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Page 1 CCM6 & CCM6+ UNIT 13 Statistics and Data 2016-2017

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CCM6/CCM6+ Unit 13

Collect, Analyze, and Display Data

2016 – 2017

Name: ____________________________________

Math Teacher: _____________________________

Projected Quiz Date: __________________________

Projected Test Date: __________________________

Main Idea Page(s)

Unit 13 Vocabulary 2 - 3

Statistical Questions 4 - 5

Find and Compare Statistical Measures 6 - 9

More with Mean, Median, Mode, Range 10 - 18

Histograms 19 - 27

Mean and Measures of Center 28 - 30

Box Plots 31 - 35

Mean Absolute Deviation (MAD) 36 - 38

MAD and IQR as Measures of Variability 39 - 49

More Practice with Histograms and Box Plots 50 - 56

Unit 13 Study Guide 57 - 59


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CCM6/CCM6+ Unit 13: Collect, Analyze, Display Data

Vocabulary

Statistics The practice of collecting and analyzing data in large quantities

Data Values such as counts, ratings, measurements, or opinions that are gathered to answer

questions.

Mean A value that represents the "evening out" of the values in a set of data

Median The number that is the midpoint of a set of data

Mode The data value that occurs the most

Range The difference between the least value and the greatest value in a data set

Line Plot A quick, simple way to organize data along a number line where the X's (or other symbols)

above a number represent how often each value is mentioned

Measures of

Center

Establish a central location in the data set

Measures of

Variability

Establish the degree of variability (or scatter) of the individual data values and their deviations

from the measures of center

Histogram A display that shows the distribution of numeric data. The range of data values, divided into

intervals, is displayed on the horizontal axis. The vertical axis shows frequency.

Frequency Table A list of items or intervals that shows the number of times, or frequency, with which they

occur.

Interval is a set of real numbers with the property that any number that lies between two numbers in the

set is also included in the set

Distribution The arrangement of values in a data set

Gap a break or opening

Cluster a group of things or persons close together

Peak being at the point of maximum frequency, intensity, use, etc.

Box Plot a method of visually displaying a distribution of data values by using the median, quartiles,

and extremes of the data set. A box shows the middle 50% of the data

Quartiles one of the values of a variable that divides the distribution of the variable into four groups

having equal frequencies

Lower Quartile for a data set with median m, the first quartile is the median of the data values less than m

Example: for the data set {1, 3, 6, 7, 10, 12, 14, 15, 22,120}, the first quartile is 6.

Upper Quartile

for a data set with median m, the third quartile is the median of the data values greater than m

Example: for the data set {2, 3, 6, 7, 10, 12, 14,15, 22, 120}, the third quartile is 15.


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Outlier a value that lies far from the "center" of a distribution

Inter-Quartile

Range

a measure of variation in a set of numerical data, the inter-quartile range is the distance

between the first and third quartiles of the data set

Example:for the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the interquartile range is 15–6= 9.

Five-Number

Summary

The minimum value, lower quartile, median, upper quartile, and maximum value

Minimum Value The lowest value in a data set

Maximum Value The greatest value in a data set

Mean Absolute

Deviation

(M.A.D.)

The average distance of all data values from the mean of the set

Variability Degree to which data are spread out around a center value

Skewed asymmetry in a frequency distribution

Summary

statistics

include quantitative measures of center (median and median) and variability (interquartile

range and mean absolute deviation) including extreme values (minimum and maximum),

mean, median, mode, range, and quartiles

Symmetrical characterized by or exhibiting symmetry; well-proportioned, as a body or whole; regular in

form or arrangement of corresponding parts


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Intro to Statistics

What is Statistics? ________________________________________________________________________________

________________________________________________________________________________

Why is it important? ________________________________________________________________________________

What are statistical questions? ________________________________________________________________________________

________________________________________________________________________________

The teacher will show you a variety of questions. Determine whether the question is statistical or

non-statistical and write the questions under the correct heading.

Statistical Non-Statistical

Chose one of the statistical questions from the list above and determine how you might find

the answer to this question.

What are ways in which you can collect data?

1.

2.

3.

4.

5.

6.


5


6

The mean, the median, the mode, and the range are measures that

describe a set of data. This picture will help you learn about each measure.

MEAN (average)

Add all the values, then divide by the total

number of values.

MEDIAN (middle)

The middle value, when the values are

arranged in order.

MODE (most)

The value that occurs most often

in a set of data. There can be one mode,

more than one mode, or no mode.

RANGE

The difference between the

greatest number and the

least number in a set of data.

Tim’s bowling scores for the past 5 games are listed below:

89 98 110 98 105 Write mean, median, mode, or range to answer the following questions about Tim’s scores.

1. The number 98 indicates which measure? ___________________________

2. Tim added up all his scores and divided by 5. Which measure did he find?________________

3. Tim found that the difference between his highest and lowest score was 21 points.

That measure is called the ______________________

4. Tim noticed that he got a score of 98 twice. Which measure is Tim focusing on? ___________

Terry’s test scores for the past 5 assessments are listed below. Calculate the range, mean,

median, and mode.

Range: ___________ Mean: _________ Median: _____________ Mode: ___________

Terry’s Test Scores

76

81

94

81

78

Measures that

Describe a Set of Data


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8


9


10

Finding Statistical Measures and Comparing them to Understand Data

BIG IDEAS….

