TOPICS

Unit-14(Examples)

Statistics

**Example-1 :-** The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in table below. Find the mean of the marks obtained by the students.

Marks obtained (x_{i}) | Number of students (f_{i}) | f_{i}x_{i} |

10 | 1 | 10 |

20 | 1 | 20 |

36 | 3 | 108 |

40 | 4 | 160 |

50 | 3 | 150 |

56 | 2 | 112 |

60 | 4 | 240 |

70 | 4 | 280 |

72 | 1 | 72 |

80 | 1 | 80 |

88 | 2 | 176 |

92 | 3 | 276 |

95 | 1 | 95 |

Total | Σf_{i} = 30 | Σf_{i}x_{i} = 1779 |

**Example-2 :-** : The table below gives the percentage distribution of female teachers in the primary schools of rural areas of various states and union territories (U.T.) of India. Find the mean percentage of female teachers by all the three methods discussed in this section.

Percentage of female teachers | No. of states/U.T. (f_{i}) | x_{i} | d_{i} = x_{i} - 50 | u_{i} =(x_{i} - 50)/10 | f_{i}x_{i} | f_{i}d_{i} | f_{i}u_{i} |

15 - 25 | 6 | 20 | -30 | -3 | 120 | -180 | -18 |

25 - 35 | 11 | 30 | -20 | -2 | 330 | -220 | -22 |

35 - 45 | 7 | 40 | -10 | -1 | 280 | -70 | -7 |

45 - 55 | 4 | 50 | 0 | 0 | 200 | 0 | 0 |

55 - 65 | 4 | 60 | 10 | 1 | 240 | 40 | 4 |

65 - 75 | 2 | 70 | 20 | 2 | 140 | 40 | 4 |

75 - 85 | 1 | 80 | 30 | 3 | 80 | 30 | 3 |

Total | Σf_{i} = 35 | Σf_{i}x_{i} = 1390 | Σf_{i}d_{i} = -360 | Σf_{i}u_{i} = -36 |

**Example-3 :-** : The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does the mean signify?

Number of wickets taken | Number of bowlers (f_{i}) | x_{i} | d_{i} = x_{i} - 200 | u_{i} =(x_{i} - 200)/20 | f_{i}u_{i} |

20 - 60 | 7 | 40 | -160 | -8 | -56 |

60 - 100 | 5 | 80 | -120 | -6 | -30 |

100 - 150 | 16 | 125 | -75 | -3.75 | -60 |

150 - 250 | 12 | 200 | 0 | 0 | 0 |

250 - 350 | 2 | 300 | 100 | 5 | 10 |

350 - 450 | 3 | 400 | 200 | 10 | 30 |

Total | Σf_{i} = 45 | Σf_{i}u_{i} = -106 |

**Example-4 :-** The wickets taken by a bowler in 10 cricket matches are as follows:

2, 6, 4, 5, 0, 2, 1, 3, 2, 3

Find the mode of the data.

Let us form the frequency distribution table of the given data as follows:

No. of wickets | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

No. of matches | 1 | 1 | 3 | 2 | 1 | 1 | 1 |

**Example-5 :-** A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household:
Find the mode of this data.

Family size | No. of families |

1 - 3 | 7 |

3 - 5 | 8 |

5 - 7 | 2 |

7 - 9 | 2 |

9 - 11 | 1 |

**Example-6 :-** : The marks distribution of 30 students in a mathematics examination are given in Table 14.3 of Example 1. Find the mode of this data. Also compare and interpret the mode and the mean.

Refer to Table 14.3 of Example 1.

Class interval | No. of students |

10 - 25 | 2 |

25 - 40 | 3 |

40 - 55 | 7 |

55 - 70 | 6 |

70 - 85 | 6 |

85 - 100 | 6 |

**Example-7 :-** A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data was obtained:
Find the median height.

Class interval | Frequency | Cumulative Frequency (cf) |

Below 140 | 4 | 4 |

140 - 145 | 7 | 4 + 7 = 11 |

145 - 150 | 18 | 11 + 18 = 29 |

150 - 155 | 11 | 29 + 11 = 40 |

155 - 160 | 6 | 40 + 6 = 46 |

160 - 165 | 5 | 46 + 5 = 51 |

Total | n = 51 |

**Example-8 :-** The median of the following data is 525. Find the values of x and y, if the total frequency is 100.

Class intervals | Frequency | Cumulative Frequency (cf) |

0 - 100 | 2 | 2 |

100 - 200 | 5 | 7 |

200 - 300 | x | 7 + x |

300 - 400 | 12 | 19 + x |

400 - 500 | 17 | 36 + x |

500 - 600 | 20 | 56 + x |

600 - 700 | y | 56 + x + y |

700 - 800 | 9 | 65 + x + y |

800 - 900 | 7 | 72 + x + y |

900 - 1000 | 4 | 76 + x + y |

Total | n = 76 + x + y |

**Example-9 :-** The annual profits earned by 30 shops of a shopping complex in a locality give rise to the following distribution:
Draw both ogives for the data above. Hence obtain the median profit.

We first draw the coordinate axes, with lower limits of the profit along the horizontal axis, and the cumulative frequency along the vertical axes. Then, we plot the points (5, 30), (10, 28), (15, 16), (20, 14), (25, 10), (30, 7) and (35, 3). We join these points with a smooth curve to get the ‘more than’ ogive, as shown in Fig. Now, let us obtain the classes, their frequencies and the cumulative frequency from the table above.

Classes | No. of shops | Cumulative Frequency (cf) |

5 - 10 | 2 | 2 |

10 - 15 | 12 | 2 + 12 = 14 |

15 - 20 | 2 | 14 + 2 = 16 |

20 - 25 | 4 | 16 + 4 = 20 |

25 - 30 | 3 | 20 + 3 = 23 |

30 - 35 | 4 | 23 + 4 = 27 |

35 - 40 | 3 | 27 + 3 = 30 |

Total | n = 30 |

CLASSES