Problems on measures of central tendency

1. The following numbers give the weights of 55 students of a class. Prepare a suitable frequency table:  42, 74, 40, 60, 82, 115, 41, 61, 75, 83, 63, 53, 110, 76, 84, 50, 67, 65, 78, 77, 56, 95, 68, 69, 104, 80, 79, 79, 54, 73, 59, 81, 100, 66, 49, 77, 90, 84, 76, 42, 64, 69, 70, 80, 72, 50, 79, 52, 103, 96, 51, 86, 78, 94, 71.

          (a) Draw the histogram and frequency polygon of the above data. 
          (b) For the above heights, prepare a cumulative frequency table and draw the less than ogive. 

   Sol: Given the weights of 55 students, we can prepare a frequency table by categorizing the data into class intervals. Let's use intervals of size 10 to create the frequency table.

• Firstly, find the minimum and maximum weight to determine the range. MIN = 40, MAX = 115.
• Let's choose class intervals of size 10: 40-49, 50-59, 60-69, etc.
• Count the number of students in each interval.

    Frequency table:

                                               

   

              (a) Histogram: 

    

              (a) Frequency polygon:

     

             (b)  Cumulative frequency table:

Interval	Cumulative Frequency
40-49	7
50-59	14
60-69	24
70-79	40
80-89	47
90-99	51
100-109	54
110-119	55

     (b) Ogive curve:

    2. What are the points to be borne in mind in the formation of a frequency table?
        Choosing appropriate class intervals, from a frequency table for the following data: 10.2, 0.5, 5.2, 6.1, 3.1, 6.7, 8.9, 5.4, 3.6, 9.2, 6.1, 7.3, 2.0, 1.3, 6.4, 8.0, 4.3, 4.7, 12.4, 8.6, 13.1, 3.2, 9.5, 7.6, 4.0, 5.1, 8.1, 1.1, 11.5, 3.1, 6.8, 7.0, 8.2, 2.0, 3.1, 6.5, 11.2, 12.0, 5.1, 10.9, 11.2, 8.5, 2.3, 3.4, 5.2, 10.7, 4.9, 6.2.

Sol:  When forming a frequency table, there are several points that should be borne in mind:

 

•  Understand the purpose: The table should be suitable for the objective of the analysis. For instance, detailed classifications may be required for an in-depth study, while broader categories may suffice for an overview.
• Minimum and Maximum values: Determine the range of the data by identifying the minimum and maximum values. This helps in defining the classes.
• Class Intervals: It should be uniform, meaning that they should have equal width, unless the data necessitates variable width due to its nature. Choose a suitable class width based on the range and the number of observations. The width should neither be too small nor too large. The starting point of the first-class intervals is often a convenient number slightly less than or equal to the smallest observation.
• The number of classes typically ranges from 5 to 20. The formula 2 power k > n, where n is the number of observations and k is the number of classes, can be a good starting point.
• The class intervals should be distinct and not overlap.
• If there are exceptionally high or low values, consider using open-ended classes like "more than or less than".
• Determine the class mid points, boundaries, and limits. This helps in plotting graphs and further analysis.
• They can be used for raw data to count the frequency for each class.
• If required, include a column for cumulative frequency.

  Frequency table:

                                          

Interval	Frequency
0.5 - 3.0	6
3.0 - 5.5	15
5.5 - 8.0	10
8.0 - 10.5	9
10.5 - 13.0	7

 

3. The following table shows the distribution of the number of students per teacher in 750 colleges:

  Students: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28

Frequency: 7, 46, 165, 195, 189, 89, 28, 19, 19, 9, 3

Draw histogram for the data

4. Descriptive statistics for age wise view on electrical vehicles.

Row	Age	Frequency	Percentage
1	Below 25	243	68.0672268907563
2	25 - 35	55	15.406162464
3	35 – 45	34	9.5238095238095237
4	Above 45	25	7.0028011204481793

5. Descriptive statistics for occupation wise view on electrical vehicles

Row	Occupation	Frequency	Percentage
1	Student	203	56.862745098039213
2	Government Employee	27	7.56302521008
3	Private Employee	91	25.490196078431371
4	Self-employee	24	6.7226890756302522
5	Un-employee	12	3.3613445378151261

   6. Data visualization for global warming wise view on electrical vehicles  

                  

Row	Global warming	Frequency	Percentage
1	Strongly Agree	135	37.815126050420169
2	Agree	182	50.980392156862742
3	Disagree	28	7.8431372549019605
4	Strongly Disagree	12	3.361344537

   7. Data visualization for knowledge about electrical vehicles     

     

Row	Knowledge about Ev	Frequency	Percentage
1	Newspapers	46	12.88515406162465
2	Magazine	57	15.96638655
3	Television	55	15.406162464985995
4	Internet	199	55.742296918767508

 

8. Data visualization for charging points for electrical vehicles  

Row	Charging Points	Frequency	Percentage
1	Each 50 KM	129	36.134453781512605
2	Each 100 KM	141	39.495798319327733
3	Each 150 KM	54	15.12605042
4	Each 200 KM	33	9.2436974789915958