Shape of a set of data o Symmetrical data…mean and median are close to the same

o Skewed data…leans left or right…mean and median are different…mean is pulled

left or right

Outliers o When there is an outlier, it pulls the mean up or down

o Outliers barely affect the median

o If there is an outlier, the median will be more typical for that set of data than the

mean

Names:

Maxi Swanson

(11)

Thomas Petes

(11)

Michelle Hughes

(14)

Shoshana White

(13)

Deborah Black

(12)

Tonya Stewart

(12)

Tony Tung

(8)

Richard Mudd

(11)

Janice Wong

(10)

Bobby King

(9)

Charlene Greene

(14)

What part of the data above is most important?

Create a dot plot (line plot) of the data.

Can you make any conclusions about this data? ________________________________________

Describe the SHAPE of the data… Symmetrical? Skewed? Random?

Write a statistical question about the data above. _________________________________________


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Investigating Statistics…What numbers affect the MEDIAN?

1. Using the names provided, place them in order from least to greatest. What is the middle

number? How many numbers are to the left of the middle number? How many numbers are

to the right of the middle number?

Middle Number: ______

# of numbers to the left: ______

# of numbers to the right: ______

2. The median is the number that is the midpoint of a set of data. The same number of data

values occur before and after the median. What is the median for these data?

3. Remove two names from your data set so that:

The median stays the same: _______ when you remove ___________________________

The median increases: _______ when you remove ____________________________

The median decreases: _______ when you remove ____________________________

4. Maxi Swanson is moving. When she leaves, what will be the new median for the data set?

5. What would happen to the median of the data set if you add a name with 16 letters?

6. What would happen to the median of the data set if you add a name with 89 letters?

**What do you call a data point far away from all others? ____________________________

7. What would be the length of two names that you could add to the data set so that

The median stays the same:

The median increases:

The median decreases:


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Investigating Statistics…What numbers affect the MEAN?

Susan has six pets. She made the line plot to show the

lengths of her pets’ names.

Describe the SHAPE of the data.

What is the mean of the data above?

If you add a 1 letter pet name, what will be the new mean?

What will be the new median?

How did a low data value change the mean?

How did a low data value change the median?

If you remove the 1 and replace it with a 25, what will be the new mean?

What will be the new median?

How did a high data value change the mean?

How did a high data value change the median?

Do very high or very low data values far away from the other data (outliers) change the mean or

median more?

When there is an outlier, which measure of center would be better to use for that set of data?

When there is no outlier, which measure of center would be better to use for that set of data?


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Investigating Statistics:How does the SHAPE of the data affect the center?

PEAK … CLUSTER … SKEWED … SYMMETRICAL … GAP

Describe the shape of the data in the line plot

to the right.

Mean = ______________

Median = _____________

Mode = ____________

Range = ____________

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Describe the shape of the data in the

line plot to the right.

Mean = ______________

Median = _____________

Mode = ____________

Range = ____________

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


to the right.

Mean = ______________

Median = _____________

Mode = ____________

Range = ____________


to the right.

Mean = _________ Median = _________

Mode = __________ Range = ________

Which measures are affected most by the outlier? ___________________________________


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PRACTICE

1.

Range: ________________________________________

Mean: _________________________________________

Median: ________________________________________

Mode: _________________________________________

Which measure of center is best for Jessica—mean or median? Explain.

2. On an exam, three students scored 75, four students scored 82, three students scored 88, four

students scored 93, and one student scored 99. If the answer is 88, what is the question? Hint:

Write the scores out!

3. Find the mean, median, mode, and range of the dot plot.

Mean: ___________

Median: ___________

Mode: ___________

Range: __________

For the line plot, is mean or median a better measure of center? Explain.

Jessica’s History Test Scores

81

97

99

89

91

50

http://www.google.com/imgres?safe=active&sa=X&rls=com.microsoft:en-us:IE-SearchBox&biw=1080&bih=619&tbm=isch&tbnid=s0m9h1LN702yTM:&imgrefurl=http://math.tutorvista.com/statistics/line-plots.html&docid=YlpR87YLomsmyM&imgurl=http://image.tutorvista.com/content/feed/u769/eg 3.PNG&w=261&h=184&ei=f9edUtDaOIiUkQf41YDABw&zoom=1&iact=hc&vpx=266&vpy=275&dur=109&hovh=147&hovw=208&tx=111&ty=91&page=1&tbnh=147&tbnw=196&start=0&ndsp=18&ved=1t:429,r:14,s:0,i:123


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The two graphs below show students’ name lengths in two classes.

Ms. Campo’s class:

7

6

5

4

3

2

1

0

9 10 11 12 13 14 15 16 17 18 19

Mr. Young’s class:

X

X X X

X X X X

X X X X X X X X

8 9 10 11 12 13 14 15 16 17 18 19 20

1. What is the typical name length for:

a) Ms. Campo’s class. b) Mr. Young’s class.

2. Find the mean, median, mode, and range for both classes in the boxes above.

3. How does the data distribution (SHAPE) compare between these two classes?

3. Since Mr. Young’s class data has an outlier, which measure best represents his data?

Mean = _____

Median = _____

Mode = _____

Range = _____

Mean = _____

Median = _____

Mode = _____

Range = _____


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1. The line plot below represents the number of letters written to overseas pen pals by the students at

Waverly Middle School. Each "x" represents 10 students. How many students wrote 18 or more

letters?

A) 250 B) 8 C) 30 D) 80


Waverly Middle School. Each "x" represents 10 students. How many students wrote more than 6

and fewer than 12 letters?

A) 110 B) 120 C) 100 D) 90

3. Thirteen bowlers were asked what their score was on their last game. The scores are shown

below.

183, 152, 155, 181, 176, 193, 171, 170, 186, 170, 187, 159, 183

Find the range of the bowlers' scores.

