Mathematics

Key unit competence

By the end of this unit, I will be able to collect, present and interpret grouped data.

Unit outline

• Definition and examples of grouped data.

• Grouping data into classes.

• Frequency distribution tables for grouped data.

• Commulative frequency distribution tables.

• Measure of central tendency for grouped data.

• Graphical presentation of grouped data (polygon, histogram).

10.1 Grouped data

In unit 8 of S1, you were introduced to basic statistics that involved collection, organization and analysis of ungrouped data. In this unit, we are going to continue with the same using grouped data

10.1.1 Definition of grouped data

Activity 10.1

Given masses (in kg) of 30 pupils as follows:

1. Draw the following frequency distribution table in your exercise books. Count and fill the number of pupils whose mass fall in the given groups of masses:

2.  Find the total number of pupils (∑ f) and fill it in the bottom cell of the frequency column.

3.  Answer the following questions from the table:

(a)  Which group has the highest   number of pupils? (b) Which group has the lowest   number of pupils?

4.  Suppose you had pupils with masses of 50.4 kg and 50.9 kg. Discuss with your classmates in  groups where you would place them, giving reasons for you choice of group.

The data in this table represents marks scored by a group of 40 students in a mathematics test. We can analyse this performance by putting these students in groups according to their performance. For example, in this test, 2 students scored between 30 and 39, 9 scored between 40 and 49, 14 scored between 50 and 59, 7 scored between 60 and 69 and 8 scored between 70 and 79.

This information can be presented as in Table 10.3 below

When data is presented as in table 10.3 above, it is said to be a grouped data.

Grouped data is data that has been sorted into classes or categories.

Table 10.3 is an example of a frequency table. From table 10.3 it means that all scores from 30 to 39 inclusive, are in one group; all scores from 40 to 49 inclusive, are in the next group and so on.

Each of these groups is called a class or class interval.  The values 30 and 39, 40 and 49, 50 and 59, etc. are called class limits for the respective class.

Note that, when the number of items in a data distribution is small, it is easy to deal with them; but when the number of items is large, it becomes necessary to group the data.

10.1.2 Frequency distribution table for grouped data

Activity 10.2

Table 10.4 below shows fifty scores for 25 basketball games.

Using the data in table 10.4;

1. Identify  

(i)  the lowest score   

(ii) the highest score

2. Find the range of the scores.

3. Set these scores into 10 groups or class intervals beginning with 1 – 10, 11 – 20…. 91 – 100 in a frequency table.

4. For each group, state the class limits.

5. For each class frequency, denote the frequencies f1 = …, f2 = … and so on upto the 10th group.

6. How do you think the range can be useful in determining a suitable number of classes or the size of the class interval?

1. Consider the following data on the diameters of 40 ball bearings that were recorded in mm.

      

 (i) the lowest diameter is 27 and

(ii) the highest diameter is 67.

• The range of the scores is the difference between the highest score and the lowest score. 

Range =  highest value  –  lowest        

value  =  67 – 27 = 40

2. Using classes 26 – 30, then 31 – 35,  36 – 40, 41 – 45 and so on, we get the following frequency table.

• Thus, for the class 26-30, 26 is called the Lower Class Limit and 30 is called the Upper Class Limit.

• Table 10.5 is called a frequency distribution table for Grouped Data. Similarly we can state the class limits for the rest of the groups.

• The number of observations in each class is the class frequency, denoted by f for example, the frequency for the class 41- 45 is f = 9.

•  For this data the group size was already determined for us. Generally, the range and the number of classes can be used to estimate the class width or size.

     

Usually class sizes are better in multiples of 5 or 10. For example, to work with about 10 classes

we can then use a convenient class of 5.

 Similarly we can estimate the number of classes as follows:

From this type of data presentation, we can draw better conclusions about the data than before.

Some of these conclusions are:

(a) The number of balls whose diameters fall between 31 and 35 is 3.

(b) No ball measures less than 26 mm.

(c) Nine balls have a diameter between 41 and 45 and so on.

Example 10.1

Table 10.6 shows the masses (in grams) of 50 carrots taken from a plot of land on which the effect of a new fertiliser was being investigated.

 

Make a frequency distribution table for this data

Solution

The smallest mass is 63 g and the largest is 118 g.  The difference between the largest value and smallest value is called the range.

Thus, range = 118 g – 63 g = 55 g.

