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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

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• Mean The average of a group of numbers is called the mean.

• MedianThe middle number of the group is called the median.

• ModeThe number that appears the most often in a listing of number

Lets review what we have

learned in our lesson

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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.

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“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.

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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

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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

* *X X

BOX

WhiskersOutliers

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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.

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Let us find the Median, the Lower quartiles (LQ) and the Upper quartiles (UQ)

1 2 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

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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 =77M = 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.

77 83 85

LQ M UQ

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Practice

Example 1: Find the median, and upper and lower quartiles of this set:

23, 18, 20, 30, 28, 21, 32, 16, 33Solution: 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

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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

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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.

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• 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

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First we shall compute the inner fences from the inner quartile range (IQR).

The inner fences would be placed at1.5*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

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Now let us mark the LIF and the UIF on our Plot.

LIF = 65UIF = 97

LIF LQ M UQ UIF

12 12

^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110 65 97

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Now we compute the outer fences

• Outer Fences is computed as under3*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

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Now let us also mark the LOF and the UOF on our Plot.

L0F = 53UOF = 109

LOF LIF LQ M UQ UIF UOF

^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110

24 24

53 109

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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 be50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95,100

LIF =65 UIF =97

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Now let us draw the whiskers on our plot

LOF LIF LQ M UQ UIF UOF

^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110

X X

73 95

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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.

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Now let us mark the Outliers on our Box Plot.

LOF LIF LQ M UQ UIF UOF

^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110

X X

73 95

* *

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So after removing the fences we get our Box Plot as under.

LQ M UQ

^.........^.........^.........^.........^.........^.........^ 50 60 70 80 90 100 110

X X

73 95

* *

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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.

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QuartileWhat 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.

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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.

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PracticeExample : 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

Bus Car Bicycle walk

No. of students

400 300 200 100

4001000

= a360°

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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.

1001000

= d360°

3001000

= b360°

2001000

= c360°

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Practice

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

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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 1 2 3 4 5 6 7 8 9 1011 12 13

Your Turn

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Your TurnFrom 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

20 25 30 35 40 45

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8. 11, 10, 23, 34 Answer: LQ= 10.5; UQ = 28.59. 45, 56, 89, 40, 60 Answer: LQ= 42.5; UQ = 72.510.11, 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 = 157. 4, 14, 2, 30, 8, 12 Answer: LQ= 3; UQ = 22

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Let us play a game

Click here to play

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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:

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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.

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Solution:2) By comparing the box of the plot we can say that

class 2 has better set of scores.

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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.

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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.

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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.

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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.

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You did Great Today!!

Be sure to practice what we have learned today.