A) 20 B) 41 C) 53 D) 31

4. A group of friends tested themselves to see how many times each person could hit a tennis ball

against the wall without missing. The results are below:

7 15 28 8 21 30 30 10

22 4 17 7 17 22 10 8

Find the range of the data set.

A) 26 B) 16 C) 36 D) 23


below.

190, 150, 154, 194, 182, 190, 170, 151, 190, 170, 178, 161, 180


A) 56 B) 44 C) 34 D) 23


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Waverly Middle School. Each "x" represents 10 students. How many students wrote 14 or more

letters?

A) 140 B) 160 C) 0 D) 14


Waverly Middle School. Each "x" represents 10 students. How many students wrote more than 6

and fewer than 20 letters?

A) 250 B) 240 C) 230 D) 220



8 13 22 8 18 28 28 12

25 6 15 8 15 25 12 8


A) 19 B) 22 C) 32 D) 12



7 11 25 7 25 23 23 15

21 7 12 7 12 21 15 7


A) 28 B) 8 C) 18 D) 15


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below.

192, 158, 154, 195, 180, 183, 188, 151, 180, 185, 184, 166, 184


A) 56 B) 34 C) 23 D) 44

11. Which measure is affected MOST by an outlier?

A) mean B) median C) mode

Why? ________________________________________________________________

12. Which measure is affected LEAST by an outlier?

A) mean B) median C) mode

Why? _________________________________________________________________

13. How is a bar graph like a line plot (dot plot)?

14. What kind of data is best for the mode? the median? the mean?


19

Histograms

Histogram:

____________________________________________________________________

____________________________________________________________________

Constructing a Histogram

Mrs. Pittman gave her class a history test. The class of 16 students had the following

scores: 75, 80, 65, 80, 95, 85, 65, 80, 90, 80, 70, 85, 90, 70, 85, 70

Construct a histogram to represent this data.

Frequency Chart

Interval Tally Frequency


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1. Listed below are the daily high temperatures (◦F) for the first 20 days of

April. Choose appropriate intervals to group the data, make a frequency table

for the data, and construct a histogram for the data.

55 62 68 75 69 78 82 79 85 88

65 60 58 75 80 82 74 78 78 72 Frequency Chart



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2. Thirty people in Max’s neighborhood participated in a Walk-A-Thon fundraiser.

The ages of the walkers were as follows:

12 8 32 35 15 47 9 15 52 55 70 18 36 29 12

11 16 45 44 19 62 60 8 23 27 10 34 74 13 59

a. Make a histogram for the set of data.

Frequency Chart


b. Determine the mean and median for this data set.

c. Explain how the median for this data relates to the graph of the data.

d. If the seven youngest participants did not walk and seven members of the Golden Oldies Club

(over 70 years of age) took their place, how would this change the graph of the data?

Determine the mean and median for this new data set.


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23

More Practice with Histograms

MIGRAINES: HISTOGRAMS

Below are data collected from patients who suffer from migraine headaches.

The patients were instructed to take their assigned drugs as soon as their headaches began and to

record how much time passed before the drugs gave relief.

Drug A is a traditional drug, and Drug B is an experimental drug.

Each value is the number of minutes (rounded to the nearest two minutes) that elapsed before a

patient got relief.

Drug A (100 patients)

16, 18, 18, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 36, 38, 38,

40, 42, 44, 44, 46, 46, 48, 50, 54, 56, 56, 58, 58, 62, 62, 64, 64, 66, 68, 68,

70, 70, 70, 72, 72, 74, 76, 76, 76, 78, 78, 80, 80, 80, 82, 82, 84, 84, 86, 88,

88, 88, 88, 90, 90, 90, 90, 90, 92, 92, 92, 92, 94, 94, 94, 96, 96, 98, 98, 98,

98, 100, 100, 100, 102, 102, 102, 104, 104, 106, 106, 108, 108, 108, 110,

110, 112, 114, 118, 120

Drug B (50 patients)

18, 20, 20, 22, 24, 24, 24, 26, 26, 30, 30, 30, 34, 34, 34, 36, 36, 36, 38, 38,

40, 40, 44, 44, 46, 50, 52, 52, 56, 56, 58, 62, 62, 66, 74, 74, 78, 88, 94, 98,

98, 100, 104, 106, 110, 116, 120, 120, 121, 121

1. From examining these data, which drug do you think gave faster relief from headache pain?____

Explain______________________________________________________________________

On the next page you will create a frequency table for each drug. Draw dividing lines in the

data above (for each drug) to separate into these intervals:

10-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89, 90-99, 100-109, 110-119, 120-129

Then, fill out the frequency table on the next page according to the amount per interval.


24

More Practice with Histograms

Migraines: Histograms

Time to Effect of Drug A (100 patients)

Time (minutes) Frequency Percentage of

Total (100)

10 - 19

20 – 29

30 – 39

40 – 49

50 – 59

60 – 69

70 – 79

80 – 89

90 – 99

100 – 109

110 – 119

120 – 129

Time to Effect of Drug B (50 patients)

Time

(minutes)

Frequency Percentage of

Total (50)

10 - 19

20 – 29

30 – 39

40 – 49

50 – 59

60 – 69

70 – 79

80 – 89

90 – 99

100 – 109

110 – 119

120 – 129

**To calculate the percent: # in the interval

# 𝑝𝑎𝑡𝑖𝑒𝑛𝑡𝑠 𝑓𝑜𝑟 𝑡ℎ𝑎𝑡 𝑑𝑟𝑢𝑔 • 100 = ______%


25

2. Construct a histogram for each data set on the axes provided below. Title your display.

Drug A

# 20

of 18

P 16

A

T 14

I

E 12

N

T 10

S

8

6

4

2

0

Drug B 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 100-109 110-119 120-129

# 20

of 18

P 16

A

T 14

I

E 12

N

T 10

S

8

6

4

2

0 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 100-109 110-119 120-129


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3. From examining the histograms, which drug do you think gave the most relief from headache

pain? ________________ Explain._________________________________

___________________________________________________________________

___________________________________________________________________

4. A relative-frequency histogram uses the % in each interval rather than the frequency.

Why is this useful for our data sets? What is the advantage of a relative-frequency histogram for

our data sets?

