Download UNIT 5-STATISTICS AND PROBABILITY

MATHEMATICS
GRADE 7
UNIT 5
STATISTICS AND
PROBABILITY
1
VOCABULARY FOR FLASHCARDS
A.) Biased Sample: Tickets from a basketball game are placed in a
container. The collector of the tickets does not mix
the tickets around. Then a winner is chosen for a
prize. This is a biased sample. A person from the top
is most likely to win the prize because the people who
bought the tickets first are on the bottom.
B.) Counting Principle: The counting principle states the following: “If one
event can occur in “m” ways and another event can
occur in “n” ways, then the two events can occur
(m)(n) ways.”
C.) Dot plot: A dot plot is a type of graphic display used to compare
frequency categories or groups.
D.) Experimental Probability: This is the probability that an event actually
occurred after doing an experiment. For
example, if a person flips a coin 100 times
and heads came up 70 times, the experimental
probability would 0.7. Experimental and
theoretical probability will not necessarily
always equal each other.
E.) Mean: The mean of data is better known as the average. To find the
mean (average), the sum of all the data must be taken and then
divided by the total number of items of data.
F.) Median: The median is the value that falls in the middle of the data.
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VOCABULARY FOR FLASHCARDS
G.) Mode: The mode is the number that occurs most often.
H.) Probability: Probability is a numerical measurement that allows someone
to know how often an event will occur. This numerical value
is measured on a scale from 0 to 1. The probability of an
event to occur can be written as the following ratio:
Probability of Event =
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝑭𝒂𝒗𝒐𝒓𝒂𝒃𝒍𝒆 𝑶𝒖𝒕𝒄𝒐𝒎𝒆𝒔
𝑻𝒐𝒕𝒂𝒍 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝑶𝒖𝒕𝒄𝒐𝒎𝒆𝒔
I.) Random Sample: A random amount of items taken from a
whole group or population. This random sample represents the
whole group or population.
J.) Range: The range of a set of data is the difference between the
greatest number and the least number.
K.) Theoretical Probability: This is the probability that an event is
supposed to occur. For example, the
theoretical probability of heads coming up when
flipping a coin is 0.5 or half the time. The
theoretical probability of tails coming up when
flipping a coin is 0.5 or half the time.
L.) Whole group or Population: This can be represented by
people, marbles, paperclips, cookies, etc.
M.) With Replacement: An event where an item is replaced which returns
the total number of items back to the original
amount.
3
VOCABULARY FOR FLASHCARDS
N.) Without Replacement: An event where an item is not replaced which in
turn decreases the total number of items.
4
A.) Random Sample: A random amount of items taken from a
whole group or population. This random sample represents the
whole group or population.
Example 1: Tickets from a basketball game are placed in a container. The
collector of the tickets puts there hand in and mixes the tickets
around. Then a winner is chosen for a prize. This is random
because the person first mixed the tickets around before
drawing the winner.
EXAMPLES OF WAYS TO OBTAIN RANDOM SAMPLES:
1. In a mall, interview every 10th person who walks in the door.
2. Draw names out of a container.
3. Divide a school into all the different grade levels and then choose a
person from each group.
B.) Biased Sample: Tickets from a basketball game are placed in a
container. The collector of the tickets does not mix the
tickets around. Then a winner is chosen for a prize.
This is a biased sample. A person from the top is most
likely to win the prize because the people who bought
the tickets first are on the bottom.
EXAMPLES OF WAYS TO OBTAIN BIASED SAMPLES:
1. Ask only the girls in a study hall what the school’s favorite subject is.
2. Ask only the boys in physical education class what the school’s favorite
subject is.
3. Ask the people at a fishing derby what their favorite outdoor activity is.
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C.) Whole group or Population: This can be represented by
people, marbles, paperclips, cookies, etc.
Example 1: A survey is being done for a school to determine what
their favorite lunch meal is.
Example 2: A survey is being done for New York State to determine the
favorite leisure activity.
Example 3: All the names of a football team are placed in container to
determine who will be the one to put the footballs away each
night.
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Statistical Measures
D) Mean: The mean of data is better known as the average. To find the
mean (average), the sum of all the data must be taken and then
divided by the total number of items of data.
