Cumulative frequency revision card
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The document provides information about cumulative frequency curves and distributions: 1. A cumulative frequency curve is drawn from a cumulative frequency table by plotting the upper class boundary against the cumulative frequency. 2. The curve can be used to estimate measures of central tendency and spread, like the median and quartiles, by tracing the values on the y-axis that correspond to the fraction of total frequency, like the 25th percentile. 3. Examples show how to calculate values like the median and interquartile range from curves and tables, and how to determine probabilities from the curve values.
Cumulative frequency revision card
- 1. Cumulative Frequency The cumulative frequency curve is drawn from a cumulative frequency table. We plot the upper class boundary against the cumulative frequency
- 2. WASSCE MAY/JUNE 2018 𝑀𝑎𝑟𝑘𝑠 0 − 9 10 − 19 20 − 29 30 − 39 40 − 49 50 − 59 60 − 69 70 − 79 80 − 89 90 − 99 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 10 28 40 92 𝑦 + 60 90 50 30 15 5 The table shows the distribution of marks scored by 500 candidates in an examination. a) Find the value of 𝑦 b) Construct a cumulative frequency table and use it to draw a cumulative frequency curve. c) Use the curve to estimate the probability of selecting a candidate who scored not more than 45%. Solution 𝑇𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 500 10 + 28 + 40 + 92 + 𝑦 + 60 + 90 + 50 + 30 + 15 + 5 = 500 𝑦 = 80 Class boundary Upper class boundary Frequency Cumulative Frequency −0.5 − 9.5 9.5 10 10 9.5 − 19.5 19.5 28 38 19.5 − 29.5 29.5 40 78 29.5 − 39.5 39.5 92 170 39.5 − 49.5 49.5 140 310 49.5 − 59.5 59.5 90 400 59.5 − 69.5 69.5 50 450 69.5 − 79.5 79.5 30 480 79.5 − 89.5 89.5 15 495 89.5 − 99.5 99.5 5 500 Tracing from the graph, 250 students scored not more than 45% P(Selecting a candidate who scored not more than 45%)= 250 500 = 1 2
- 3. JAMB 2001 The graph above shows the cumulative frequency of the distribution of masses of fertilizer for 48 workers in one institutio Which of the following gives the interquartile range? A. Q3 – Q1 B. Q3 – Q2 C. Q2 – Q1 D. ½ (Q3 – Q1) Solution The interquartile range is the third quartile minus the first quartile The correct answer is A
- 4. JAMB 1984 The cumulative frequency function of the data below is given by the equation 𝑦 = 𝑐𝑓(𝑥). What is cf(5)? A. 30 B. 32 C. 55 D. 62 E. 92 Solution 𝑐𝑓(5) Means adding up the score of 5 𝑐𝑓 5 = 30 + 32 + 30 = 92 The correct answer is E. 𝑺𝒄𝒐𝒓𝒆(𝒏) 𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚(𝒇) 3 30 4 32 5 30 6 35 8 20
- 5. JAMB 2004 The graph above shows the cumulative frequency curve of the distribution of marks in a class test. What percentage of th students scored more than 20 marks? A. 68% B. 28% C. 17% D. 8% Solution 𝑡ℎ𝑒 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑤ℎ𝑜 𝑠𝑐𝑜𝑟𝑒𝑑 𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛 20 marks= 25−18 25 × 100% = 7 25 × 100% = 28%
- 6. 1. The table below is a distribution of ages of pupils. Which of the following points will be plotted when drawing a cumulative frequency curve for the above distribution? A. (4,2) B. (3,5) C. (15.5,15) D. (7,10) E. (0.5,2) The correct answer is (15.5,15) 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4
- 7. 1. The table below is a distribution of ages of pupils. Which of the following points will be plotted when drawing a cumulative frequency curve for the above distribution? A. (4,2) B. (3,5) C. (15.5,15) D. (7,10) E. (0.5,2) The correct answer is (15.5,15) 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4 Solution We plot the upper class boundary against the cumulative frequency. Upper Class Boundary Cumulative frequency 5.5 2 10.5 10 15.5 15 20.5 18 25.5 22
- 8. 2. The table below is a distribution of ages of pupils. Which of the following points will be the highest point on the cumulative frequency curve? A. (4,22) B. (24,22) C. (21,22) D. (23,10) E. (0.5,2) The correct answer is (23,22) 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4
- 9. 2. The table below is a distribution of ages of pupils. Which of the following points will be the highest point on the cumulative frequency curve? A. (4,22) B. (24,22) C. (21,22) D. (25.5,22) E. (25.5,4) The correct answer is (23,22) 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4 Solution We plot the upper class boundary against the cumulative frequency. Upper Class Boundary Cumulative frequency 5.5 2 10.5 10 15.5 15 20.5 18 25.5 22
