Collection, Tabulation & Presentation of Grouped Data SS1 Mathematics Lesson Note

COLLECTION, TABULATION AND PRESENTATION OF GROUPED DATA  

1. GROUPED DATA:

Data are said to be 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 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, below is a list of 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, the table will show 35 different scores, 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 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 the table below:

NOTE:  

In 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, and 20 – 30 because 10 appears in both the 1st and the 2nd class intervals and 20 in the 2nd and 3rd class intervals. 

However, when we group discrete data, we are treating it as though it were continuous.

1. 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.

III. CLASS LIMITS 

The end numbers of each class interval are known as the class limits of that interval. In the table above the 1st class 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 2nd class, the class interval is 45 – 49. 45 is the lower class limit and 49 is the upper class limit.

1. CLASS BOUNDARIES 

When 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 1st 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 2nd class 

Lower class boundary = 49 +452=44.5kg

Upper-class boundary 49 +502=49.5kg

For the 3rd class 

Lower class boundary = 49.5kg

Upper class boundary = 54 +552=54.5kg

and so on.

Notice that in this case for each class interval: To obtain the lower class boundary, subtract 0.5 from the lower class limit.

To obtain the upper class boundary add 0.5 to the upper class limit.

1. CLASS WIDTH OF A CLASS INTERVAL

The Class width is also called the class size. 

Class width = upper-class boundary – lower-class boundary 

For example, for 1st interval,

Class width = 44.5 – 39.3 = 5

For 2nd interval,

Class width = 49.5 – 44.5 = 5 

1. 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 halfway between the lower limit and the upper limit.

Or class mid-value = lower class boundary + upper-class boundary2

For Example, the class mid-value of the 1st interval = 39.5 +44.42=42

2nd interval  =44.5 +49.52=47, etc.

CLASSWORK: 

1. In a particular company, the amount of money to the nearest naira spent by workers on transportation to work daily was 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. a) Construct a grouped frequency distribution of this data taking equal intervals 0 – 49, 50 – 99, …
2. b) Find the class boundaries and the class marks of each class interval 
3. c) Use the frequency distribution to find the class interval with the highest frequency 
4. d) State the width of each class interval

2. The weights of some students in a class of a 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, Show the class boundaries, class marks, tally and the frequencies, in that order.

ASSIGNMENT: 1. The weights to the nearest kg of a group of people are shown in the table below.

Use the table to answer questions 1 –4

1. Copy and complete the table.
2. What is the modal class?   A. 51 – 60    B. 61 – 70    C. 71 – 80    D. 31 – 40 
3. Find the class widths of the last two class intervals A. 90.5 – 80.5 = 10    B. 50.5 – 40.5 = 10    C. 41 – 40 = 1      D. 70 – 51 = 19
4. A. 90.5 – 70.5 = 20    B. 80.5 – 70.5 – 10    C. 100.5 – 90.5 = 10    D. 50.5 – 20.5 = 30 

4. Estimate the mode of the frequency distribution