E) Median: The median is the value that falls in the middle of the data.
To find the median of given data, the following must be done:
Odd Number Of Data Items:
1. Arrange the data in order from highest to lowest or vice-versa
2. Find the number that falls in the middle, this is the median number
Even Number Of Data Items:
1. Arrange the data in order from highest to lowest or vice-versa
2. Find the 2 numbers that fall in the middle and take the average of
them, this will be the median number
F) Mode: The mode is the number that occurs most often. A set of data
can have one of three modes:
1. Only one mode
2. More than one mode
3. No mode
G) Range: The range of a set of data is the difference between the
greatest number and the least number
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Examples of Statistical Problems
1. The Whitesboro girls basketball team scored the following points in their
most recent games: 48, 65, 72, 54, 52, 65, 58, 60, 49, and 57. Find
the mean, median, mode and range of the data.
Mean =
48  65  72  54  52  65  58  60  49  57 580

 58
10
10
Median = 48, 49, 52, 54, 57, 58, 60, 65, 65, 72
57  58
 57.5
2
Mode = 65
Range = 72 – 48 = 24
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Examples of Statistical Problems
2. C.C. Sabatha a pitcher for the New York Yankees gave up the following
earned runs in his most recent games: 4, 2, 3, 5, 1, 3, 3, 5 ,6, 2, and
4. Find the mean, median, mode and range of the data.
Mean =
4  2  3  5  1  3  3  5  6  2  4 38

 3.45
11
11
Median = 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6
Median is 3
Mode = 3
Range = 6 - 1 = 5
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Examples of Statistical Problems
3. Given the following set of data, find the mean, median, mode and range:
14, -9, -11, 23, 2, 0, -9, 0, 8
14  9  11  23  2  0  9  0  9 18

2
Mean =
9
9
Median = -11, -9, -9, 0, 0, 2, 8, 14, 23
Median = 0
Mode = 0
Range = 23 – (-11) = 34
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Constructing a Dot Plot
H.) Dot plot: A dot plot is a type of graphic display used to compare
frequency categories or groups.
Method:
1. Draw a “number line” but have it end with line segments instead of
arrows.
2. Increment the “number line” appropriately.
3. Write a title above the “number line.”
4. Label below the “number line” what the information is pertaining to and
the units of measure.
5. Plot the values above the number using “x’s”. If any values repeat,
draw another “x” and stack the “x’s” above each other.
Example 1: Mr. Crowe’s class took a 10 point quiz on calculating the mean
absolute deviation of data. The following scores are what his
students received on the quiz:
7, 5, 8, 8, 9, 10 ,10, 7, 6, 10
A.) Construct a dot plot of the following information.
Mr. Crowe’s Quiz Scores (out of 10)
x
x x
x
x
x x x x x
5
6 7 8 9 10
Quiz Score
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B.) Calculate the mean.
𝟕 + 𝟓 + 𝟖 + 𝟖 + 𝟗 + 𝟏𝟎 + 𝟏𝟎 + 𝟕 + 𝟔 + 𝟏𝟎 𝟖𝟎
=
=𝟖
𝟏𝟎
𝟏𝟎
C.) Calculate the median.
5, 6, 7, 7, 8, 8, 9, 10, 10, 10
median =
𝟖+𝟖
𝟐
=
𝟏𝟔
𝟐
D.) Calculate the range.
10 – 5 = 5
E.) Determine the mode.
10
12
=𝟖
Example 2: Given the following dot plot below, calculate the mean, median,
range and mode.
Ages of Volunteers at Newburgh Town Park
x
x
x x x
x
x
x x x x
x
25 26 27 28 29 30
Ages (years)
A.) Calculate the mean. If necessary, round your answer to the nearest
year.
𝟐𝟓 + 𝟐𝟓 + 𝟐𝟔 + 𝟐𝟕 + 𝟐𝟕 + 𝟐𝟖 + 𝟐𝟖 + 𝟐𝟗 + 𝟐𝟗 + 𝟑𝟎 + 𝟑𝟎 + 𝟑𝟎 𝟑𝟑𝟒
̅
=
= 𝟐𝟕. 𝟖𝟑
𝟏𝟐
𝟏𝟐
= 28
B.) Calculate the median.
