2. Confidential2 Warm Up Find the mean, mode (s), and median for each set of data 1.90, 92, 94, 91, 90, 94, 95,98 93, 90 and 94, 93 2. 8.0, 9.1, 8.9, 9.0,9.3, 9.4 8.95, none, 9.05 3. 5, 0, 9, 9, 3, 0, 5, 5, 4 4.4, 5, 5 4. 31, 18, 19, 18, 18, 17, 12 19, 18, 18 5. 14, 80, 78, 25, 30, 59, 69, 55, 25, 59, 50, 59 50 ¼, 59, 57

3. Confidential3 Mean The average of a group of numbers is called the mean. Median The middle number of the group is called the median. Mode The number that appears the most often in a listing of number Lets review what we have learned in our lesson

4. Confidential4 Box and Whisker Plots:  A box and whisker plot is used to display a set of data.  By this plot we can easily see where most of the numbers are.  To create this plot we first find out median, first quartile and second quartile.

5. Confidential5 “Box and Whisker Plot” Box plot is mostly used to find out outliers in a set of data points. Data points are the data values or a collection of some numbers on which we want to construct a box plot Outliers are data points, out of line with the rest of the data set. These are points which are too far from the reasonable central value, so the outliers are those data values that don't seem to "fit“ in the data collected.

6. Confidential6 Now let me explain these terms by taking help of an example Suppose we want to measure the exact weight of an apple. Every body in the class is made to take the reading of the weight of the apple. In the end when everybody has finished we will take the average of these readings to get the exact weight of the apple. So now let us suppose we have these 15 readings as given below (These are the Data Points) 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95,100

7. Confidential7 If you observe closely some students must not have taken the readings properly, that is why we have values ranging from 50 to 100. So we need to construct a Box Plot of these data points and discard the totally wrong values before we can take the average, to get the exact weight of the apple. ( These totally wrong values are the outliers ) 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95,100 Have a look at the data points closely * * XX BOX Whiskers Outliers

8. Confidential8 Now let us see the steps required to construct the Box Plot To construct the ‘Box Plot’ we must first arrange the data points in ascending order and then find the following Median The median is the number in the middle of our set. It divides the data set into two halves the upper and the lower half. **If there are even number of data points then we need to take the average of the two middle numbers to get the value of the median upper and lower quartiles. The upper and lower quartiles are the medians of the upper half and the lower half of the data set.

9. Confidential9 Let us find the Median, the Lower quartiles (LQ) and the Upper quartiles (UQ ) 12 3 4 5 6 7 8 9 10 11 12 13 14 15 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100 ^ ^ ^ L.Q. Median U.Q. Upper HalfLower Half Center number of complete data set Center number of Upper half Center number of Lower half

10. Confidential10 Now we will mark these values of median, LQ and UQ on a scale. We first draw a horizontal line from 50 to 110 as all our data values fall between this range. LQ =77 M = 83 UQ =85 ^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110 Now we mark these three values on our scale and draw a box, from the LQ to the UQ. 778385 LQ M UQ

11. Confidential11 Practice Example 1: Find the median, and upper and lower quartiles of this set: 23, 18, 20, 30, 28, 21, 32, 16, 33 Solution: First write all the data points into ascending order. 16, 18, 20, 21, 23, 28, 30, 32, 33 There are 9 numbers so fifth number (the middle one) will be the median. Median = 23

12. Confidential12 Practice Now we will find the LQ and UQ. LQ is the median of the of lower half of the ordered data 16, 18, 20, 21 LQ = (18+20)/2 = 38/2 = 19 UQ is the median of the of upper half of the ordered data 28, 30, 32, 33 UQ = (30+32)/2 = 62/2 = 31

13. Confidential13 Now we will calculate the Inner Quartile Range.. Inner Quartile Range (IQR) is the difference between the Upper Quartile and the Lower Quartile IQR = UQ - LQ. In our previous example it will be equal to IQR = 85 - 77 = 8.

14. Confidential14 Fences are the limits till which we will accept the values to be correct and any data points outside these fences will be the ‘outliers’ and hence discarded by us Now we will go further and add Inner and Outer "fences." on both the sides of the box

15. Confidential15 First we shall compute the inner fences from the inner quartile range (IQR). The inner fences would be placed at 1.5*IQR left of the L.Q. 1.5*IQR right of the U.Q. ** 1.5 is used as a standard value for calculating the inner fences in a Box Plot As IQR = 8 so substituting we get 1.5 * 8 = 12 In our example, the inner fences will be at Lower Inner Fence (LIF) = 77 - 12 = 65 Upper Inner Fence (UIF) = 85 + 12 = 97

16. Confidential16 Now let us mark the LIF and the UIF on our Plot. LIF = 65 UIF = 97 LIF LQ M UQ UIF 12 12 ^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110 65 97

17. Confidential17 Now we compute the outer fences Outer Fences is computed as under 3*IQR left of the L.Q. 3*IQR right of the U.Q. ** 3 is used as a standard value for calculating the outer fences in a Box Plot As IQR = 8 so substituting we get (3*8 = 24) In our example, the outer fences will be at Lower Outer Fence (LOF) 77 - 24 = 53 Upper outer Fence (UOF) 85 + 24 = 109

18. Confidential18 Now let us also mark the LOF and the UOF on our Plot. L0F = 53 UOF = 109 LOF LIF LQ M UQ UIF UOF ^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110 24 53 109

19. Confidential19 Now we need to add the "whiskers." Find the first value above (to the right of) the Lower Inner Fence. Mark it with an X and draw a line connecting it to the box. Similarly, we find the first value below (to the left of) the Upper Inner Fence. Mark it with an X and draw a line connecting it to the box as well. In our data set this would be 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95,100 LIF =65 UIF =97

20. Confidential20 Now let us draw the whiskers on our plot LOF LIF LQ M UQ UIF UOF ^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110 XX 7395

21. Confidential21 Marking the Outliers Values between the inner and outer fences are called "suspect outliers." We mark them with an asterisk “ * ". Values outside the outer fences are called "highly suspect outliers." We mark them with an "o". In our example, we have two suspect outliers: the data points 60 and the 100. We also have one highly suspect outlier: the data point 50.