We need to group the data into a convenient number of classes.  Usually, the reasonable number of classes varies from 4 to 12. By dividing the range by class size, we get the number of classes. Thus a class size of 5 will give us
55/5 i.e. 11 classes. A class size of 8 will give us 55/8 i.e. 7 classes

A class size of 10 will give us 55/10 i.e. 6
classes.

Let us use a class size of 10. Table 10.7 is the required frequency table.

Note:  If the range is small, it is more convenient to use class sizes which are even. If it is large, multiples of 5 or 10 are more convenient.  This is helpful especially if there is need to represent the data graphically.

Exercise 10.1

1. A handspan is the distance (length) from the end of the thumb to the end of the small finger when the hand is fully open. Table 10.8 shows the handspans of some 21 children measured in centimetres.

Make a frequency distribution table, grouping the data into four classes starting with 14.0 – 15.9.

2. The lengths of 36 pea pods were measured to the nearest millimetre and recorded in Table 10.9.

Put the data into a grouped frequency table by choosing a convenient number of classes.

3. The percentage burns for 70 fire accident victims treated in a hospital in two years were recorded in the hospital records as shown in Table 10.10.

 Make a grouped frequency table using classes  10 – 19,  20 – 29,    30 – 39, etc.

4.  A pupil measured the amount of ink in biro pens used in his class by measuring the length (in cm) of the ink column that could be seen. He obtained the results shown in table 10.11 below.  Make a grouped frequency table with 6 classes

10.2 Data presentation

10.2.1 Class boundaries

Consider the grouped distribution in Table 10.12. Suppose the data represents the masses to the nearest kg of a group of 40 boys.

Generally, for any data obtained from measurements, the practice is to record them to the nearest value of the accuracy chosen. For example, a recorded value of say 30 kg represents a number in the interval 29.5 to 30.5. Thus 29.5 ≤ 30 < 30.5

The class 30 – 39 includes all values equal to or greater than 29.5 but less than 39.5. Thus the class interval stretches from 29.5 to 39.5. The point half way between the upper limit of the first class and the lower limit of the next class is called the class boundary. i.e. class boundary between first and second class

Similarly, the class 40 – 49 includes all values equal to or greater than 39.5 but less than 49.5

The values 29.5, 39.5, 49.5, etc. are called class boundaries for Table 10.11. Thus, the class 30–39 can be represented as 29.5 – 39.5.  The value 29.5 is the lower class boundary and 39.5 is the upper class boundary for this class

Similarly, the class 40 – 49 can be extended to 39.5 – 49.5, etc.

The difference between the upper class boundary and the lower class boundary is called the class interval, class width, or class size, i.e. class interval  =  upper class boundary –  lower class boundary. This knowledge is essential for the construction of a histogram.

10.2.2 Histogram

Activity 10.3

Table 10.13 below shows the frequency distribution table for the heights of a group of men, to the nearest centimetres.

Use the given information to do the following:

1. Identify the boundaries of the classes.

2. Rewrite the table using boundaries rather than class limits.

3. In your own words, distinguish between class limits and class boundaries.

From Activity 10.4, the class boundary between the first and the second class is given by the mean of upper limit of the first class and lower limit of the second class

Between one class and the next, the class limits have a gap between them. There is a disconnect between any two consecutive classes. In table 10.13, class boundaries are such that the end of one class to the beginning of the next. In other words, there is a sense of continuity when the frequency table is written using class boundaries table 10.14. This is the format used when constructing a histogram of grouped data.

A histogram is a bar diagram that represents the frequency distribution of a continuous data. When class intervals are of equal width, a histogram resembles a bar graph the difference is that there are no spaces between bars. In a histogram, each rectangle is drawn above each respective class interval such that the area of each rectangle is proportional to the frequency of the observations falling in the corresponding interval.

If the class intervals are equal, then the heights of the rectangles are proportional to the corresponding frequencies. If the classes intervals are not equal the heights of rectangles are represented by a unit called relative frequency or frequency density.

Activity 10.4

1. Using a dictionary or the internet, find the meaning of the terms:

(i) frequency density

(ii) relative frequency

2. Copy table 10.14 and create another column for the frequency density (fd).

3. Using an appropriate scale, draw a pair of axes. On the vertical axis, mark the frequency density.

4. On the horizontal axis, draw appropriate rectangles whose width equals the class width, and whose height equals the corresponding frequency density.