Construct two relative-frequency histograms below (one for each Drug).

Drug A

% 25

of 20

P

A 15

T

I 10

E

N 5

T

S 0 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 100-109 110-119 120-129

Time in Minutes To Effect

Drug B

% 25

of 20

P

A 15

T

I 10

E

N 5

T

S 0 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 100-109 110-119 120-129


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Time in Minutes To Effect

5. On the basis of your examination of the relative-frequency histograms, which drug do you think

gave faster relief from headache pain? _____________ Explain.

6. Some students get different answers for questions 3 on the previous page and 5 on this page.

Why do you think that happens?

7. How does changing the display (frequency histogram or relative-frequency histogram) change

the information you can read from the graph?

8. What advantage does the histogram have over the relative-frequency histogram?

9. What advantage does the relative-frequency histogram have over the histogram?

10. What have you learned from this activity?


28

A group of students made the table below.

A. Make stacks of cubes to show the size of each household.

1. How many people are in the six households altogether? Explain

2. What is the mean number of people per household? Explain.

3. How does the mean here compare to the mean for the data on p. 30?

B. What are other ways to determine the mean of a set of data other than using cubes?

C. Make a set of 6 data values that have a mean of 8.

D. Make a set of 5 data values that have a median of 8.

E. Make a set of 5 data with a range of 6, a mean of 6, and a median of 6.

Household Size

NAME # of People

Reggie 6

Tara 4

Brendan 3

Felix 4

Hector 3

Tonisha 4


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For questions 1 and 2, use the line plot below.

Number of Children in a Household

0 1 2 3 4 5 6 7 8 9 10 11 12

1. a. What is the median number of children for the 16 households? Explain how to find

the median. What does the median tell you?

b. Do any of the 16 households have the median number of children? Explain.

2. a. What is the mean number of children per household for the 16 households? Explain

how to find the mean. What does the mean tell you?

b. Do any of the 16 households have the mean number of children? Explain.

For exercises 3&4, the mean number of people per household for eight households is 6 people.

3. What is the total number of people in the eight households?

a) 11 b) 16 c) 48 d) 64

4. a. Make a line plot showing one possible arrangement for the numbers of people in the

eight households. (Remember, the MEAN is 6.)


30

b. Make a line plot showing a different possible arrangement for the numbers of people

in the eight households. (Remember, the MEAN is 6.)

c. Are the medians the same for the two arrangements you made?

5. The students in Mr.Wilson’s study hall spent the following amounts of time on their HW.

3

4 ℎ𝑜𝑢𝑟

1

2 ℎ𝑜𝑢𝑟 1

1

4 ℎ𝑜𝑢𝑟𝑠

3

4 ℎ𝑜𝑢𝑟

1

2 ℎ𝑜𝑢𝑟

What is the mean time his students spent on HW?

6. Use the data from question 5. What is the median time Mr. Wilson’s students spent on

HW?

a) 1

2 ℎ𝑜𝑢𝑟 b)

3

4 ℎ𝑜𝑢𝑟 c) 1 hour d) 1

1

4 ℎ𝑜𝑢𝑟𝑠

7. Six students each had a different number of pens. They put them all together and then

distributed them so that each student had the same number of pens.

a. Choose any of the following that could be the number of pens they had

altogether. Explain your reasoning.

A. 12 B. 18 C. 46 D. 48

b. Use your response from part a. How many pens did each person have after the

pens were distributed evenly?

c. Your classmate says that finding the mean number of pens per person is the

same as finding the number of pens each person had after the pens were

distributed evenly. Do you agree or disagree? Explain.


31

Box Plots (aka Box-and-Whisker Plots)

WARMUP:

1. Create a set of data with 5 data values that have a median of 4.

2. Create another set of data with 5 data values that have a median of 4.

3. Do you use any “tricks” to help with these questions?

4. Create a set of data with 6 data values that have a mean of 5.

5. Create another set of data with 6 data values that have a mean of 5.

6. A trick in creating a set of data with the same mean is:


32

Box Plots (aka Box-and-Whisker Plots)

FIVE MAGIC NUMBERS

1. The ______________ _______________ is the smallest data value.

This starts the first “whisker.”

2. The ______________ _______________ is the middle of the lower half of the data.

_______% of the data values are below this number and ________% are above.

This ends the first “whisker” and starts the “box.”

3. The ______________ is the middle of all of the data.


This is a vertical line inside the “box.”

4. The ______________ _______________ is the middle of the upper half of the data.


This ends the “box” and starts the last “whisker.”

5. The ______________ _______________ is the largest data value.

This is the end of the last “whisker.”

**The INTER-QUARTILE RANGE is the __________ of the ___________. This is the

middle half (50%) of the data.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

NOW, here’s a set of data….you have to ______________ it to find the 5 magic numbers!

8, 20, 12, 16, 18, 22, 24, 14, 25, 20, 21

Rewrite the numbers:

Circle the five magic numbers in the list you made! Now, put each point above the number line below. Make your box and whiskers!


33

Here’s another set of data….do it again!

