Statistics

Statistics helps us summarise large sets of data with a single representative value. In Class 9 you found the mean, median and mode of ungrouped data. In this chapter you will find these three measures of central tendency for grouped data, where observations are arranged in class intervals such as 0-10, 10-20 and so on. Each class is represented by its class mark, the midpoint (upper limit + lower limit)/2, on the assumption that the frequency of a class is centred at its class mark. The mean is found by the direct method, the assumed mean method or the step-deviation method, which simplify calculation when the numbers are large. The mode lies in the modal class, the class with the highest frequency, and is given by l + ((f1 - f0)/(2f1 - f0 - f2)) x h. The median uses cumulative frequencies and the formula l + ((n/2 - cf)/f) x h. You will also use the empirical relation 3 Median = Mode + 2 Mean and decide which measure best suits a given situation.

Detailed Worksheet: Statistics Section A - Definitions (10 marks) 1. What is a class mark? Write the formula to find it. (2 marks) 2. Write the formula for the mean of grouped data by the direct method and by the step-deviation method, explaining each symbol. (2 marks) 3. Define the modal class. Write the formula for the mode of grouped data, explaining l, h, f1, f0 and f2. (2 marks) 4. What is cumulative frequency? Write the formula for the median of grouped data, explaining each symbol. (2 marks) 5. State the empirical relationship between the three measures of central tendency. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the mean of the following data using the direct method. Marks: 0-10, 10-20, 20-30, 30-40, 40-50; Number of students: 5, 8, 15, 16, 6. (3 marks) 7. Find the mean height of 40 plants using the step-deviation method. Height (cm): 100-120, 120-140, 140-160, 160-180, 180-200; Number of plants: 4, 6, 10, 12, 8. (3 marks) 8. Find the mode of the following data. Class: 0-20, 20-40, 40-60, 60-80, 80-100; Frequency: 10, 15, 25, 30, 20. (3 marks) 9. Find the median of the following data. Class: 10-20, 20-30, 30-40, 40-50, 50-60; Frequency: 4, 6, 10, 12, 8. (3 marks) 10. The mean of the following distribution is 27. Find the missing frequency p. Class: 0-10, 10-20, 20-30, 30-40, 40-50; Frequency: 5, 8, p, 16, 6. (3 marks) Section C - Diagrams (10 marks) 11. Draw a histogram for the distribution of heights of 40 plants given in Question 7 (100-120: 4, 120-140: 6, 140-160: 10, 160-180: 12, 180-200: 8). Mark the modal class on the histogram and estimate the mode from the figure. (4 marks) 12. Prepare a cumulative frequency table (less than type) for the data: Class 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 6, 9, 15, 12, 8. Identify the median class from the table and show it clearly. (3 marks) 13. Draw a frequency polygon for the data: Class 10-20, 20-30, 30-40, 40-50, 50-60 with frequencies 4, 6, 10, 12, 8, by joining the class marks. Mark the highest point and state which class is modal. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. For the data Class 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 16, 6, find the mean, the median and the mode. Check whether the empirical relation 3 Median = Mode + 2 Mean holds approximately and comment on the result. (5 marks) 15. The median of the following data is 28.5 and the total frequency is 60. Find the values of x and y. Class: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60; Frequency: 5, x, 20, 15, y, 5. (5 marks) 16. A shoe shop owner, a school principal studying average marks, and a town planner studying household incomes each need one measure of central tendency. Which measure (mean, median or mode) should each use? Give reasons, and explain why the median is often preferred for income data. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the complete working table (class marks, deviations, products or cumulative frequencies) for every calculation. Give answers correct to two decimal places.