to calculate the value of the mean of the squares of the first 10 natural numbers.
We know the formula of the mean of n numbers is given by: sum of all the n observations / n
So, the sum of the squares of the first n natural numbers is given by: $\dfrac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$
We need to find the sum of squares of the first 10 natural numbers i.e., we need to find the value when n = 10.
$ \Rightarrow $ sum of the squares of the first 10 natural numbers = $\dfrac{{10\left( {10 + 1} \right)\left( {2\left( {10} \right) + 1} \right)}}{6} = \dfrac{{10\left( {11} \right)\left( {21} \right)}}{6}$
Now, the mean of the squares of the first 10 natural numbers = sum of the squares of the first 10 natural numbers / 10
$ \Rightarrow $ Mean = $\dfrac{{\dfrac{{10\left( {11} \right)\left( {21} \right)}}{6}}}{{10}} = \dfrac{{10\left( {11} \right)\left( {21} \right)}}{{6\left( {10} \right)}} = \dfrac{{11\left( {21} \right)}}{6} = \dfrac{{231}}{6} = 38.5$
Therefore, the mean of the squares of the first 10 natural numbers is 38.5.

Note: In this question, you may get confused while starting the solution with the formula of sum of first n natural numbers. You can also solve this question by directly calculating the mean of the square of the first 10 natural numbers namely 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Their squares will be 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Using the formula of mean, we get $\dfrac{{1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100}}{{10}} = \dfrac{{385}}{{10}} = 38.5$. Hence, the answer is verified as well.