In the previous section we saw the method for preparing Discrete Frequency distribution tables, Grouped Frequency distribution tables, Bar graphs and Histograms. In this section we will see some solved examples.
Solved example 1.1
There are 44 students in a class. The list below shows the distance (in km) that they travel from their home to school.
| 6 | 2 | 7 | 12 | 1 | 9 | 2 | 6 |
|---|---|---|---|---|---|---|---|
| 5 | 7 | 3 | 4 | 1 | 5 | 4 | 4 |
| 5 | 8 | 6 | 5 | 2 | 5 | 9 | 5 |
| 11 | 12 | 1 | 9 | 2 | 14 | 4 | 7 |
| 9 | 6 | 6 | 7 | 3 | 2 | 6 | 3 |
| 4 | 7 | 9 | 3 |
Make a frequency table and answer the following questions:
1) How many students are from exactly 1 km away?
2) How many students travel more than 5 kms?
3) How many travel between 5 and 10 kms?
4) How many travel more than 10 kms?
Solution:
 |
| Fig.1.21 |
The frequency distribution table is shown in fig.1.21 below. It is a Discrete frequency distribution table. The distances in the raw data list are written in ascending order in the column 1.
Based on the table, we can answer the given questions.
1. The answer 3 is obtained from the first row
2. More than 5 kms means 5 km is not to be included. So we consider all the rows below the one corresponding to 5 km. Thus we get the answer as 21 students
3. Between 5 km and 10 km means both 5 and 10 are to be included. So we consider all the rows below 4 and above 11. Thus we get the answer as 23 students.
4. More than 10 km means 10 km is not included. So we consider all the higher values than 10. That is ., row 11 and all the rows below 11. Thus we get the answer as 4.
Solved example 1.2
The weights of the members of the school health club are given below:
| 38 | 37.5 | 40.5 | 59 | 48 | 48 | 37.7 |
|---|---|---|---|---|---|---|
| 58 | 50 | 54.5 | 39 | 40 | 40.5 | 49 |
| 32 | 43 | 45 | 53 | 37 | 44 | 51 |
| 50.5 | 32.5 | 46 | 55 | 36 | 44.5 | 47 |
| 42.5 | 33 |
Prepare a Grouped frequency distribution table.
Solution:
The smallest value is 32, and the largest value is 59. So we need the region between 32 and 59 on the number line. This is shown in the fig below:
The smallest value is 32, and the largest value is 59. So we need the region between 32 and 59 on the number line. This is shown in the fig below:
The length of the required portion is 59 -32 = 27. If we decide to fix the width of class intervals to be equal to 10, then the number of class intervals will be equal to 27/10 = 2.7. So we can make utmost 3 class intervals. This is rather low. When the number of class intervals are low, the values in the raw data list will fall within a small area in the number line. It will appear like the raw data list itself. It will not serve the purpose of an ‘organised distribution’.
On the other extreme, if the class interval is very low, the values will fall within a large area in the number line. More rows will be required in the table, and it will appear like a ‘Discrete frequency distribution table’.
So instead of 10, let us choose a 'class interval' of 5. Then the number line should also show the intervals of 5. That is., number line should show 0, 5, 10, 15 etc., This is shown below:
In the above fig., we can see that
• Some portion before 32, and some portion after 59 has to be included, in order to make the intervals 'equal'. So we have to extend the red portion towards the left upto 30, and towards the right upto 60.
• Also note that, in this present problem, no values are falling in the region from 0 to 30. So that region can be avoided in our diagrams. But when such a region is avoided, a 'cut line' should be shown.
The following fig. shows these two modifications
So now we have the required region, and it is divided into 6 equal class intervals, each of width 5. We can make the frequency distribution table as shown in fig.1.25. It is a Grouped frequency distribution table. kk
The Histogram based on the table is shown in fig.1.26 below:
[It may be noted that, the Histograms are usually drawn on Graph papers. They show the units clearly. The divisions are shown with thin lines and the sub divisions are shown with thinner lines. So, for a histogram drawn on graph paper, we do not have to show the horizontal white dotted lines as in fig.1.26.]
From the above table and histogram, we can make the following conclusions:
• Only 3 students have a heavy weight of 55 to 60 kg
• Only 3 students have a light weight of 30 to 35 kg
• Most students have a weight ranging from 35 to 55 kg. And in this, the range possessed by the largest number of students is 40 to 45. It is possessed by 7 students.
Now let us take a look at the extreme cases, when the class interval is too small or too large. First we will see the case when the class is too small. If we take the class as '2', the histogram will be as shown in the fig.1.27 below.
From the histogram, we can make the following conclusions:
• Only 2 students have heavy weights of 58 to 60 kg
• Only 3 students have light weights of 32 to 34 kg
• Most students have a weight ranging from 36 to 52 kg. And in this, the range possessed by the largest number of students is 36 to 38. It is possessed by 4 students.
We can see that this histogram gives more 'detailed' results. But it closely resembles a Bar graph, and requires more space. On many occasions, such high level of details may not be required.
Next we will see the other extreme case when the class is too large. If we take the class as '10', the histogram will be as shown in the fig.1.28 below:
From the histogram, we can make the following conclusions:
• 9 students have weights ranging from 30 to 40
• 9 students have weights ranging from 50 to 60
• 12 students have weights ranging from 40 to 50
We can see that this histogram gives less detailed results. On many occassions, such low level of details may not be sufficient. It closely resembles the 'raw data' itself because, the number of students with particular weights are distributed randomly in the three classes.
