Calculate the mean for the following frequency distribution:

Marks	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	3	7	10	6	8	2	4

Hint: First of all, we will find the middle value of the marks intervals which we are going to name as ${{x}_{i}}$. The middle value of each marks distribution is calculated by adding the lower value and upper value of the interval and divides them by 2. Then we are going to multiply each middle value to their corresponding frequency. Now, we are going to add all these multiplications and then divide this result in addition to the sum of all the frequencies. The result of this division will give us the mean of the above frequency distribution.

Complete step-by-step solution:
The marks and frequency distribution is given in the above problem is as follows:

Now, we are going to find the middle values of all the marks intervals.
The formula that we are using for finding the middle value of the marks interval is as follows:
$\dfrac{\text{Lower Limit}+\text{Upper Limit}}{2}$
The middle value for 10-20 is equal to:
$\begin{align}
  & \dfrac{10+20}{2} \\
 & =\dfrac{30}{2}=15 \\
\end{align}$
Similarly, we can find all the middle values of the marks interval and in the below, we are tabulating the middle value.

We are naming the general name for middle value as ${{x}_{i}}$ in which if “i” equals 1 then we get 15, if “i” equals 2 then 25 and so on till “i” equals 7 when the middle value is 75. Similarly, we are naming the general frequency as ${{f}_{i}}$.

Now, we are multiplying ${{f}_{i}}\text{ }and\text{ }{{x}_{i}}$ together and writing it in the tabular form we get,

Adding all the values of ${{f}_{i}}{{x}_{i}}$ we get,
\[\begin{align}
  & \sum\limits_{i=1}^{7}{{{f}_{i}}}{{x}_{i}}=45+175+350+270+440+130+300 \\
 & \Rightarrow \sum\limits_{i=1}^{7}{{{f}_{i}}}{{x}_{i}}=1710 \\
\end{align}\]
The summation of all the frequencies is equal to:
$\begin{align}
  & \sum\limits_{i=1}^{7}{{{f}_{i}}}=3+7+10+6+8+2+4 \\
 & \Rightarrow \sum\limits_{i=1}^{7}{{{f}_{i}}}=40 \\
\end{align}$
Now, dividing 1710 to the summation of all the frequencies we get,
$\begin{align}
&\dfrac{\sum\limits_{i=1}^{7}{{{f}_{i}}}{{x}_{i}}}{\sum\limits_{i=1}^{7}{{{f}_{i}}}}=\dfrac{1710}{40} \\
&\Rightarrow\dfrac{\sum\limits_{i=1}^{7}{{{f}_{i}}}{{x}_{i}}}{\sum\limits_{i=1}^{7}{{{f}_{i}}}}=42.75 \\
\end{align}$
Hence, we got the mean of the frequency distribution is 42.75.

Note: You can check the mean value that you are getting is correct or not. The mean value that we are getting in the above solution is 42.75. And this value which 42.75 lies in the marks intervals given in the above problem. The interval in which 42.75 is lying is 40-50.
Now, if you might have got the mean as 425.66 then quickly you can understand that the mean is wrong because this value is not lying in any interval given above. This is how you can check the mean.