Mean or average or arithmetic mean is one of the representative values

of data. We can find the mean of observations by dividing the sum of all the

observations by the total number of observations.

**Mean of raw data:**

If x_{1}, x_{2}, x_{3}, ……. x_{n} are n observations, then

**Arithmetic Mean** = (x_{1}, x_{2}, x_{3}, ……. x_{n})/n

= (∑x_{i})/n

∑ (Sigma) is a Greek letter showing

summation

**1.**

Weights of 6 boys in a group are 63, 57, 39, 41, 45, 45. Find the mean weight.

**Solution:**

Number of observations = 6

Sum of all the observations = 63 + 57 + 39 + 41 + 45 + 45 = 290

Therefore, arithmetic mean = 290/6 = 48.3

**Mean of tabulated data:**

If x_{1}, x_{2}, x_{3}, x_{4}, ……. x_{n} are n observations, and f_{1}, f_{2}, f_{3}, f_{4}, ……. f_{n} represent frequency of n observations.

Then mean of the tabulated data is given by

= (f_{1} x_{1} + f_{2} x_{2} + f_{3} x_{3} + ……. f_{n} x_{n})/(f_{1} + f_{2} + f_{3} + …… f_{n}) = ∑(f_{i}x_{i})/∑f_{i}

**2.** A die is

thrown 20 times and the following scores were recorded 6, 3, 2, 4, 5, 5, 6, 1, 3,

3, 5, 6, 6, 1, 3, 3, 5, 6, 6, 2.

Prepare the frequency table of scores on the upper face of

the die and find the mean score.

**Solution:**

Number on the upperface of die |
Number of times it occurs(frequency) |
f_{i}x_{i} |

1 | 2 | 1 × 2 = 2 |

2 | 2 | 2 × 2 = 4 |

3 | 5 | 3 × 5 = 15 |

4 | 1 | 4 × 1 = 4 |

5 | 4 | 5 × 4 = 20 |

6 | 6 | 6 × 6 = 36 |

Therefore, mean of the data = ∑(f_{i}x_{i})/∑f_{i}

= (2 + 4 + 15 + 4 + 20 + 36)/20

= 81/20

= 4.05

**3.** If the mean of the following distribution is 9, find the value of p.

X |
4 | 6 | p + 7 | 10 | 15 |

f |
5 | 10 | 10 | 7 | 8 |

**Solution:**

Calculation of mean

x_{i} |
f_{i} |
x_{i}f_{i} |

4 | 5 | 20 |

6 | 10 | 60 |

p + 7 | 10 | 10(p + 7) |

10 | 7 | 70 |

15 | 8 | 120 |

∑f_{i} = 5 + 10 + 10 + 7 + 8 = 40

∑ f_{i}x_{i} = 270 + 10(p + 7)

Mean = ∑(f_{i}x_{i})/∑f_{i}

9 = {270 + 10(p + 7)}/40

⇒ 270 + 10p + 70 = 9 × 40

⇒ 340 +10p = 360

⇒ 10p = 360 – 340

⇒ 10p = 20

⇒ p = 20/10

⇒ p = 2

** Mean of grouped data: **

While calculating the mean of the grouped data, the values x_{1}, x_{2}, x_{3}, ……. x_{n} are taken as the mid-values or the class marks of various class intervals. If the frequency distribution is inclusive, then it should be first converted to exclusive distribution.

**4. **The following table shows the number of plants in 20 houses in a group

Number of Plants |
0 – 2 | 2 – 4 | 4 – 6 | 6 – 8 | 8 – 10 | 10 – 12 | 12 – 14 |

Number of Houses |
1 | 2 | 2 | 4 | 6 | 2 | 3 |

Find the mean number of plans per house

**Solution: **

We have

Number of Plant |
Number of Houses (f _{i}) |
Class Mark (x _{i}) |
f_{i}x_{i} |

0 – 2 | 1 | 1 | 1 × 1 = 1 |

2 – 4 | 2 | 3 | 2 × 3 = 6 |

4 – 6 | 2 | 5 | 2 × 5 = 10 |

6 – 8 | 4 | 7 | 4 × 7 = 28 |

8 – 10 | 6 | 9 | 6 × 9 = 54 |

10 – 12 | 2 | 11 | 2 × 11 = 22 |

12 -14 | 3 | 13 | 3 × 13 = 39 |

∑f_{i} = 1 + 2 + 2 + 4 + 6 + 2 + 3 = 20

∑f_{i} x_{i} =1 + 6 + 10 + 28 + 54 + 22 + 39 = 160

Therefore, mean = ∑(f_{i}x_{i})/ ∑f_{i} = 160/20 = 8 plants

**● Statistics**

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