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CHAPTER 39 Cumulative Frequency Cumulative Frequency Cumulative Frequency Tables The cumulative frequency is the running total of the frequency up to the end of each class interval. Cumulative Frequency Graph To draw a cumulative frequency graph: 1. Draw and label the variable on the horizontal axis and the cumulative frequency on the vertical axis 2. Plot the cumulative frequency against the upper class boundary of each class and join the points with a smooth curve. Using Cumulative Frequency Graphs Median When we draw a cumulative frequency graph we can use it to calculate the MEDIAN of the frequency distribution. Work out what half the total frequency is, go to this point on the cumulative frequency axis (vertical axis). Read across to the curve and down to the horizontal axis, this value is the median of the frequency distribution. The Interquartile Range To find the LOWER QUARTILE, work out what one quarter of the total frequency is, go to this point on the cumulative frequency axis. Read across to the curve and down to the horizontal axis, this value is the LOWER QUARTILE of the frequency distribution. To find the UPPER QUARTILE, work out what three quarters of the total frequency is, go to this point on the cumulative frequency axis. Read across to the curve and down to the horizontal axis, this value is the UPPER QUARTILE. Interquartile Range = Upper Quartile – Lower Quartile Cumulative Frequency Graph Example The following table shows the marks obtained in a GCSE mathematics exam: Number of a) Draw a cumulative frequency curve Marks b) Use your graph to 1 - 10 find an estimate for: (i) the median number 11 – 20 of marks 21 - 30 (ii) the interquartile range of the marks 31 – 40 (iii) the percentage of students 41 – 50 who scored less than 36 marks c) If the pass mark is set at 25 marks, estimate how many students pass d) If 65% of the students pass, estimate what the pass mark is Frequency 15 58 92 30 5 Cumulative Frequency Graph Answer Number of Marks (less than) Cumulative Frequency 10.5 15 20.5 15 + 58 = 73 30.5 73 + 92 = 165 40.5 165 + 30 = 195 50.5 195 + 5 = 200 b) (i) The median is the middle value. The total cumulative frequency is 200. So the middle value will be at approximately 100 Median = 22 marks Cumulative Frequency Graph (ii) The lower quartile is the one quarter value. The total cumulative frequency is 200. So the one quarter value will be at approximately 50 Lower quartile = 17.5 Marks The upper quartile is the three quarters value. The total cumulative frequency is 200. So the three quarters value will be at approximately 150. Upper quartile = 28 Marks Interquartile range = 28 – 17.5 = 10.5 Marks (iii) The number of students who score less than 36 Marks = 185 % of students who score less than 36 marks = 185 x 100 200 = 92.5% Cumulative Frequency Graph c) The number of students who score less than 25 marks = 128 The number of students who pass = 200-128 = 72 d) 65% of 200 = 65 x 200 100 130 students score more than the pass mark so 70 students score less than the pass mark Pass mark = 20 Marks Box Plots BOX PLOTS (or BOX AND WHISKER DIAGRAMS) are a convenient way of representing the MINIMUM VALUE, LOWER QUARTILE, MEDIAN, UPPER QUARTILE and MAXIMUM VALUE of a set of data. It is easy to compare two (or more) distributions by comparing their BOX PLOTS. It is very easy to draw a BOX PLOT underneath the CUMULATIVE FREQUENCY CURVE representing the data. A BOX is used to display the LOWER QUARTILE, MEDIAN and UPPER QUARTILE. Lines join this box to the Minimum Value and Maximum Value. A BOX PLOT shows how the data is spread out and how the middle 50% of the data is clustered.