16, 18, 12, 10, 14, 10, 8, 12

Rewrite in order from least to greatest:

What do I do for the five magic numbers if they are “between” data values?

Find the five magic numbers from the data set above.Then make your box-and-whisker plot.

Analysis: 25% of the data is below _________.

50% of the data is above _________.

75% of the data is below _________.

50% of the data is between ___________ and _________.

50% of the data is in the ___________________.

50% of the data is in the ___________________.

How can a box-and-whisker plot help you to understand a set of data? _________________

What is the Inter-Quartile Range (IQR)?

If there are 12 numbers in a set of data, how many numbers are in each part of the box-and-

whisker plot?


34

16. Create 2 box plots above the number line below (one for each set of data).Label them!

Midwestern States (area in 1,000 mi2): 45, 36, 58, 97, 56, 65, 87, 82, 77

Southern States (area in 1,000 mi2): 52, 59, 48, 52, 42, 32, 54, 43, 70, 53, 66

What conclusions can you make by looking at the plots above?


35

Warmup:

For the following types of data, which measure of central tendency is BEST—mean, median, or

mode? Explain.

1. most popular movie in the last month

2. favorite hobby

3. class sizes in a school

4. ages of members in a club

****CHALLENGE****

Each person has taken four tests and has one more test to take. Find the score that each person must

make to change the mean or median as shown.

5. Barry has scores of 93, 84, 86, and 75. He wants to raise the mean to 86.

Hint: When you find the mean, first you add, then divide…now undo (first multiply then subtract).

6. Liz has scores of 87, 75, 82, and 93. She wants to raise the median to 87.

Hint: 87 will be the MIDDLE point (like the fulcrum on a balanced scale).

7. Li’s bowling scores are: 129, 136, 201, 146, 154.

Make a box plot (find those 5 magic numbers and graph them). What is the IQR?


36

Mean Absolute Deviation (MAD)

MAD NOTES: Describing Data through Measures of Center & Spread

There are two ways to describe a set of data:__________________ & ____________________

Today we will focus on the numerical descriptions.

MEASURES OF CENTER

Mean: A numerical measure of center that is the arithmetic average of the data. Affected by

outliers.

Mean: �̅� = 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎=

∑𝑥

∑𝑛

Median: A numerical measure of center that describes the middle value of a data set. The median is

not affected by outliers. Note that the median does not have to be one of the values in the data set, but a

value that divides the data set in half so that 50% of the data values lie above the median and 50% of the data

set lie below the median.

MEASURES OF SPREAD describe________________________________________________

______________________:

______________________:

MEAN ABSOLUTE DEVIATION EXAMPLE

STEP 1:

STEP 2:

STEP 3:

STEP 4:

DATA DIFFERENCE

Data minus Mean

ABSOLUTE

VALUE

MEAN:

Symbols to represent the mean. The large “E”

type symbol indicates a “sum”. So the top is the

sum of the data numbers and the bottom is the

sum of how many pieces of data there are.

STEP 3 STEP 2 STEP 1

STEP 4


37

_________________________________: A numerical measure of spread that shows how much data

values vary from the mean. A low mean absolute deviation indicates that the data points tend to be very

close to the mean and not spread out very far so the mean is an accurate description of “typical”, and a

high mean absolute deviation indicates that the data points are spread out over a large range of values.

BACK TO PPT SLIDE 9…..

Who do you think is it BEST STUDENT? Explain.

Data Set: 30, 38, 40, 42, 48 Find the mean: �̅� = _________

SUM of Values: Mean Absolute Deviation (MAD) =

DATA DIFFERENCE

Data minus Mean

ABSOLUTE VALUE

Li

MEAN:

DATA DIFFERENCE

Data minus Mean

ABSOLUTE

VALUE

MAD:

Bessie

MEAN:

DATA DIFFERENCE

Data minus Mean

ABSOLUTE

VALUE

MAD:

Jamal

MEAN:

DATA DIFFERENCE

Data minus Mean

ABSOLUTE

VALUE

MAD:


38

Mean Absolute Deviation Homework

Find the mean absolute deviation for each set of data.

1. 80, 82, 82, 88, 90, 94, 102, 104, 106

5. 160, 166, 170, 172, 178, 180, 190, 204, 260

More MAD and measures of variability

The following data set represents the size of 9 families.

3, 2, 4, 2, 9, 8, 2, 11, 4

What is a mean? ________________________________________________________________

What is the mean for this set of data? __________

What is mean absolute deviation?

the ____________ _____________of each data value from the _____________

What is the MAD for this data set? Make a chart and figure it out! MAD = _________

MEAN: MAD:

DATA DIFFERENCE

Data minus Mean

ABSOLUTE VALUE

Sum = Sum =

MEAN: MAD:

DATA DIFFERENCE

Data minus Mean

ABSOLUTE VALUE


39

3 4 5 6 7 8 9

Based on the picture above, what are the “FIVE MAGIC NUMBERS” and what do they mean?

1.

2.

3.

4.

5.

Looking at the boxplot below, name and give the value of the FIVE MAGIC NUMBERS.

1.

2.

3.

4.