From Activity 10.4, you notice that:

• Frequency density, also known as relative frequency is a statistical data that compares class frequency to the class width, for the purposes of constructing a histogram of a given set of data.

                 

The required table that includes the frequency density column is as shown below i.e. Table 10.15

Note: class interval = 5 for all classes.

A histogram (or a frequency histogram) is a graph that consists of a series of rectangles:

(a) drawn on a continuous scale (i.e. no gaps between the rectangles) and

(b) with areas being proportional to class frequencies. The height of the rectangles are obtained as,  , f being the frequency and w the class width.  The value is called frequency density.

Solution

Table 10.17 represents the same data as in Table 10.16, but with the class limits having been changed to class boundaries and the column for the frequency density included

Choose a suitable scale and use it to represent marks on the horizontal axis, indicating all the class boundaries along this axis.

Since the class interval is constant, the rectangles must have the same width. Fig. 10.2 below shows the required histogram.

Note that:

(i)  The class boundaries mark the boundaries of the rectangular bars in the histogram.

(ii) In Fig. 10.2, the horizontal axis is compressed. We put  to show that there is no information being displayed on the lower part of the axis.

(iii) Where the widths of the rectangles of a histogram are equal (as in Fig. 10.2), the histogram is similar to a bar chart. Therefore the height of the bars is also proportional to the respective frequencies.

Example 10.3

In a certain function, the ages of the people present were recorded as shown in Table 10.18.

Draw a histogram for this data

Solution

Remember that in a histogram, the area of each rectangle represents the frequency and the width of a rectangle the class interval. Thus, we get the height of the bar by dividing the frequency by the class interval. This result is known as the frequency density (Table 10.19).

The height of each column (Table 10.19) represents the average number of people in each age group.  It is assumed that there is a uniform distribution within the class intervals. Fig. 10.3 shows the required histogram.

Exercise 10.2

1. Table 10.20 represents the number of patients who attended a mobile clinic grouped by age.

Exercise 10.2

1. Table 10.20 represents the number of patients who attended a mobile clinic grouped by age.

        

Using a scale of 2 cm to represent 1 unit on y-axis, draw a histogram to represent the distribution.

2. In a high altitude weather station, wind speeds were observed for a period of 100 days

 Using a scale of 4 cm to represent 1 unit on the y-axis, draw a histogram to represent the distribution.

3. Table 10.22 shows the heights to the nearest centimetre of a sample of seedlings in a tree nursery.

Construct a histogram to represent the distribution. Who do you think would be interested in a tree nursery project? To whose benefit would it be?

4. Use the data in Table 10.23 and a suitable number of class to make a frequency distribution table.

10.2.3 Frequency polygon

Fig. 10.4 shows a frequency polygon obtained using information in Table 10.24 shows the same information that was in Table 10.3 on page 178 but this time, the mid-points or class mid marks of the classes are included. Each class mark is obtained as half the sum of the class limits (or boundaries). These classmarks are useful in the construction of the frequency polygons, calculation of the mean and the standard deviation of grouped data.

In order to construct a frequency polygon of a given set of data, we plot the class mid-points against the  corresponding frequency densities. Therefore, the class mid-points represent the corresponding class intervals. These mid-points are calculated using class limits as follows

When the frequency densities are plotted against the corresponding midpoints and the points joined with straight line segments, we obtain a frequency polygon (Fig. 10.4).

Thus, a frequency polygon is a graph in which frequency densities are plotted against class mid-points and the points are joined with straight line segments.

Note that in Table 10.24, the first point given is the mid-point of the first class whose frequency is 2. Normally, it is taken that the previous class

(i.e 20 – 29) has frequency 0 (zero). So the graph is extended to the mid-point of this class.  Similarly, it is taken that the next class, after the last, has frequency 0 (zero). The graph is also extended to this class. Sometimes, more than one frequency polygon may be drawn on the same axes for purposes of comparing frequency distributions.

Activity 10.5

Table 10.25 shows the performance of two quizes. The maximum possible mark that could be scored was 50.

1. Copy table 10.25 and on it introduce a 4th column with the heading “class mid-values".

2. On the same axes and using an appropriate scale on each axis, draw a frequency polygon for each class.

3. Explain how you would use these frequency polygons.

From this activity, you should have observed the following:

(i) A frequency polygon is obtained by plotting class-mid values against the corresponding frequency densities.

(ii) You needed to add another two columns for the frequency density for each set.