Statistics

This chapter is about presenting data graphically so that it can be understood at a glance. You will revise how raw data is organised into a frequency distribution table with class intervals, and the meaning of terms such as class limits, class size or width, and class mark, which is the mid-point of a class interval found as the average of its upper and lower limits. You will study three ways of presenting data: bar graphs, in which bars of uniform width are drawn with equal spacing and heights proportional to the values; histograms, which are drawn for continuous class intervals with rectangles touching each other; and frequency polygons. You will learn how to draw a histogram when the class intervals have varying widths, by finding the adjusted frequency, or length of each rectangle, so that areas are proportional to frequencies. You will draw frequency polygons by joining the mid-points of the tops of the rectangles of a histogram, or directly by plotting class marks against frequencies and joining the points, including imaginary classes with zero frequency at each end.

Detailed Worksheet: Statistics Section A - Definitions (10 marks) 1. What is a frequency distribution? What are class intervals? (2 marks) 2. Define class size and class mark. Find the class mark of the interval 30-40. (2 marks) 3. How is a histogram different from a bar graph? (2 marks) 4. What is a frequency polygon? How is it drawn from a histogram? (2 marks) 5. When are adjusted frequencies needed in drawing a histogram? (2 marks) Section B - Calculations and Applications (15 marks) 6. The marks of 20 students are: 12, 25, 8, 17, 32, 21, 15, 29, 4, 18, 36, 27, 11, 22, 19, 30, 6, 24, 14, 38. Make a frequency distribution table with classes 0-10, 10-20, 20-30 and 30-40 (upper limit not included). (3 marks) 7. Find the class marks of the classes 150-155, 155-160, 160-165, 165-170 and 170-175, and the class size. (3 marks) 8. The class marks of a distribution are 47, 52, 57, 62, 67 and 72. Find the class size and the class limits of the first and last classes. (3 marks) 9. A distribution has classes 1-2, 2-3, 3-5, 5-7 and 7-10 with frequencies 6, 8, 12, 10 and 9. Taking the minimum class size as 1, find the adjusted frequency (length of the rectangle) for each class. (3 marks) 10. The number of students in five sections of a school is: A 45, B 38, C 42, D 50, E 35. Find the total, the section with the most students and the difference between the largest and smallest sections. Choose a suitable scale for a bar graph. (3 marks) Section C - Diagrams (10 marks) 11. Draw a histogram for the following weekly wages of workers: 800-900: 8, 900-1000: 12, 1000-1100: 15, 1100-1200: 10, 1200-1300: 5. (4 marks) 12. Draw a bar graph for the following data on the modes of transport used by 60 students: bus 22, cycle 15, walking 10, car 8, auto 5. (3 marks) 13. Draw a frequency polygon for the following data without drawing a histogram: class marks 5, 15, 25, 35, 45 with frequencies 4, 10, 16, 8, 2. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. The following table gives the lengths of 40 leaves in mm: 118-126: 3, 127-135: 5, 136-144: 9, 145-153: 12, 154-162: 5, 163-171: 4, 172-180: 2. Convert the classes into continuous form, draw a histogram and state the class with the maximum number of leaves. (5 marks) 15. The marks of two sections in a test out of 50 are: classes 0-10, 10-20, 20-30, 30-40, 40-50; Section P: 3, 9, 17, 12, 9; Section Q: 5, 19, 15, 10, 1. Draw frequency polygons for both on the same graph and compare the performance of the two sections. (5 marks) 16. The distribution of ages of 100 people is: 0-10: 10, 10-20: 15, 20-40: 30, 40-60: 25, 60-90: 20. Calculate adjusted frequencies using the minimum class size, draw a histogram, and explain why drawing rectangles with the given frequencies as heights would be misleading. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use graph paper, choose suitable scales and label both axes.