5.

http://www.google.com/imgres?start=171&safe=active&sa=X&rls=com.microsoft:en-us:IE-SearchBox&biw=1080&bih=619&tbm=isch&tbnid=8q4xEVWAmNQP6M:&imgrefurl=http://www.astarmathsandphysics.com/gcse_maths_notes/gcse_maths_notes_five_figure_summaries_and_boxplots.html&docid=jvDojWjVCjUFnM&imgurl=http://www.astarmathsandphysics.com/gcse_maths_notes/gcse_maths_notes_five_figure_summaries_and_boxplots_html_6e035693.gif&w=634&h=241&ei=JYGgUuGaFM3okAf044CYBw&zoom=1&iact=hc&vpx=242&vpy=363&dur=1641&hovh=138&hovw=364&tx=150&ty=38&page=9&tbnh=101&tbnw=258&ndsp=21&ved=1t:429,r:73,s:100,i:223

http://www.google.com/imgres?start=253&safe=active&sa=X&rls=com.microsoft:en-us:IE-SearchBox&biw=1080&bih=619&tbm=isch&tbnid=BxOWzLKWNtIUmM:&imgrefurl=http://www.shodor.org/interactivate/discussions/BoxPlot/&docid=hp6i97SVJkm3tM&imgurl=http://www.shodor.org/media/N/j/g/xNmVlMGNjNDMxNjZiNGRmODYyZWJlMDM0NGI3NmY.jpg&w=300&h=116&ei=9IGgUvVCh52RB6_pgegN&zoom=1&iact=hc&vpx=616&vpy=326&dur=2235&hovh=92&hovw=240&tx=126&ty=47&page=13&tbnh=92&tbnw=240&ndsp=20&ved=1t:429,r:71,s:200,i:217


40

Variability

To find a RANGE you need which two “MAGIC NUMBERS?”

To find the InterQuartile Range (IQR), you need which two “MAGIC NUMBERS?”

Use the data displayed in the box plots below to answer the questions.

Amount Spent by Each Customer (in dollars) at Casual Café

Amount Spent by Each Customer (in dollars) at Bountiful Bistro

1) Find the following for each set of data.

a. median

b. range

c. interquartile range

2) Use the medians of the data to compare the amounts spent by customers at each restaurant.

3) Use the ranges and interquartile ranges of the data to compare how amounts spent by customers

at each restaurant vary. Which restaurant would you rather eat at based on this data? Explain.

4) Is there symmetry or lack of symmetry in each box plot?


41

Measures of Variability: MAD and IQR

Variability means ______________________________________________________________

Why is MAD a measure of variability?

Why is IQR a measure of variability?

Math Exam Grade

Review Class

No Review

Grade as a Percent

1) What conclusions can you draw from looking at the plot about how effective the math

exam review class was?

2) What is the difference in the medians between the sets of data?

3) The mean absolute deviation for both groups of students is 6.2. Compare that value to

the difference in medians. What does that tell you about the data?

4) What is the interquartile range for each set of data? What does that tell you about the

data?

CHALLENGE: Determine which 2 sets of data will overlap more.

Set A has a mean of 12 and a mean absolute deviation of 5.1.

Set B has a mean of 23 and a mean absolute deviation of 4.9.

Set C has a mean of 10 and a mean absolute deviation of 4.8. Day 9 HW (MAD and IQR = Measures of Variability)

Amount

of

Review

Before

the

Exam


42

Directions: Solve the following problems by finding the median, interquartile range (IQR), or

range. You may use a calculator, but you must show your work!

1. {13, 15, 9, 35, 25}

Median = ________

Q1 = ________

Q3 = ________

IQR = ________

2. {6, 1, 3, 8, 5, 11, 1, 5}

Median = ________

Q1 = ________

Q3 = ________

IQR = ________

3. Jason and Jill are two students in Mr. White’s math class. On the last five quizzes, Jason

scored an 80, 90, 95, 85, and 70. Jill scored a 70, 75, 90, 100, and 95. Find the median and

IQR for each student.

Jason’s Median :______ Jill’s Median = _______

Jason’s IQR = ______ Jill’s IQR = ________ Whose quiz grades are more consistent? _______


43

Find the mean absolute deviation for each set of data. Round to the nearest hundredth if

necessary. Then describe what the mean absolute deviation represents.

4. Number of Daily Visitors to a Web Site 5. Zoo Admission Prices ($)

112 145 108 160 122 9.50 9.00 8.25 9.25 8.00 8.50

Mean = ___________ Mean = ___________

MAD = ________ MAD = ________

6. 7.

Data Values Difference from Mean Absolute

Value

Data Values Difference from Mean Absolute

value


44


45


46


47


48


49


50

More practice with Histograms and Box Plots Multiple Choice. Identify the choice that best completes the statement or answers the question.

____ 1. The number of pieces of candy in randomly selected jars are 20, 1, 29, 21, 40, 52, 41, 44, 60, 67.

Display the data using a histogram.

a. Pieces of Candy in Jars c. Pieces of Candy in Jars

b. Pieces of Candy in Jars d. Pieces of Candy in Jars

____ 2. Display the data using a histogram.

The numbers of students in different classes at a community college:

25, 15, 28, 52, 22, 38, 42, 44, 24, 32, 19, 28, 29, 20, 31

a.

c.

1–20 21–40 41–60 61–80Pieces of candy

Fre

qu

ency

1

2

3

4

5

6

7

8

9

10

1–20 21–40 41–60 61–80Pieces of candy

Fre

qu

ency

1

2

3

4

5

6

7

8

9

10

1–20 21–40 41–60 61–80Pieces of candy

Fre

qu

ency

1

2

3

4

5

6

7

8

9

10

1–20 21–40 4–-60 61–80Pieces of candy

Fre

qu

ency

1

2

3

4

5

6

7

8

9

10

15-19 20-29 30-39 40-54

Fre

qu

ency

S tudents

Community College Class Sizes

1

2

3

4

5

6

7

15-24 25-34 35-44 45-54

Fre

qu

ency

S tudents


1

2

3

4

5

6

7


51

b.

d.

