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**Hint:**We will first make the table again with one row or column consisting of the product of the marks and frequency of students. Now, sum up all the values. Using all this information and the formula of the mean of data, where mean represents the average, we will find the mean.

**Complete step by step answer:**

We are given the following data in tabular form:-

Marks | 4 | 6 | 10 | 15 | 18 | 20 |

Frequency | 2 | 4 | 8 | 12 | 3 | 1 |

Let us make the table in vertical format like as follows:-

Marks | Frequency |

4 | 2 |

6 | 4 |

10 | 8 |

15 | 12 |

18 | 3 |

20 | 1 |

Let us now just add a column to it which contains the product of both marks and frequency.

So, we will get the table as follows:-

Marks \[({x_i})\] | Frequency \[({f_i})\] | \[{f_i}{x_i}\] |

4 | 2 | 8 |

6 | 4 | 24 |

10 | 8 | 80 |

15 | 12 | 180 |

18 | 3 | 54 |

20 | 1 | 20 |

Now, we know that the formula of mean is given by the following expression:-

$\bar x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$, where $\bar x$ is the mean which is required.

Let us first find the value of $\sum {{f_i}{x_i}} $.

$ \Rightarrow \sum {{f_i}{x_i}} = 8 + 24 + 80 + 180 + 54 + 20$

Solving this, we will get as follows:-

$ \Rightarrow \sum {{f_i}{x_i}} = 366$

Now, let us find the value of $\sum {{f_i}} $.

$ \Rightarrow \sum {{f_i}} = 2 + 4 + 8 + 12 + 3 + 1$

Solving this, we will get as follows:-

$ \Rightarrow \sum {{f_i}} = 30$

Now, putting both $\sum {{f_i}{x_i}} = 366$ and $\sum {{f_i}} = 30$ in the formula $\bar x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$.

We will then get:-

$ \Rightarrow \bar x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }} = \dfrac{{366}}{{30}}$

Solving this, we will then get:-

$ \Rightarrow \bar x = 12.2$

**$\therefore $ The mean of the given data is 12.2.**

**Note:**

The students must know what the word “mean” represents. Mean means the average of the numbers. For example, take the question, we solved above. In that we found mean. Now, mean in these questions will represent, if we distribute the obtained marks to each of 30 students, then how many marks each student will score. This basically gives an idea of how good the whole class together is scoring.

Fun Fact:- Do you know that if we calculate the mean of all the persons in the world and their sleep time. It is 7 minutes!!!!!

There are basically three types of mean in Mathematics which are arithmetic mean, the geometric mean, and the harmonic mean. The relation between them is $A.M. \geqslant G.M. \geqslant H.M.$

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