25, 25, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30
𝟐𝟖 + 𝟐𝟖 𝟓𝟔
=
= 𝟐𝟖
𝟐
𝟐
C.) Calculate the range.
D.) Determine the mode.
30 – 25 = 5
30
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I.) Mean Absolute Deviation (MAD): The mean absolute deviation (MAD)
describes the average distance between the data values in a distribution
and the mean of the distribution. This value helps compare estimates of
spread against a formal measurement. Mean absolute deviation (MAD)
helps to determine if the mean is useful. The smaller the mean absolute
deviation (MAD) the more relevant the mean value is.
METHOD TO FIND MEAN ABSOLUTE DEVIATION (MAD):
a) Calculate the mean of the data.
b) Subtract the mean from each data point.
c) Average the absolute values of all values from Part b to get the mean
absolute deviation (MAD).
PLEASE NOTE: WE WILL USE A CHART TO ORGANIZE THE INFORMATION
Example 1: An advanced mathematics classroom has 10 students in it. The
students took a test and the results were the following:
90, 75, 85, 100, 97, 80, 70, 90, 99, 83
a) Calculate the mean.
𝟗𝟎 + 𝟕𝟓 + 𝟖𝟓 + 𝟏𝟎𝟎 + 𝟗𝟕 + 𝟖𝟎 + 𝟕𝟎 + 𝟗𝟎 + 𝟗𝟎 + 𝟖𝟑 𝟖𝟔𝟎
=
= 𝟖𝟔
𝟏𝟎
𝟏𝟎
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b) Complete the table to find the absolute deviation from the mean.
Data Points, x
70
75
80
83
85
90
90
90
97
100
Deviation from Mean
x – mean
70 – 86 = -16
75 – 86 = -11
80 – 86 = -6
83 – 86 = -3
85 – 86 = -1
90 – 86 = 4
90 -86 = 4
90 – 86 = 4
97 – 86 = 11
100 – 86 = 14
Absolute Deviation from
Mean, |𝐱 − 𝐦𝐞𝐚𝐧|
|−𝟏𝟔| = 𝟏𝟔
|−𝟏𝟏| = 𝟏𝟏
|−𝟔| = 𝟔
|−𝟑| = 𝟑
|−𝟏| = 𝟏
|𝟒| = 𝟒
|𝟒| = 𝟒
|𝟒| = 𝟒
|𝟏𝟏| = 𝟏𝟏
|𝟏𝟒| = 𝟏𝟒
c) Using the absolute deviation from the mean, find the mean absolute
deviation (MAD).
𝟏𝟔 + 𝟏𝟏 + 𝟔 + 𝟑 + 𝟏 + 𝟒 + 𝟒 + 𝟒 + 𝟏𝟏 + 𝟏𝟒 𝟕𝟒
=
= 𝟕. 𝟒
𝟏𝟎
𝟏𝟎
d) What does the result for the mean absolute deviation (MAD) tell you
about the mean? Is it relevant or irrelevant?
Seeing how the mean absolute deviation (MAD) is 7.4, this shows the mean is
irrelevant because the data values are so far away from the mean.
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Example 2: The chart below represents the points Lawrence and Howard
scored for the first 5 games of the basketball season.
Game
1
2
3
4
5
Lawrence
18
20
8
25
13
Howard
20
16
15
9
15
a) Calculate the mean for Lawrence and Howard.
Lawrence =
Howard =
𝟏𝟖+𝟐𝟎+𝟖+𝟐𝟓+𝟏𝟑
𝟓
𝟐𝟎+𝟏𝟔+𝟏𝟓+𝟗+𝟏𝟓
𝟓
=
=
𝟖𝟒
𝟕𝟓
𝟓
𝟓
= 𝟏𝟔. 𝟖
= 𝟏𝟓
b) Complete the table to find the absolute deviation from the mean for
Lawrence.
