Transcript Math Review - Cobb Learning

Math Review
Data Analysis, Statistics, and
Probability
Data Interpretation


Your primary task in these questions is to
interpret information in graphs, tables, or
charts, and then compare quantities,
recognize trends and changes in the date,
or perform calculations based on the
information you have found.
You should be able to understand info
presented in a table or various types of
graphs.
Data Interpretation

Schools
Circle Graphs (Pie Charts)
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This pie chart shows the
percentage of its total
6%
expenditures that Weston
spends on various types of
15%
expenses. Suppose you are
given that the total expenses
in 2004 were $10 million, and
you are asked for the amount
of money spent on the police
dept and fire dept combined. 18%
You can see that together
these two categories account
6%
for 23% + 12%, so 35% total.
12%
Therefore, 35% of $10 million
is $3.5 million.
Police
Dept
20%
Fire Dept
23%
Water
Supply
Other
Expense
s
Sanitatio
Data Interpretation
Line Graphs
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This graph shows the high and low temps in Weston for the first
7 days of Feb.
From the graph you can see that the high on Feb 5 was 25
degrees and the low was 10 degrees. You can also see the
difference between high and low temps that day was 15.
45
40
35
30
25
20
15
10
5
0
High
Low
2/
1/
20
2/ 10
2/
20
2/ 10
3/
20
2/ 10
4/
20
2/ 10
5/
20
2/ 10
6/
20
2/ 10
7/
20
10
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Temp (Fahrenheit)
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Date
Data Interpretation
Bar graphs
This one shows the amount of snow that fell each day
in Weston for the first 7 days of Feb. For example,
you can see that no snow fell on Feb 2 and that 6
inches of snow fell the next day.
7
6
5
4
3
2
1
0
eb
7F
eb
6F
eb
5F
4`
eb
Fe
b
3F
eb
2F
eb
Column 1
1F
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Amount of Snow (inches)
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Data Interpretation
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Pictographs
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Presents data using
pictorial symbols.
So you can see that 40
snowmen were built
on Feb 7 and 20 were
built the day before.
Date
Feb
Feb
Feb
Feb
Feb
Feb
Feb
1
2
3
4
5
6
7
= 10
snowmen
Number of
Snowmen
Data Interpretation
Scatterplot
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Compares two characteristics
of the same group of people
or things. You can see a lot of
info from this plot. There are
20 people. 4 with 1 year of
experience, 3 with 2 years, 4
with 3 years, 2 with 4 years, 3
with 5 years, and 4 with 6
years. The median level of
experience is 3 years. Salary
tends to increase with
experience. 3 people make
$700 a week and 3 others
make $850 week. The median
salary is $787.50
Weekly Salary (dollars)

900.00
850.00
800.00
750.00
700.00
650.00
600.00
550.00
0
2
4
6
Years of Experience
8
Data Interpretation

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Questions may ask to read the info on the
chart/graph, or to identify specific pieces of
information (data), compare data from different
parts of the graph, and manipulate the data.
Make sure to
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Look at the graph to make sure you understand it and
what info is being displayed
Read the labels
Make sure you know the units
Understand what is happening to the data as you
move through the table, graph, or chart.
Read the question carefully.
Data Interpretation
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Example
The graph below shows
profits over time. The
greater the profits the
higher the point on the
vertical axis will be. Each
tick mark is another
$1000. As you move right
along the horiz. axis,
months are passing.
Comparison of Monthly Profits
Profit (thousands of dollars)
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5
4
3
Company X
Company Y
2
1
0
Jan
Feb
Mar
Apr
May
Data Interpretation
Example
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In what month or
months did each
company make the
greatest profit?
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For company x it was
April
For company y it was
May
Comparison of Monthly Profits
Profit (thousands of dollars)
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5
4
3
Company X
Company Y
2
1
0
Jan
Feb
Mar
Apr
May
Data Interpretation
Example
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Between which two
consecutive months did
each company show the
greatest increase in profit?
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Increase/Decrease in
profit is shown by the
slope (steepness of line)
Just looking at it, you can
tell for company x it was
between march and april
For company y it is
between jan and feb
Comparison of Monthly Profits
Profit (thousands of dollars)
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5
4
3
Company X
Company Y
2
1
0
Jan
Feb
Mar
Apr
May
Data Interpretation
Example

