Week 8 – SS1 Third Term Mathematics Notes

 
 WEEK 8
TOPIC:Collection, Tabulation and Presentation of Grouped Data

 GROUPED DATA:Data are said to grouped,if two or more values are put together as one under one cell.In this case the variable column(first column) is known as Class interval,there are other parameters associated with grouped data and they are as listed below
Class Interval: 1-10, 11-20, 21-30………..
Class Boundaries:It is the possible extra length,created for the class interval:
0.5-10.5, 10.5-20.5, 20.5-30.5………
Class mark/Mid mark(x):1+10,11+20,21+30…………
2 2 2
Therefore,class marks are:5.5, 15.5, 25.5………
This is also known as class size,It is the difference between the UPPER class interval and the LOWER class interval.
When a given data has a large number of values, it is cumbersome to prepare its frequency table. For example, the table below show scores out of 60 obtained by SS 3 students in a test:
30    12    58    23    25    14    8    20    5    35
27    38    53    32    36    15    14    37    13    50
31    19    34    51    25    30    39    10    42    33
55    16    45    18    56
If the above data is organised in a frequency table as in example 22.1, the table will show 35 differentsocres, each of them occurring 1 time except 25 and 14 which occur 2 times. So the frequency table and the bar chart of this data would not be very useful because the result would show no pattern. To overcome this problem, we can organise the data into groups or classes. Before we group the data, we consider the range first which is 5 – 58. With this range the data can be grouped into class intervals such as: 1 – 10, 11 – 20, 21 – 30, 31 – 40, 41 – 50, 51 – 60.

 When a data is divided into groups it is called a grouped frequency distribution. The groups or classes into which the data are arranged are called class intervals. The first class interval is 1 – 10, the second class interval is 11 – 20, etc. since each class interval covers 10 possible marks, we say that the class width is 10 marks. The frequency distribution table for this data is shown in table (a) below:

 Table (a)

 Note that in a grouped discrete data, the data are usually whole numbers. For this reason, the class intervals do not overlap because each mark can only appear in each interval. So it is wrong to use intervals such as 1 – 10, 10 – 20, 20 – 30 because 10 appears in both the 1^st and the 2^nd class intervals and 20 in the 2^nd and 3^rd class intervals. However, when we group discrete data, we are actually treating it as though it was continuous.

 
 Grouped Continuous data
When dealing with continuous data, the variable is measured on a continuous scale. It is important to know where to place values that appear to be between groups or classes. For example, the frequency distribution in table (b) below shows the weight of 50 students to the nearest kg.

 This data is continuous, so we need to find the class boundaries (or the class mid value) and the width of class intervals.

 Class limits
The end numbers of each class interval are known as the class limits of that interval. In the table above the 1^stclass is 40 – 44. These figures give the class interval.The end numbers 40 and 44 are called the class limits. 40 is the lower class limit and 44 is the upper class limit.
Similarly, for the 2^nd class the class interval is 45 – 49. 45 is the lower class limit and 49 is the upper class limit.

 Class boundaries
When a data is given to the nearest unit, the class interval 40 – 44 theoretically includes all weights from 39.5kg to 44.5kg. we say that the 1^st class interval has class boundaries of 39.5kg and 44.5kg.
39.5kg is the lower class boundary and 44.5kg is the upper class boundary.

 Each class boundary can be found by adding the upper limit of one class to the lower limit of the next class and dividing the result by 2.

 For the 2^nd class
Lower class boundary =
Upper class boundary

 For the 3^rd class
Lowe class boundary = 49.5kg
Upper class boundary =
and so on.
Noticethat in this case for each class interval:

1. To obtain the lower class boundary, subtract 0.5 from the lower class limit.
2. To obtain the upper class boundary add 0.5 to the upper class limit.

 Class width of a class interval
Theclass width is also called the class size.
Class width = upper class boundary – lower class boundary
For example, for 1^st interval,
Class width = 44.5 – 39.3 = 5
For 2^nd interval,
Class width = 49.5 – 44.5 = 5

 Class mid-value (class mark)
The mid-value of a class is known as the class mark. For a given class interval, the class mid-value is exactly half way between the lower limit and the upper limit.
Or class mid-value
=
Forexample, the class mid-value of the 1^st interval =
2^ndinterval =, etc.

 EVALUATION
The weights of some students in a class of group of students to the nearest kg are given below:
65, 70, 60, 46, 51, 55, 59, 63, 68, 53, 47, 53, 72, 53, 67, 62, 64, 70, 57, 56
73, 56, 48, 51, 58, 63, 65, 62, 49, 64, 53, 59, 63, 50, 48, 72, 67, 56, 61, 64
With the class intervals 45-49,50-54,55-59 etc,Showing the class boundaries, class marks, tally and the frequencies, in that order.

 GENERAL EVALUATION

1. In a particular company, the amount of money to the nearest naira spent by workers on transportation to work daily were recorded as follows:

   30        60    120    200    80    90    74    240    236    125    40    75    110    120
   220    130    180    60    90    112    150    210    245    135    140    80    100    125
   215    240    50    60    180    190    180    148    120    88    138    195    248    130
   140    150    154    208    225    65    145

   1. Construct a grouped frequency distribution of this data taking equal intervals 0 – 49, 50 – 99, …
   2. Find the class boundaries and the class marks of each class interval
   3. Use the frequency distribution to find the class interval with the highest frequency
   4. State the width of each class interval

 READING ASSIGNMENT
Essential Mathematics for Senior Secondary Schools 1 page 348 – 350

 WEEKEND ASSIGNMENT
The weights to the nearestkg of a group of people are shown in the table below.

Use the table to answer question 1 –5

1. Copy and complete the table.
2. What is the modal class? A. 51 – 60    B. 61 – 70    C. 71 – 80    D. 31 – 40

Find the class widths of the last two class intervals

1. A. 90.5 – 80.5 = 10    B. 50.5 – 40.5 = 10    C. 41 – 40 = 1 D. 70 – 51 = 19
2. A. 90.5 – 70.5 = 20    B. 80.5 – 70.5 – 10    C. 100.5 – 90.5 = 10    D. 50.5 – 20.5 = 30
3. Estimate the mode of the frequency distribution

 THEORY

1. (a) Copy and complete the table below for the length of leaves given to the nearest cm.

1. Find the class widths of the 1^st and 2^nd classes.
2. Estimate the mode of the frequency distribution
3. Find the median class.

1. The distribution below shows the number of workers in a farm with their daily earnings:
   

1. How many workers are there in the farm?
2. What is the mode?