Data Points, x
8
13
18
20
25
Deviation from Mean
x – mean
8 – 16.8 = -8.8
13 – 16.8 = -3.8
18 – 16.8 = 1.2
20 – 16.8 = 3.2
25 – 16.8 = 8.2
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Absolute Deviation from
Mean, |𝐱 − 𝐦𝐞𝐚𝐧|
|−𝟖. 𝟖| = 𝟖. 𝟖
|−𝟑. 𝟖| = 𝟑. 𝟖
|𝟏. 𝟐| = 𝟏. 𝟐
|𝟑. 𝟐| = 𝟑. 𝟐
|𝟖. 𝟐| = 𝟖. 𝟐
c) Complete the table to find the absolute deviation from the mean for
Howard.
Data Points, x
Deviation from Mean
x – mean
9 – 15 = -6
15-15 = 0
15 - 15 = 0
16 – 15 = 1
20 – 15 = 5
9
15
15
16
20
Absolute Deviation from
Mean, |𝐱 − 𝐦𝐞𝐚𝐧|
|−𝟔| = 𝟔
|𝟎| = 𝟎
|𝟎| = 𝟎
|𝟏| = 𝟏
|𝟓| = 𝟓
d) Find the mean absolute deviation (MAD)for each person. Round to the
nearest hundredths place if needed.
Lawrence =
Howard =
𝟖.𝟖+𝟑.𝟖+𝟏.𝟐+𝟑.𝟐+𝟖.𝟐
𝟓
𝟔+𝟎+𝟎+𝟏+𝟓
𝟓
=
𝟏𝟐
𝟓
=
𝟐𝟓.𝟐
𝟓
= 𝟓. 𝟎𝟒
= 𝟐. 𝟒𝟎
e) Who has the greater mean absolute deviation (MAD)? What does this
mean?
Lawrence has the higher mean absolute deviation (MAD). This statistic means
that Howard’s mean is more relevant than Lawrence’s due to fact that his
scores are closer to the mean.
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J.) Probability: Probability is a numerical measurement that allows someone
to know how often an event will occur. This numerical value
is measured on a scale from 0 to 1. The probability of an
event to occur can be written as the following ratio:
Probability of Event =
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝑭𝒂𝒗𝒐𝒓𝒂𝒃𝒍𝒆 𝑶𝒖𝒕𝒄𝒐𝒎𝒆𝒔
𝑻𝒐𝒕𝒂𝒍 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝑶𝒖𝒕𝒄𝒐𝒎𝒆𝒔
K.) Theoretical Probability: This is the probability that an event is supposed
to occur. For example, the theoretical
probability of heads coming up when flipping a
coin is 0.5 or half the time. The theoretical
probability of tails coming up when flipping a
coin is 0.5 or half the time.
L.) Experimental Probability: This is the probability that an event actually
occurred after doing an experiment. For
example, if a person flips a coin 100 times
and heads came up 70 times, the experimental
probability would 0.7. Experimental and
theoretical probability will not necessarily
always equal each other.
Describing Probabilities in Words
a) certain: when an event will definitely happen
b) impossible: when an event will definitely not happen
c) more likely than not: event occur more than not
d) as likely as not: event to occur or not occur equally
e) less likely than likely: event not to occur than occur
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Example 1: A person has a bag with 12 marbles in it. The number of red
marbles are 6, the number of white and blue marbles
respectively are 3 and 3. What is the theoretical probability of
pulling out a white marble?
P(white marble) =
𝟑
𝟏𝟐
𝟏
= =. 𝟐𝟓 = 𝟐𝟓%
𝟒
Example 2: In the game of hockey, a team can win, lose or tie. What is
the theoretical probability of the team winning.
𝑷(𝒘𝒊𝒏𝒏𝒊𝒏𝒈) =
𝟏
̅%
= ̅̅̅̅
. 𝟑 = 𝟑𝟑. 𝟑
𝟑
Example 2b: A team plays 100 games and wins 60. What is the
experimental probability of winning?
P(winning) =
𝟔𝟎
𝟏𝟎𝟎
=
𝟑
𝟓
= 𝟎. 𝟔 = 𝟔𝟎%
PLEASE NOTE: BASED ON THE THEORETICAL PROBABILITY, THE TEAM
SHOULD HAVE WON APPROXIMATELY 33 GAMES BUT
THEY ACTUALLY WON 60. IT IS OBVIOUS THAT THE
THEORETICAL AND EXPERIMENTAL PROBABILITY DID
NOT COME OUT THE SAME. THIS IS
MATHEMATICALLY ACCEPTABLE.