- 10. 3. The graph below is the cumulative frequency curve that represents marks scored in a class test. Estimate the median mark from the graph. A. 12.5 B. 13.5 C. 14.5 D. 11.5 E. 10.5 The correct answer is 11.5
- 11. 3. The graph below is the cumulative frequency curve that represents marks scored in a class test. Estimate the median mark from the graph. A. 12.5 B. 13.5 C. 14.5 D. 11.5 E. 10.5 The correct answer is 11.5 Solution median mark= 1 2 𝑓th value = 1 2 (22)th value = 11th value Tracing from the graph it corresponds to 11.5
- 12. 4. The graph below is the cumulative frequency curve that represents marks scored in a class test. Estimate the third quartile from the graph. A. 17.5 B. 18.5 C. 14.0 D. 15.5 E. 16.5 The correct answer is 17.5
- 13. 4. The graph below is the cumulative frequency curve that represents marks scored in a class test. Estimate the third quartile from the graph. A. 17.5 B. 18.5 C. 14.0 D. 15.5 E. 16.5 The correct answer is 17.5 Solution Third quartile= 3 4 𝑓th value = 3 4 (22)th value = 16.5th value Tracing from the graph it corresponds to 17.5
- 14. 5. The graph below is the cumulative frequency curve that represents marks scored in a class test. How many students sat for the test?. A. 18 B. 19 C. 20 D. 21 E. 22 The correct answer is 22
- 15. 5. The graph below is the cumulative frequency curve that represents marks scored in a class test. How many students sat for the test?. A. 18 B. 19 C. 20 D. 21 E. 22 The correct answer is 22 Solution The number of students is is equal to the total frequency which is The y-value of the highest point =22
- 16. 6. The graph below is the cumulative frequency curve that represents marks scored in a class test. What is the probability of scoring at least 13?. A. 7 11 B. 2 22 C. 3 11 D. 4 11 E. 7 22 The correct answer is 7 22
- 17. 6. The graph below is the cumulative frequency curve that represents marks scored in a class test. What is the probability of scoring at least 13?. A. 7 11 B. 2 22 C. 3 11 D. 4 11 E. 7 22 The correct answer is 7 22 Solution P(scoring at least 13)= 22−15 22 = 7 22
- 18. 8. The table below is a distribution of ages of pupils. Which of the following points will the cumulative frequency curve pass through? A. (13,15) B. (12,15) C. (4,2) D. (5.5,8) E. (5.5,2) The correct answer is (5.5,2) 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4
- 19. 8. The table below is a distribution of ages of pupils. Which of the following points will the cumulative frequency curve pass through? A. (13,15) B. (12,15) C. (4,2) D. (5.5,8) E. (5.5,2) The correct answer is (5.5,2) 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4 Solution We plot the upper class boundary against the cumulative frequency. Upper Class Boundary Cumulative frequency 5.5 2 10.5 10 15.5 15 20.5 18 25.5 22
- 20. 9. The table below is a cumulative frequency distribution table. What is the value of 𝑓? A. 11 B. 12 C. 13 D. 14 E. 15 The correct answer is 12 Marks Frequency Cumulative frequency 1 − 4 3 𝑥 5 − 8 𝑦 7 9 − 12 2 9 13 − 16 𝑥 12
- 21. 9. The table below is a cumulative frequency distribution table. What is the value of 𝑓? A. 11 B. 12 C. 13 D. 14 E. 15 The correct answer is 12 Marks Frequency Cumulative frequency 1 − 4 3 𝑥 5 − 8 𝑦 7 9 − 12 2 9 13 − 16 𝑥 12 Solution 𝑓= the final cumulative frequency=12
- 22. 10. The table below is a cumulative frequency distribution table. What is the value of 𝑥 + 𝑦? A. 8 B. 7 C. 9 D. 10 E. 11 The correct answer is 7 Marks Frequency Cumulative frequency 1 − 4 3 𝑥 5 − 8 𝑦 7 9 − 12 2 9 13 − 16 𝑥 12
- 23. 10. The table below is a cumulative frequency distribution table. What is the value of 𝑥 + 𝑦? A. 8 B. 7 C. 9 D. 10 E. 11 The correct answer is 7 Marks Frequency Cumulative frequency 1 − 4 3 𝑥 5 − 8 𝑦 7 9 − 12 2 9 13 − 16 𝑥 12 Solution 𝑓= 12 3 + 𝑦 + 2 + 𝑥 = 12 𝑥 + 𝑦 = 7
- 24. Summary 1. Plot the marks against the cumulative frequencies 2. You can estimate the measures of spread from the cumulative frequency curve 3. Trace 1 4 𝑓 𝑡ℎ value to estimate first quartile 4. Trace 1 2 𝑓 𝑡ℎ value to estimate first median 5. Trace 3 4 𝑓 𝑡ℎ value to estimate third quartile
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