____ 3. The number of laps around a track for five runners are 14, 8, 7, 4, 11.

Display the data using a box-and-whisker plot.

a.

b.

c.

d.

____ 4. Make a histogram for the numbers of students in different classes at a community college.

25, 15, 28, 52, 22, 38, 42, 44, 24, 32, 19, 28, 29, 20, 31

a.

c.

15-24 25-34 35-44 45-54

Fre

qu

ency

S tudents


1

2

3

4

5

6

7

15-24 25-34 35-44 45-54

Fre

qu

ency

S tudents


1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20 22 240

0 2 4 6 8 10 12 14 16 18 20 22 240

0 2 4 6 8 10 12 14 16 18 20 22 240

0 2 4 6 8 10 12 14 16 18 20 22 240

15-24 25-34 35-44 45-54

Fre

qu

ency

S tudents


1

2

3

4

5

6

7

15-24 25-34 35-44 45-54

Fre

qu

ency

S tudents


1

2

3

4

5

6

7


52

b.

d.

____ 5. The number of calls answered by a paramedic team over an 8-day period are given. Use the data to

make a box-and-whisker plot.

12, 6, 8, 15, 14, 6, 14, 10

a.

b.

c.

d.

____ 6. Average prices for 35 different models (types) of car from the 3 most popular automobile brands in

the United States are shown.

Make a dot plot of the data and explain what the distribution means.

Average Suggested Retail Prices, 2011 Automobile Models (in thousands of dollars) Price Range 10- 14.9 15- 19.9 20- 24.9 25- 29.9 30- 34.9 35- 39.9 40- 44.9 45- 49.9

# of Models 1 3 5 6 9 2 2 3

Price Range 50- 54.9 55- 59.9 60- 64.9 65- 69.9 70- 74.9 75- 79.9 80- 84.9

# of Models 2 0 0 1 0 0 1

a.

This data distribution is skewed to the left. Most car models cost less than the mean.

15-19 20-29 30-39 40-54

Fre

qu

ency

S tudents


1

2

3

4

5

6

7

15-24 25-34 35-44 45-54

Fre

qu

ency

S tudents


1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170


53

b.

The data distribution is skewed to the right. Most car models cost more than the mean.

c.

The data distribution is skewed to the right. Most car models cost more than the mean.

d.

This data distribution is skewed to the left. Most car models cost less than the mean.

____ 7. The high temperatures in Concord, CA, for October 1–15, 2005, are given below.

Look at the histogram of these data below. What is the error in this histogram?

a. The bar for 74–77 is too short. c. The bar for 82–85 is too tall.

b. The bar for 78–81 is too tall. d. The bar for 86–89 is too short.


54

____ 8. The high temperatures for Concord, CA, for October 1–15, 2005, are given below.

Look at the box-and-whisker plot of these data below. What, if anything, is wrong with this box-and-whisker

plot?

a. The value of Q1 is incorrect. c. The value of Q3 is incorrect.

b. The median is incorrect. d. The box-and-whisker plot is correct.

____ 9. The ages of the U.S. Presidents that were inaugurated during the 1900s are given below.

Look at the box-and-whisker plot of these data below. What, if anything, is wrong with this box-and-whisker

plot?

a. The value of Q1 is incorrect. c. The value of Q3 is incorrect.

b. The median is incorrect. d. The box-and-whisker plot is correct.

____ 10. Which data set could be used to create the box-and-whisker plot shown below?

a. {4, 4, 4, 4, 6, 6, 8, 9, 11, 11, 11} c. {2, 3, 4, 4, 5, 7, 8, 9, 11, 13, 14}

b. {3, 4, 4, 4, 5, 7, 8, 10, 11, 12, 13} d. {3, 3, 3, 4, 7, 7, 11, 11, 13, 13, 13}

66 68 70 72 74 76 78 80 82 84 86 88 90


55

____ 11. A weatherperson records how much rain, in inches, falls each day in the first week of a month. She

makes a box-and-whisker plot of the data, shown below.

Which could be the data the weatherperson recorded?

a. {0, 1, 1, 2, 4, 4, 6} b. {0, 0, 1, 2, 2, 4, 6}

____ 12. An oceanographer is measuring high tide against a mark he’s made on a post. The results of several

days of measurements are shown below.

Which data set shows the heights of high tide over this period?

a. { , , 0, 0, 0, 0, 1, 1, 3.5} c. { , , , , 0, 0, 0.9, 1.1, 3.5}

b. { , , , 0, 0, 1, 1, 2, 3.5} d. { , , , , 0, 1.5, 2, 2.5, 3.5}

____ 13. Make a box-and-whisker plot of the data. Find the interquartile range.

a.

Interquartile range: 5.5

b.

Interquartile range: 5

c.

Interquartile range: 5.5

d.

Interquartile range: 5

6 7 8 9 10 11 12 13 14 15 16 17 18 19

6 7 8 9 10 11 12 13 14 15 16 17 18 19

6 7 8 9 10 11 12 13 14 15 16 17 18 19

6 7 8 9 10 11 12 13 14 15 16 17 18 19


56

____ 14. Marisa recorded the heights, in inches, of the 17 students in her homeroom. The data set she gathered

was 66, 63, 60, 55, 73, 59, 59, 63, 71, 61, 65, 64, 64, 69, 65, 66, 64. She created a histogram to display the

data. Which of the following are possible intervals she could use in her histogram?

a. 60-64, 65-69, 70-74

b. 55-59, 60-66, 67-69, 70-74

c. 55-59, 60-64, 65-69, 70-74

d. 60-69, 70-79

____ 15. Seung used box-and-whisker plots to show the points he scored in his basketball games this season.

He used different plots for the home and away game data, and produced the graph below.

