Box Plots & Cumulative Frequency Graphs

Box Plots AKA: box and whiskers plots Graphical display of the 5 number summary: Minimum value Lower quartile (Q1) Median (Q2) Upper quartile (Q3) Maximum value

Box Plots Example. Draw a box plot for the following data: Determine 5 number summary. Create an appropriate number line. Be sure to label the axis. 3. Plot the 5 number summary & connect the appropriate points using a straight edge. 8 2 3 9 6 5 4 Min: 2 Q1: 2.5 Q2: 5 Q3: 6 Max: 9

Outliers Outliers are noted on box plots with an asterisk or a dot Use 1.5 X IQR criteria It is possible to have more than one outlier at either end Each whisker extends to the last value that is not an outlier

Outliers Example. Draw a box plot for the following data. Determine the 5 number summary. Test for outliers. 3. Create an appropriate number line. Be sure to label the axis. 4. Plot the 5 number summary & connect the appropriate points using a straight edge. 1 3 5 6 7 8 9 10 12 13 14 16

Outliers Determine the 5 number summary. Test for outliers. 3. Create an appropriate number line. Be sure to label the axis. 4. Plot the 5 number summary & connect the appropriate points using a straight edge. Min: 1 Q1: 6.5 Q2: 8 Q3: 10 Max: 16 Lower Boundary Upper Boundary Q1 – 1.5 X IQR Q3 + 1.5 X IQR 6.5−1.5×3.5=1.25 10+1.5×3.5=15.25 1 is below lower boundary 16 is above upper boundary

Box Plots How can we make the calculator do this for us? “Stat” “Edit” Enter data into L1 “2nd” “y=” (“Stat Plot”) Select the plot you’d like to use Turn plot “on” Select the type you’d like to use List: L1 (or applicable list) “Graph” Tips: -Make sure there aren’t equations in “y=” that will interfere with your box plot -You can use the calculator to check the box plot that you create

Interpreting Box Plots 25% of values are between smallest value & lower quartile (lower whisker) 25% of values are between lower quartile & median (lower small rectangle) 25% of values are between median & upper quartile (upper small rectangle) 25% of values are between upper quartile & largest value (upper whisker) 50% of values lie between lower & upper quartiles (entire rectangle)

Interpreting Box Plots A set of data with a symmetric distribution will have a symmetric box plot. The whiskers are the same length and the median is in the center of the box.

Interpreting Box Plots A set of data which is positively skewed will have a positively skewed box plot. The right whisker is longer than the left whisker and the median line is to the left of the box.

Interpreting Box Plots A set of data which is negatively skewed will have a negatively skewed box plot. The left whisker is longer than the right whisker and the median line is to the right of the box.

Parallel Box Plots A visual comparison of the distribution of two data sets Can easily compare descriptive statistics, such as median, range & IQR

Parallel Box Plots Example. A hospital is trialing a new anesthetic drug and has collected data on how long the new and old drugs take before the patient becomes unconscious. They wish to know which drug is more reliable. Using the below box plots, compare the two drugs for speed and reliability.

Parallel Box Plots Speed: Using the median, 50% of the time the new drug takes 9 seconds or less, compared with 10 seconds for the old drug. We conclude that the new drug is generally a little quicker. Reliability: Old drug: Range = 21 – 5 = 16; IQR = 12.5 – 8 = 4.5 New drug: Range = 12 – 7 = 5; IQR = 10 – 8 = 2 The new drug times are less “spread out” than the old drug times. The new drug is more reliable.

Cumulative Frequency Graphs Cumulative frequency: the sum of all the frequencies up to and including the new value Race finishing time t Frequency Cumulative Frequency 2 h 26 ≤ t < 2 h 28 8 2 h 28 ≤ t < 2 h 30 3 8 + 3 = 11 2 h 30 ≤ t < 2 h 32 9 11 + 9 = 20 2 h 32 ≤ t < 2 h 34 11 20 + 11 = 31 2 h 34 ≤ t < 2 h 36 12 31 + 12 = 43 2 h 36 ≤ t < 2 h 38 7 43 + 7 = 50 2 h 40 ≤ t < 2 h 42 5 50 + 5 = 55 2 h 42 ≤ t < 2 h 48 55 + 8 = 63 2 h 48 ≤ t < 2 h 56 6 63 + 6 = 69

Cumulative Frequency Graphs To draw a cumulative frequency graph: scale and label the axes correctly (variable on the x-axis & cumulative frequency on the y-axis) plot the first point (lowest bound, 0) plot the middle points (upper bound, corresponding cumulative frequency) plot the last point (highest bound, total frequency) connect the points with a smooth curve

Cumulative Frequency Graphs

Cumulative Frequency Graphs Example. A supermarket is open 24 hours a day and has free parking. The number of parked cars each hour is monitored over a period of several days. Organize this information into a cumulative frequency table. Then draw a graph of the cumulative frequency. # of cars parked per hour (n) Frequency 0 ≤ n < 50 6 50 ≤ n < 100 23 100 ≤ n < 150 41 150 ≤ n < 200 42 200 ≤ n < 250 30 250 ≤ n < 300 24 300 ≤ n < 350 9 350 ≤ n < 400 5

Cumulative Frequency Graphs # of cars parked per hour (n) Frequency Cumulative frequency 0 ≤ n < 50 6 50 ≤ n < 100 23 29 100 ≤ n < 150 41 70 150 ≤ n < 200 42 112 200 ≤ n < 250 30 142 250 ≤ n < 300 24 166 300 ≤ n < 350 9 175 350 ≤ n < 400 5 180

Interpreting Cumulative Frequency Graphs To find the median, find the frequency amount that translates to 50% of the cumulative frequency. Follow this amount on the y-axis over to the curve, and then down to the x-axis to find the median. To find the lower quartile, Q1, find the frequency amount that translates to 25% of the cumulative frequency. Follow this amount on the y-axis over to the curve, and then down to the x-axis to find Q1. To find the upper quartile, Q3, find the frequency amount that translates to 75% of the cumulative frequency. Follow this amount on the y-axis over to the curve, and then down to the x-axis to find Q3. To find the interquartile range subtract the lower quartile from the upper quartile: IQR = Q3 – Q1.

Interpreting Cumulative Frequency Graphs Example. Use the cumulative frequency graph to estimate the: median finishing time number of competitors who finished in less than 2 hours 35 minutes percentage of competitors who took more than 2 hours 39 minutes to finish time taken by a competitor who finished in the top 20% of runners completing the marathon

Interpreting Cumulative Frequency Graphs

Interpreting Cumulative Frequency Graphs i. Median = 50th %ile. 50% of 69 = 34.5. Start with cumulative frequency of 34.5 & find corresponding time. 2 hours 34.5 minutes

Interpreting Cumulative Frequency Graphs ii. Start on x-axis at 2 hours 35 minutes & find corresponding cumulative frequency. 37 people

Interpreting Cumulative Frequency Graphs iii. Start on x-axis at 2 hours 39 minutes & find corresponding cumulative frequency. 52 people took < 2 h 39 min. 69−52 69 = 𝟐𝟔.𝟒%

Interpreting Cumulative Frequency Graphs iv. 20% of 69 = 13.8. Find time corresponding to cumulative frequency of 13.8. 2 hour 31 min