* Lesson one*

* Class: SS2A&B *

* Subject: Mathematics*

* *

*Topic: Histograms of grouped Data*

Data are numerical facts or information that can be measured or quantified. Examples are students’ scores in Mathematics test, record of experimental observations, heights or weights of students and so on. These raw data are collected, organized and represented on a frequency table in a manner that make them meaningful and easy to interpret.

Data are represented on the frequency table in to two ways namely:

- As ungrouped data: Here the data are represented singly or individually with their respective frequencies.
- As grouped data: Here the data are organized into groups called class intervals with their frequencies. This happen when the number of entries in a given data is large, the construction of a frequency distribution may be difficult, hence, there will be need to group the data. For instance, you have 1000 different data to represent on a frequency table. If the data are not grouped you may have to construct a frequency table with 1000 rows to accommodate the data which is practically not feasible because it consumes time and space.But before grouping the data, the number of groups that the whole data will be divided into must be ascertained.

In grouping any numerical data, the following statistical parameters must be understood.

__CLASS INTERVAL__: Is the number of groups that a particular data is divided ito. E.g 10-19, 20-29, 30-39 etc.

__CLASSS LIMITS__: The class limits of grouped data are the end numbers of the class intervals. For instance in the class interval 10-19. 10 and 19 are the lower and upper class limits respectively.

__CLASS MARK :__ It is the centre or midpoint of any class intervals. It is obtained by adding the lower and upper class limits together and dividing the result by 2. The class mark of the class interval 10-19 is 14.5 I.e (10+19)/2=14.5.

__CLASS BOUNDARY__: Is regarded as the true limit of each class because it includes all values that approximate to the whole number. It is divided into lower and upper class boundaries.

Lower class boundary is the average value of the lower class limit of a given class and the upper limit of the class before it. While the upper class boundary is the average of the upper class limit of a given class and the lower class limit of the next class. For example, consider the following class intervals. 10 – 19, 20 – 29, 30 – 39 etc.

The lower class boundary (LCB) of the class interval 20 – 29 is = (19 + 20)/2 = 19.5 and the upper class boundary (UCB) is (29 + 30)/2 = 29.5. Therefore the class boundary of the class interval 20 – 29 is 19.5 – 29.5

__CLASS WIDTH/SIZE__: Is the difference between the Upper and lower class boundaries. For instance, the class width of the class 20 – 29 is 10. i.e 29.5 – 19.5 =10.

* FREQUENCY*: Is the number of times a particular score appears in a given data.

__CUMULATIVE FREQUENCY__: The cumulative frequency of a particular class interval is the sum of the frequency of that class and the frequencies of all the classes before it.

__FREQUENCY DENSITY__: This is used when the class intervals are not equal e.g 10-19, 20-24, 25-39.etc. It is the ratio of frequency to class width.

frequency Density= Frequency/Class width.

Ex1: The following are the weights of 40 men randomly picked in a village.

82 80 82 79 78 75 73 71

76 86 96 73 67 37 31 74

65 63 64 61 62 73 59 58

57 65 45 52 43 35 29 40

75 36 46 64 43 37 95 85

Using the class intervals 27-30, 31-34, 35-38…..construct a frequency table showing the following columns: class interval, class mark, class boundary, frequency and cumulative frequency.

*Solution*

Class interval | Class mark | Class boundary | frequency | Cumulative frequency |

27-30 | 28.5 | 26.5-30.5 | 1 | 1 |

31-34 | 32.5 | 30.5-34.5 | 2 | 3 |

35-38 | 36.5 | 34.5-38.5 | 4 | 7 |

39-42 | 40.5 | 38.5-42.5 | 1 | 8 |

43-46 | 44.5 | 42.5-46.5 | 3 | 11 |

47-50 | 48.5 | 46.5-50.5 | 0 | 11 |

51-54 | 52.5 | 50.5-54.5 | 1 | 12 |

55-58 | 56.5 | 54.5-58.5 | 2 | 14 |

59-62 | 60.5 | 58.5-62.5 | 3 | 17 |

63-66 | 64.5 | 62.5-66.5 | 5 | 22 |

67-70 | 68.5 | 66.5-70.5 | 1 | 23 |

71-74 | 72.5 | 70.5-74.5 | 5 | 28 |

75-78 | 76.5 | 74.5-78.5 | 4 | 32 |

79-82 | 80.5 | 78.5-82.5 | 4 | 36 |

83-86 | 84.5 | 82.5-86.5 | 2 | 38 |

87-90 | 88.5 | 86.5-90.5 | 0 | 38 |

91-94 | 92.5 | 90.5-94.5 | 0 | 38 |

95-98 | 96.5 | 94.5-98.5 | 2 | 40 |

__Histogram__: Is a graphical representation of data. It consists of rectangular bars whose heights are proportional to the frequencies with no gap in between the bars. It can be used to represent ungrouped and grouped data. Histogram is plotted frequency on the y- axis against the class boundary on the x- axis. The mode of the distribution can also be determine geometrically from a histogram of a grouped frequency distribution as follows:

(a) Identity the modal class by inspection, the class with the highest frequency.

(b) Using the rectangular bar of of the modal class, draw two diagonal lines from the corners of the rectangle to the corners of the two adjacent bars left and right of the rectangular bar of the modal class.

(c) From the point of the intersection of two diagonal lines, drop a line to touch the x-axis to obtain the mode.

Example 2: The table below shows the distribution of marks in percentages scored by a class of 180 students in examination.

Class interval | 20-24 | 25-29 | 30-34 | 35-39 | 40-44 | 45-49 | 50-54 | 55-59 | 60-64 | 65-69 |

Frequency | 1 | 2 | 7 | 18 | 22 | 42 | 30 | 37 | 15 | 6 |

(a) Construct a histogram of the distribution and use it to;

(b) estimate the mode of the distribution.

Solution

Class interval | Frequency | Boundaries |

20-24 | 1 | 19.5-24.5 |

25-29 | 2 | 24.5-29.5 |

30-34 | 7 | 29.5-34.5 |

35-39 | 18 | 34.5-39.5 |

40-44 | 22 | 39.5-44.5 |

45-49 | 42 | 44.5-49.5 |

50-54 | 30 | 49.5-54.5 |

55-59 | 37 | 54.5-59.5 |

60-64 | 15 | 59.5-64.5 |

65-69 | 6 | 64.5-69.5 |

Histogram showing the marks scored by 180 students in an examination

(c) From the histogram, the mode of the distribution is 52.5

__Assignment__

- The following are the length in cm of fifty planks cut by a machine in the wood processing factory of Kara Sawmills Nig Ltd.

Length | 21-30 | 31-40 | 41-5 | 51-60 | 61-70 | 71-80 | 81-90 | 91-100 |

Frequency | 2 | 6 | 9 | 9 | 11 | 6 | 4 | 3 |

(a) Construct a histogram of the distribution

(b) From your histogram, find the modal length of the plank.

- The following are the marks in percentages of fifty Students in Physics exam.

18 34 20 39 36 32 29 19 25 38

25 22 25 28 37 40 39 27 23 19

30 19 21 25 40 27 18 32 19 40

26 18 39 31 19 29 40 30 21 26

40 20 37 33 26 35 35 29 25 18

Using the class intervals 18 – 20, 21- 23, 24 – 26…. construct a frequency table showing the following columns: class interval, frequency class mark, class boundary and cumulative frequency.

N/B: You are advice to refer to your Mathematics textbooks for further studies.

Submit the assignment on the 29th May, 2020 via whatsApp @ 07035773326 or at the school’s security post

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