Unfortunately, Seung cannot remember which plot represents the home game data and which represents the

away game data. Which fact can he use to determine which set of data was used to create each box-and-

whisker plot?

a. Seung scored at least 5 points in every home game.

b. The mode of the set of away game scores was 14.

c. Seung scored 23 points in an away game last week.

d. Seung played more home games than away games.

____ 16. Shawn collected the heights, in inches, of his classmates. He rounded the measurements to the

nearest inch and used those values to create a dot plot.

The next day, Shawn revisited the data he gathered about his classmates’ heights, rounded each measurement

to the nearest multiple of 5 inches, and used this set of rounded data to create a second dot plot.

Which characteristic of the two dot plots must be the same for both sets of data?

a. the highest data value

b. the number of data points plotted

c. the lowest data value

d. the range of the data

0 2 4 6 8 10 12 14 16 18 20 22 24 260


57

Unit 13 STUDY GUIDE

Analyze a set of data (Show all work if necessary):

1. # of slices of pizza eaten by a 16-year old boy in one sitting: 7, 5, 6, 7, 5, 5, 8, 15

Mean =__________ Median =__________ Mode =_________ Range =________

Do you think there’s an outlier? ___________

Which measure of “center” is best for this set of data?________________

2. Use the data in the table to make a frequency table with intervals.

Then use the frequency table to make a histogram.

Miles per Gallon of Vehicles on the Interstate

12 45 31 9 32 19 27 23 34 29 17 25 28

Complete the frequency table: Complete the histogram:

6

F

R 5

E

Q 4

U

E 3

N

C 2

Y

1

1-10 11-20

3. Describe the shape of the data in the histogram.

4. Based on the shape of the data, which interval has the median and mean? _______________

5. Create a line plot for this set of data: 8, 11, 5, 8, 9, 6, 8, 7, 10

Miles per Gallon of Vehicles on I-95

Miles per Gallon Frequency

1-10

11-20


58

6. From the $ Spent on Snacks line plot above: 7. From the Shoe Sizes line plot above:

a) Find the range: _________ a) Find the mean: ________

b) Find the median: _________ b) Find the mode: ________

Answer questions 8-10 using the box-and-whisker plots below.

The top box plot shows # of

pencils in students’ backpacks.

The bottom box plot shows the # of

pens in students’ backpacks.

8. What percent of students had more than 7 pencils in their backpacks?

9. What percent of students had less than 10 pens in their backpacks?

10. What do the box-and-whisker plots tell you about students carrying around pens and pencils in

their backpacks? Interpret the data on the graphs.

11. Create a box-and-whisker plot from the following set of data: 8, 2, 9, 4, 6, 8, 5

Minimum=_______ Q1=________ Median=________ Q3=________ Maximum=_______

0 1 2 3 4 5 6 7 8 9

12. What is the interquartile range (IQR) of this set of data? ____________

http://www.google.com/imgres?sa=X&rls=com.microsoft:en-us:IE-SearchBox&biw=1440&bih=719&tbm=isch&tbnid=lGUMXgCd2-5CSM:&imgrefurl=https://gcps.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/6605/9184/Inferences_Using_Line_Plots_print.html&docid=FG3cVCQ61fU49M&imgurl=https://gcps.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/6605/9184/Line_Plot_2.jpg&w=775&h=399&ei=8leJUdm4D5DK9QTRtYGoDA&zoom=1&iact=hc&vpx=922&vpy=419&dur=141&hovh=161&hovw=313&tx=132&ty=91&page=3&tbnh=145&tbnw=282&start=57&ndsp=34&ved=1t:429,r:89,s:0,i:436

http://www.google.com/imgres?sa=X&rls=com.microsoft:en-us:IE-SearchBox&biw=1440&bih=719&tbm=isch&tbnid=V6M33teX9rtAUM:&imgrefurl=http://www.eduplace.com/math/mw/background/4/09/te_4_09_overview.html&docid=OSf32ScA5YUNeM&imgurl=http://www.eduplace.com/math/mw/background/4/09/graphics/ts_4_9_wi-2.gif&w=187&h=94&ei=8leJUdm4D5DK9QTRtYGoDA&zoom=1&iact=hc&vpx=644&vpy=350&dur=47&hovh=75&hovw=149&tx=93&ty=44&page=2&tbnh=75&tbnw=149&start=23&ndsp=34&ved=1t:429,r:47,s:0,i:310

http://www.google.com/imgres?start=288&sa=X&rls=com.microsoft:en-us:IE-SearchBox&biw=1440&bih=719&tbm=isch&tbnid=voEBoKyV6Qb7sM:&imgrefurl=https://gcps.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/6605/7945/boxandwhisker_print.html&docid=ktOiIIMbDUy_pM&imgurl=https://gcps.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/6605/7945/boxandwhisker4.jpg&w=532&h=180&ei=elmJUYWXGpKI9gSw2YH4Bw&zoom=1&iact=hc&vpx=781&vpy=471&dur=4312&hovh=130&hovw=386&tx=149&ty=55&page=10&tbnh=94&tbnw=272&ndsp=34&ved=1t:429,r:20,s:300,i:64


59

13. Student heights in inches: 65, 62, 59, 60, 64, 70

Find the mean absolute deviation of student heights.

Fill in the table below to find it: Mean height = _______ M. A. D. = _______

(Round to the nearest hundredth

if necessary)

If a data set has an outlier,

the MAD will be

____________ than a

data set with data values

that are all very close

together.

DATA DIFFERENCE from MEAN Absolute Value

65

62

59

60

64

70

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