22. Confidential22 Now let us mark the Outliers on our Box Plot. LOF LIF LQ M UQ UIF UOF ^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110 XX 7395 * *

23. Confidential23 So after removing the fences we get our Box Plot as under. LQ M UQ ^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110 XX 7395 * *

24. Confidential24 Now let us analyze the Box Plot that we have constructed. We shall see that Half the numbers are between 77 and 85, The middle of the data set is at 83, (Median) The "reasonable" range of the data goes from 73 to 95. These are the Whisker Points. We have three suspect data values at 50, 60, and 100. (Outliers) So now we can discard the Outliers or the suspected values from our data set and take the average of the remaining 12 values to calculate the exact weight of the Apple.

25. Confidential25 Quartile What is Quartile: The Quartiles of a data set give us break downs of the data into four groups with approximately the same number of points. The Lower Quartile (LQ) of a data set is the median of lower half of the ordered data. The Upper Quartile (UQ) of a data set is the median of the upper half of the data.

26. Confidential26 Circle Graph  A circle graph is an efficient way to present certain types of data.  The graph shows data as percent or fractions of a whole.  The total should be 100% or 1.  This graph is used to show the parts of a whole.  The angles at the center are central angles and each angle is proportional to the percent or fraction of the total.

27. Confidential27 Practice Example : Use the following data in the table to draw a circle graph. Step 1: Add the numbers to find the total. Total = 400 + 300 + 200 + 100 = 1000 Step 2: For each central angle, set up proportion to find the measure. After simplifying we will get, a = 144° Transportation Type BusCarBicyclewalk No. of students 400300200100 400 1000 = a 360°

28. Confidential28 Practice Similarly for other angles,, After simplifying we will get, b = 108°, c= 72°, d = 36° Step 3: Use a compass to draw a circle. Draw the approximate central angle with a protector. 100 1000 = d 360° 300 1000 = b 360° 200 1000 = c 360°

29. Confidential29 Practice Step 3: Label each sector and add any necessary information.

30. Confidential30 From the data given in the box and whisker plot 1. Find the Upper quartile UQ = 8 2. Find the Lower Quartile LQ = 2 3. Find the Median Median = 6 0 12345678910111213 Your Turn

31. Confidential31 Your Turn From the data given in the box and whisker plot answer the questions 4. What is the median? 30 5. What is the UQ and LQ? 45, 20 20253035 40 45

32. Confidential32 8.11, 10, 23, 34 Answer: LQ= 10.5; UQ = 28.5 9.45, 56, 89, 40, 60 Answer: LQ= 42.5; UQ = 72.5 10.11, 22, 44, 33, 66, 55 Answer: LQ= 22; UQ = 55 Find the lower and upper quartiles of the following data sets: 6.13, 17, 5, 11, 9 Answer: LQ= 7; UQ = 15 7.4, 14, 2, 30, 8, 12 Answer: LQ= 3; UQ = 22

33. Confidential33

34. Confidential34 Let us play a game Click here to play

35. Confidential35 Q1. Describe a set of data in which there is only one whisker in its box-and-whisker plot. A set in which an extreme value and a quartile are the same has no whisker on that side of the box-and- whisker plot Solution:

36. Confidential36 Q2: Two classes took the same math test. The results are shown in the box-and-whisker plots below. 1)Which class has the higher median? 2)Which class has the better set of scores? Solution: 1) Median of class 1 = 75 Median of class 2 = 85 So class 2 has the higher median.

37. Confidential37 Solution: 2)By comparing the box of the plot we can say that class 2 has better set of scores.

38. Confidential38 Q3: The circle graph represents the favorite snack for the students surveyed at a school. Which snack is the most favorite snack of the students? Solution: It is clear from the graph that 50% students stated pizza as their favorite snack at school. Most favorite snack is Pizza.

39. Confidential39 Let Us Review  A box and whisker plot is used to display a set of data.  To create this plot we first find out median, first quartile and second quartile.  Plot the given data set on a number line.  Mark the highest and lowest data points with connected black circles and make a box between the quartiles and a line through the median.

40. Confidential40 Let Us Review  To find median first write all the numbers in the ascending order.  If the number of data points is odd then the middle number will be the median.  If the number of data points in the set is even, the median is the average of the two middle numbers.  The Lower Quartile (LQ) of a data set is the median of lower half of the ordered data.  The Upper Quartile (UQ) of a data set is the median of the upper half of the data.

41. Confidential41  A circle graph is an efficient way to present certain types of data.  The graph shows data as percent or fractions of a whole.  The total should be 100% or 1.  This graph is used to show the parts of a whole Let Us Review  Inner Quartile Range (IQR) is the difference between the Upper Quartile and the Lower Quartile Fences are the limits till which we will accept the values to be correct and any data points outside these fences will be the ‘outliers’ and hence discarded.

42. Confidential42 You did Great Today!! Be sure to practice what we have learned today.