In what month did the
profits of the 2
companies show the
greatest difference?
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The point where they
are farthest apart – so
it would be in April
Comparison of Monthly Profits
Profit (thousands of dollars)
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5
4
3
Company X
Company Y
2
1
0
Jan
Feb
Mar
Apr
May
Data Interpretation
Example
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If the rate of increase or
decrease for each company
continues for the next 6
months at the same rate
shown between April and
May, which company would
have higher profits at the
end of that time?
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Need to extend the line
between Apr and May
Can easily see the answer
would be company y
Comparison of Monthly Profits
Profit (thousands of dollars)
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5
4
3
Company X
Company Y
2
1
0
Jan
Feb
Mar
Apr
May
Statistics
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Arithmetic Mean
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Average
Formula:
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(Sum of list of values)/(number of values in list)
Example:
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Three kids, aged 6, 7, and 11. Find the Arithmetic
mean.
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(6+7+11)/(3) = 24/3 = 8 years
Statistics
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Median
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The middle value of a list when the numbers are in
order.
Place the values in order (ascending or descending)
and select the middle value.
Example:
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Find the median: 200, 2, 667, 19, 4, 309, 44, 6, 1
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Order: 1, 2, 4, 6, 19, 44, 200, 309, 667
Middle value is the fifth, so median is 19
For an even list of numbers, average the two middle
values to get the median
Statistics
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Mode
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The value or values that appear the greatest
number of times.
Example:
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Find the mode: 1, 5, 5, 7, 89, 4, 100, 276, 89, 4,
89, 1, 8
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89 appears 3 times (more than any other number) – it’s
the mode!
It is possible to have more than one mode.
Statistics
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Weighted Average
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The average of 2 or more groups that do not all have
the same number of members.
Example:
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15 members of a class had an average (mean) SAT math
score of 500. The remaining 10 members had an average of
550. What is the average score of the entire class?
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Can’t take average of two numbers b/c of different amount of
students in each group. Has to be weighted toward the group
with the greater number.
So, multiply each average by its weighted factor first and then
average them.
( (500*15)+(550*10) /25) = 520
Statistics
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Average of Algebraic Expressions
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Average same way as other values.
Example:
What is the average (mean) of 3x+1 and x-3?
 Find the sum of the expressions and divide by the
number of expressions:
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( (3x+1)+(x-3) )/2 = (4x-2)/2 = 2x-1
Statistics
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Using Averages to Find Missing Numbers
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You can use simple algebra in basic average
formulas to find missing values when the
average is given:
Basic Average formula:
(sum of a list of values)/(number of values in the
list)=average
 If you have the average and the number of values,
you can figure out the sum of the values:
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Average*number of values=sum of values
Statistics
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Example
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The average (mean) of a list of 10 numbers is
15. if one of the numbers is removed, the
average of the remaining numbers is 14. what
is the number that was removed?
You can figure out the sum of all the values:
15*10=150
 Sum of values for one removed: 14*9=126
 The difference between the 2 sums will give you
the missing number. 150-126=24.
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Probability
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Some questions will involve elementary
probability.
Example:
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Find the probability of choosing an even number at
random from the set: {6, 13, 5, 7, 2, 9}
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There are 6 numbers total and only 2 even numbers, so the
probability would be 2/6 = 1/3
Remember, the probability of an event is a
number between 0 and 1, inclusive. If an event
is certain, it has probability of 1. If an event is
impossible (cannot occur) the probability is 0.
Probability
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Independent/Dependent Events
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Two events are independent if the outcome of either
event has no effect on the other. (Toss a penny and it
has ½ landing on heads and then toss a nickel and it
has ½ prob of landing on heads).
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To find the probability of 2 or more ind. events occurring
together, you multiply the different probabilities together.
(1/2 * ½)= ¼
If the outcome of one event affects the probability of
another event, they are dependent events. You must
use logical reasoning to help figure out probabilities
involving dependent events.
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So the penny landing on heads is ½ but of it then landing on
tails is 0 (can’t land on tails at the same time)
Probability
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Example
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On Monday, Anderson High School’s basketball team will play the
team from Baker High School. On Wednesday, Baker’s team will
play the team from Cole High School. On Friday, Cole will play
Anderson. In each game, either team has a 50 percent chance
of winning. A) What is the probability that Anderson will win
both its games? B) What is the probability that Baker will lose
both its games? C) What is the probability that Anderson will win
both its games and Baker will lose both its games?
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A) Anderson has ½ prob of winning 1st game and also 2nd game.
Ind events. So just multiply: ½ * ½ = ¼
B) Baker has ½ prob of losing 1st game and also 2nd game. Ind
events. So just multiply: ½ * ½ = ¼
C) For Anderson to win both games and Baker to lose both, three
games must be played out: And beats Baker, Cole beats Baker, And
beats Cole. Each game has ½ prob, and ind. So just multiply all
together: ½ * ½ * ½ = 1/8
Probability
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Geometric Probability
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Some probability questions in the math section may
involve geometric figures.
Example:
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The large circle has a radius 8 and small circle has radius 2.
If a point is chosen at random from the large circle, what is
the probability that the point chosen will be in the small
circle?
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Area of large circle is 64pie, and area of small circle is 4pie
Area of small circle/ area of large circle = 4pie/64pie = 4/64 =
1/16
So probability of choosing a point from the small circle is 1/16