EXERCISE 14.4

- The following number of goals was scored by a team in a series of 10 matches:

2, 3, 4, 5, 0, 1, 3, 3, 4, 3

Find the mean, median and mode of these scores.

Solution:

The number of goals scored by the team is 2, 3, 4, 5, 0, 1, 3, 3, 4, 3

Mean of data = ^{Sum of all observations}/_{Total number of observations} = ^{2+3+4+5+0+1+3+3+4+3}/_{10} = ^{28}/_{10 }= 2.8

= 2.8 goals

Arranging the number of goals in ascending order 0, 1, 2, 3, 3, 3, 3, 4, 4, 5

The number of observations is 10, which is an even number.

Therefore, median = ^{5th term + 6th term}/_{2} = ^{3+3}/_{2} = ^{6}/_{2} = 3

Mode of the data is the observation with the maximum frequency in data. Therefore, the mode score of data is 3 as it has the maximum frequency as 4 in the data.

- In a mathematics test given to 15 students, the following marks (out of 100) are recorded:

41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60

Find the mean, median and mode of this data.

Solution:

The marks of 15 students scored in mathematics, 41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60.

Mean of data = ^{Sum of all observations}/_{Total number of observations} =^{41+ 39+ 48+ 52+ 46+ 62+ 54+ 40+ 96+ 52+ 98+ 40+ 42+ 52+ 60}/_{15} = ^{822}/_{15 }= 54.8

Arranging the scores of 15 students in mathematics in an ascending order 39, 40, 40, 41, 42, 46, 48, 52, 52, 52, 54, 60, 62, 96, 98

As the number of observations is 15 which is odd, therefore, the median of data will be ^{(15+1)th}/_{2 }= 8^{th} observation whether the data is arranged in an ascending or descending order.

Therefore, median score of data = 52

Mode of the data is the observation with the maximum frequency in data. Therefore, the mode score of data is 52 as it has the highest frequency in the data as 3

- The following observations have been arranged in ascending order. If the median of the data is 63, find the value of x.

29, 32, 48, 50, x, x + 2, 72, 78, 84, 95

Solution:

It can be observed that the total number of observations in the given data is 10(even number).Therefore, the median of this data will be the mean of ^{10}/_{2} i.e., 5^{th} and (^{10}/_{2})+ 1 i.e., 6^{th} term.

Therefore, median of data = ^{5th observation + 6th observation}/_{2}

63 = ^{x+x+2}/_{2}

63 = ^{2x+2}/_{2}

63 = x + 1

x = 62

- Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.

Solution:

Arranging the data in an ascending order 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.

It can be observed that 14 has the highest frequency, i.e., 4 in the given data. Therefore, mode of the given data is 14.

- Find the mean salary of 60 workers of a factory from the following table:

Salary (in Rs) |
Number of workers |

3000 | 16 |

4000 | 12 |

5000 | 10 |

6000 | 8 |

7000 | 6 |

8000 | 4 |

9000 | 3 |

10000 | 1 |

Total |
60 |

Solution:

Using the table, the mean can be calculated as follows:

Salary (in Rs) |
Number of workers |
f_{i}x_{i} |

3000 | 16 | 3000 x 16 = 48000 |

4000 | 12 | 4000 x 12 = 48000 |

5000 | 10 | 5000 x 10 = 50000 |

6000 | 8 | 6000 x 8 = 48000 |

7000 | 6 | 7000 x 6 = 42000 |

8000 | 4 | 8000 x 4 = 32000 |

9000 | 3 | 9000 x 3 = 27000 |

10000 | 1 | 10000 x 1 = 10000 |

Total |
⅀f_{i} = 60 |
⅀f_{i}x_{i} = 305000 |

Mean = ^{⅀}^{fi xi}/_{⅀}_{fi} = ^{30500}/_{60} = 5083.33

Therefore, mean salary of 60 workers is Rs. 5083.33

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