Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004 AND Mathematical Studies Standard Level Peter Blythe, Jim Fensom, Jane Forrest and Paula Waldman de Tokman Oxford University Press, 2012

The mode, median and mean. These can be determined with discrete (non- grouped data) or grouped data. Discrete data takes the form of a frequency table (can find mode, median or mean). Grouped data takes a continuous (classes) form (can find the modal class and mean). Recall: Measures of Central Tendency

The cumulative frequency is the sum of all the frequencies up to and including the new value. To draw a cumulative frequency curve you need to: – construct a cumulative frequency table, with the upper boundary of each class interval in one column and the corresponding cumulative frequency in another. – Plot the upper class boundary on the x-axis and the cumulative frequency of the y-axis. Cumulative Frequency Curve

The heights of 200 students are recorded in the following table. Construct a cumulative frequency curve. Height (h) in cm Frequency 140 ≤ h < ≤ h < ≤ h < ≤ h < ≤ h < ≤ h < ≤ h < 2102 Example

Height (h) in cmFrequency Upper Bdry Cumulative F 140 ≤ h < ≤ h < ≤ h < ≤ h < ≤ h < ≤ h < ≤ h < Example First complete a table with an Upper Boundary Column and Cumulative Frequency Column.

Example Plot the Cumulative Frequency vs. Upper Boundary.

A supermarket is open 24 hrs a day and has a free car park. The number of parked cars each hour is monitored over a period of several days. Organize the following data into a cumulative frequency table and graph. Practice Number of Parked Cars / HrFrequency

A supermarket is open 24 hrs a day and has a free car park. The number of parked cars each hour is monitored over a period of several days. Organize the following data into a cumulative frequency table and graph. Practice Number Cars / HrFrequencyUpper B.Cumulative F Plot Upper Boundary vs. Cumulative Frequency

We can use cumulative frequency curves to find estimates of the percentiles and quartiles. Percentiles separate large ordered sets of data into hundredths. Quartiles separate large ordered sets of data into quarters. When the data is arranged in order: – Lower quartile = 25 th percentile – Median = 50 th percentile – Upper quartile = 75 th percentile Interpreting Cumulative Frequency Curves

For any data set: – 25% (or one quarter) of the values are between the smallest value and the lowest quartile – 25% are between the lower quartile and the median – 25% are between the median and the upper quartile – 25% are between the upper quartile and the largest value – 50% of the data lie between the lower and upper quartile Interpreting Cumulative Frequency Curves

Find the median (to cut the data in half). Find the median of each half. Another Way of Thinking About This:

The median is the second quartile, Q 2 or 50 th percentile The lower quartile, Q 1, is the median of the lower half of the data or 25 th percentile The upper quartile, Q 3, is the median of the upper half of the data or 75 th percentile The inter-quartile range is the difference in the upper quartile and the lower quartile. IQR = Q 3 – Q 1 Another Way of Thinking About This:

Find the median (to cut the data in half) Find the median of each half IQR = Q3 Q3 - Q1Q1 Half of the values are between 3.5 and 6.5. = 6.5 – 3.5 = 3 Another Way of Thinking About This:

For the data set: 9, 8, 2, 3, 7, 6, 5, 4, 5, 4, 6, 8, 9, 5, 5, 5, 4, 6, 6, 8 Find the: a) median b) lower quartile c) upper quartile d) inter-quartile range a) 5.5 b) 4.5 c) 7.5 d) 3

For the data set: 6, 4, 9, 15, 5, 13, 7, 12, 8, 10, 4, 1, 13, 1, 6, 4, 5, 2, 8, 2 Find the: a) median b) lower quartile c) upper quartile d) inter-quartile range a) 6 b) 4 c) 9.5 d) 5

50 contestants play the game Oware. In total they have to play 49 games to arrive at a champion. The average time for the 49 games are given in the table. Practice Time (t minutes)Frequency 3 ≤ t < 44 4 ≤ t < ≤ t < ≤ t < 79 7 ≤ t < 83 8 ≤ t < 92 9 ≤ t < 101 a.Construct a cumulative frequency table for the data. b.Draw a cumulative frequency graph. c.Use your graph to estimate: The lower quartile The median The upper quartile The interquartile range The 30 th percentile

Practice Time (t minutes)FrequencyUpper B.Cumulative F. 3 ≤ t < ≤ t < ≤ t < ≤ t < ≤ t < ≤ t < ≤ t < Q 1 = 4.7 min (from 12.5 on the graph) Q 2 = 5.5 min (from 25 on the graph) Q 3 = 6.4 min (from 37.5 on the graph) IQR = 1.7 min 30 th = 4.9 min (from 15 on the graph)

Practice a. 29 min b. 33 min c min