So for a particular problem, we must select the class interval according to the level of detail required. Some times the class interval that we have to use may be specified in the problem.
Solved example 1.3
The weekly wages of 30 workers in a factory are given below:
(1) Make a Grouped frequency table with intervals as 800 – 810, 810 – 820 and so on.
(2) Draw a histogram for the frequency table made for the data in Question 3, and answer the following questions.
(i) Which interval has the maximum number of workers?
(ii) How many workers earn 860 and more?
(iii) How many workers earn less than 860?
Solution:
The class intervals are given in the question itself. So we can directly make the table. It is shown in the fig. 1.29 below:
Based on the above table, the Histogram is prepared and is shown in fig.1.30 below:
Based on the above histogram, we can answer the questions.
(i) The tallest rectangle is the one corresponding to the interval 820 - 840. It has 10 workers. So the answer is : interval 820 - 840
(ii) After the 860 mark, there are two rectangles. The first has 4 workers and the second has 5 workers. So the answer is 9. [Note that, if a worker has a wage of exactly 860, then he is included in 860 - 880, and not 840 - 860. All the values greater than 860 will fall to the right of the 860 mark on the number line. So all the rectangles to the right of the 860 mark are to be considered]
(iii) There are 3 rectangles to the left of 860. From their heights, we get the answer as 5 +10 +6 =21
In the next section, we will discuss about Pie charts.
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Copyright©2016 High school Maths lessons. blogspot.in - All Rights Reserved
In the above fig., we can see that
• Some portion before 32, and some portion after 59 has to be included, in order to make the intervals 'equal'. So we have to extend the red portion towards the left upto 30, and towards the right upto 60.
• Also note that, in this present problem, no values are falling in the region from 0 to 30. So that region can be avoided in our diagrams. But when such a region is avoided, a 'cut line' should be shown.
The following fig. shows these two modifications
 |
| Fig.1.25 |
 |
| Fig.1.26 |
From the above table and histogram, we can make the following conclusions:
• Only 3 students have a heavy weight of 55 to 60 kg
• Only 3 students have a light weight of 30 to 35 kg
• Most students have a weight ranging from 35 to 55 kg. And in this, the range possessed by the largest number of students is 40 to 45. It is possessed by 7 students.
Now let us take a look at the extreme cases, when the class interval is too small or too large. First we will see the case when the class is too small. If we take the class as '2', the histogram will be as shown in the fig.1.27 below.
 |
| Fig.1.27 |
• Only 2 students have heavy weights of 58 to 60 kg
• Only 3 students have light weights of 32 to 34 kg
• Most students have a weight ranging from 36 to 52 kg. And in this, the range possessed by the largest number of students is 36 to 38. It is possessed by 4 students.
We can see that this histogram gives more 'detailed' results. But it closely resembles a Bar graph, and requires more space. On many occasions, such high level of details may not be required.
Next we will see the other extreme case when the class is too large. If we take the class as '10', the histogram will be as shown in the fig.1.28 below:
 |
| Fig.1.28 |
• 9 students have weights ranging from 30 to 40
• 9 students have weights ranging from 50 to 60
• 12 students have weights ranging from 40 to 50
We can see that this histogram gives less detailed results. On many occassions, such low level of details may not be sufficient. It closely resembles the 'raw data' itself because, the number of students with particular weights are distributed randomly in the three classes.
So for a particular problem, we must select the class interval according to the level of detail required. Some times the class interval that we have to use may be specified in the problem.
Solved example 1.3
The weekly wages of 30 workers in a factory are given below:
| 830 | 835 | 890 | 810 | 835 | 836 | 869 |
|---|---|---|---|---|---|---|
| 845 | 898 | 890 | 820 | 860 | 832 | 833 |
| 855 | 845 | 804 | 808 | 812 | 840 | 885 |
| 835 | 835 | 836 | 878 | 840 | 868 | 890 |
| 806 | 840 |
(1) Make a Grouped frequency table with intervals as 800 – 810, 810 – 820 and so on.
(2) Draw a histogram for the frequency table made for the data in Question 3, and answer the following questions.
(i) Which interval has the maximum number of workers?
(ii) How many workers earn 860 and more?
(iii) How many workers earn less than 860?
Solution:
The class intervals are given in the question itself. So we can directly make the table. It is shown in the fig. 1.29 below:
 |
| Fig.1.29 |
 |
| Fig.1.30 |
Based on the above histogram, we can answer the questions.
(i) The tallest rectangle is the one corresponding to the interval 820 - 840. It has 10 workers. So the answer is : interval 820 - 840
(ii) After the 860 mark, there are two rectangles. The first has 4 workers and the second has 5 workers. So the answer is 9. [Note that, if a worker has a wage of exactly 860, then he is included in 860 - 880, and not 840 - 860. All the values greater than 860 will fall to the right of the 860 mark on the number line. So all the rectangles to the right of the 860 mark are to be considered]
(iii) There are 3 rectangles to the left of 860. From their heights, we get the answer as 5 +10 +6 =21
In the next section, we will discuss about Pie charts.
PREVIOUS CONTENTS NEXT
Copyright©2016 High school Maths lessons. blogspot.in - All Rights Reserved
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