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Example 3: A standard game cube is rolled 60 times. How many times
should the number 5 appear?
𝟏
THEORETICAL PROBABLILITY FOR A 5 TO APPEAR IS . THEREFORE,
𝟔
USING A PROPORTION IS A MATHEMATICALLY SOUND WAY TO SOLVE
THIS PROBLEM.
𝟏(𝒓𝒐𝒍𝒍𝒊𝒏𝒈 𝒂 𝟓) 𝒙(𝒓𝒐𝒍𝒍𝒊𝒏𝒈 𝒂 𝟓)
=
𝟔(𝒕𝒐𝒕𝒂𝒍)
𝟔𝟎(𝒕𝒐𝒕𝒂𝒍)
𝟔𝒙 𝟔𝟎
=
𝟔
𝟔
x = 10
M) Counting Principle: The counting principle states the following: “If one
event can occur in “m” ways and another event can
occur in “n” ways, then the two events can occur
(m)(n) ways.”
Example 1: You have three different shirts and two different pairs of
slacks.
Counting Principle = 3 x 2 = 6 outfits
Example 2: How many different committees of 1 boy and 1 girls can be
formed from a group of 4 girls and 3 boys?
Counting Principle = 4 x 3 = 12 committees
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N.) Compound Event: Is looking at outcomes of more than one event.
O.) Tree Diagram: A tree diagram can be used to represent the total
number of possible outcomes.
P.) Sample Space: A sample space can be used to represent the total
number of possible outcomes.
Q.) Table: A table can be used to represent the total number of possible
outcomes.
R.) Listing: A list can be used to represent the total number of possible
outcomes.
Example 1: A two-person committee is to be chosen from 3 seventh-grade
and 3 eighth grade students. The committee must have one
student from each grade.
A.) Use a tree diagram to show all the possible committees.
71
72
81
83
81
82
73
83
81
82
83
82
B.) Use a table to show all the possible committees.
71
72
73
81
7181
7281
7381
82
7182
7282
7382
21
83
7183
7283
7383
C.) Use a sample space to show all the possible committees.
7181
7281
7381
7182
7282
7382
7183
7283
7383
Question 1: How many outcomes are represented? 9
Question 2: Would there be a difference if an eighth grader was chosen
first then the 7th grader? No
Question 3: What is the probability that a seventh grader will be in the
committee?
𝟗
𝟗
𝟏
= =𝟏
𝟏
Question 4: What is the probability that the same seventh grader will be in
multiple committees?
𝟑
𝟗
22
=
𝟏
𝟑
S.) With Replacement: An event where an item is replaced which returns
the total number of items back to the original
amount.
T.) Without Replacement: An event where an item is not replaced which in
turn decreases the total number of items.
Example 1: A bag has 10 marbles, 2 of which are purple, 5 are green and 3
are yellow. You pick 2 marbles, one at a time, and replace the
marble.
A.) What is the probability of selecting a purple marble and then another
purple marble?
𝟐 𝟐
𝟒
𝟏
•
=
=
𝟏𝟎 𝟏𝟎 𝟏𝟎𝟎 𝟐𝟓
B.) What is the probability of selecting a green marble then a yellow
marble?
𝟓 𝟑
𝟖
𝟐
•
=
=
𝟏𝟎 𝟏𝟎 𝟏𝟎𝟎 𝟐𝟓
Example 2: You have 10 pieces of licorice in a bag, 4 red, 3 black, 3 green.
You pick one piece of licorice, eat it, and then pick another piece
of licorice to eat.
A.) What is the probability of picking a red licorice and then a black
licorice?
𝟒 𝟑 𝟏𝟐
𝟐
• =
=
𝟏𝟎 𝟗 𝟗𝟎 𝟏𝟓
B.) What is the probability of picking a green licorice and then another
green licorice?
𝟑 𝟐
𝟔
𝟏
• =
=
𝟏𝟎 𝟗 𝟗𝟎 𝟏𝟓
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