Class5 statisticsi

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GRAPHICAL METHODS FOR DESCRIBING DATA
Frequency distributions
The frequency, or count, of a particular category or numerical value is the
number of times that the category or value appears in the data. A frequency
distribution is a table or graph that presents the categories or numerical
values along with their associated frequencies. The relative frequency of a
category or a numerical value is the associated frequency divided by the
total number of data. Relative frequencies may be expressed in terms of
percents, fractions, or decimals. A relative frequency distribution is a table
or graph that presents the relative frequencies of the categories or
numerical values.
Example: a survey was taken to find the number of children in each of 25
families. A list of the values collected in the survey follows.
1 2 0 4 1 3 3 1 2 0 4 5 2 3 2 3 2 4 1 2 3 0 2 3 1

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The table above shows the frequency distribution of the values of a variable Y.
What is the mean of the distribution?
Give your answer to the nearest 0.01

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HISTOGRAM
A bar graph representing frequency distribution for certain ranges or
intervals.
The number of data ítems in an interval is a frequency.
The bar heights represent these frequencies.

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Bar graphs
A commonly used graphical display for representing frequencies, or
counts, is a bar graph, or bar chart. In a bar graph, rectangular bars are
used to represent the categories of the data, and the height of each bar
is proportional to the corresponding frequency or relative frequency. All
of the bars are drawn with the same width, and the bars can be
presented either vertically or horizontally. Bar graphs enable
comparisons across several categories, making it easy to identify
frequently and infrequently occurring categories.

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From the graph, we can conclude …
A segmented bar graph is used to show how different subgroups or
subcategories contribute to an entire group or category. In a
segmented bar graph, each bar represents a category that consists
of more than one subcategory. Each bar is divided into segments
that represent the different subcategories. The height of each
segment is proportional to the frequency or relative frequency of the
subcategory that the segment represents.

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CIRCLE GRAPHS
Circle graphs are often called pie charts, are used to represent data with a
relatively small number of categories. They illustrate how a whole is separated
into parts. The área of the circle graph representing each category is
proportional to the part of the whole that the category represents.
Source: American Water Works Association Research
Foundation, “Residential End Uses of Water”, 1999

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Multiple choice Questions-Select one or more answer choices
The circle graph above shows the distribution of 200,000 physicians by
specialty. Which of the following sectors of the circle graph represent
more than 40,000 physicians?
Indicate all such sectors:
 Pediatrics
 Internal Medicine
 Surgery
 Anesthesiology
 Psychiatry
THE OFFICIAL GUIDE TO THE GRE revised general test 2011

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(a) Approximately what was the ratio of the value of sensitized goods to
the still-picture equipment produced in 1971 in the United States?
(b) If the value of office copiers produced in 1971 was 30 percent
greater than the corresponding value in 1970, what was the value of
office copiers produced in 1970?

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Al’s weekly net salary is $350. To how many of the categories listed was at
least $80 of Al’s weekly net salary allocated?

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The pie charts show average household expenses in 1965 and 2015. The way the
budget was divided up was very different in each of those years.
In 1965, the largest part of the budget was spent on food and housing. Thirty-five
percent of the total was spent on food and 30% on housing. This accounted for well
over half the Budget. Healthcare (15%) and clothing (12%) together took up over
one fourth of the Budget, while the smallest amounts were for transportation (5%)
and other expenses (3%).
In 2015, household spending was somewhat different. In that year, housing and
healthcare accounted for the largest part of the budget. Together they made up a
little over half of total spending. In the meantime, transportation had rised from 5%
to 15% of the Budget, while food had dropped from 35% to 15%. Spending on
clothing had dropped to 5% of the total, and other expenses had risen slightly to 5%.

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The line graph shows how many people visited the Grenby History Museum and The
Grenby Amusement Park each day for a week. For both places, the number of visitors
was higher on the weekend that it was during the week.
The number of visitors to the Grenby History Museum was low early in the week but
picked up on the weekend. On Monday, there were around 250 visitors. This number
dropped slightly on Tuesday and Wednesday, and dipped again on Thursday, to just
below 200 visitors. The number rose on Friday, then soared to 400 on Saturday, the
busiest day of the week, and then fell slightly on Sunday.
The Grenby Amusement Park also received fewer visitors during the week than on the
weekend. On Monday there were 300 visitors. This number rose slightly on Tuesday and
then fell back to 300 on Wednesday. The number climbed for the rest of the week until it
reached over 700 on both Saturday and Sunday.
Both places were busier on the weekend than during the week, although the amusement
park received more visitors every day than the museum did.

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If s is a speed, in miles per hour, at which the energy used per meter during
running is twice the energy used per meter during walking, then, according
to the graph above, s is between
A) 2.5 and 3.0
B) 3.0 and 3.5
C) 3.5 and 4.0
D) 4.0 and 4.5
E) 4.5 and 5.0
GRE Test Guide 2014

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The bar graph shows how much fast food men and women of different age groups
consumed every day in 2015. In both groups, the percentage of daily calories from fast food
decreased as age increased.
According to the graph, men consumed less fast food as they got older. The 18-34 age
group had the highest rate, getting around 17% of daily calories from fast food. There was a
significant drop, to just over 10%, between this group and the next, age 35-49. Then, there
was a smaller group, to around 9%, in the 50-64 age group. The oldest age group, age 65
and over, had a much lower consumption rate, with just over 4% of daily calories coming
from fast food.
Among women, there was a similar decrease in fast food consumption with age increase.
Women age 18-34 got 16% of their daily calories from fast food. Like men, there was a
significant drop between this and the next age group, to a little under 10%, and then a
smaller drop to the next age group. The oldest age group consumed far less, under 4%,
than the other age groups.
There was a similar decrease in fast food consumption with age increase for both men and
women, although women tended to consume slightly less than men across all age groups.

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The bar graph shows the total number of rides taken by bus and by subway in Willaimsville
for each year from 2011 to 2015. The size of the annual ridership varied in different ways
for each of the two forms of transportation.
The annual number of bus rides went up and down during the time period shown on the
graph. In 2011, annual bus ridership was a little over 400,000,000. This number rose
slightly to around 450,000,000 in 2012. It dropped a little in 2013, rose to almost
500,000,000 in 2014, and then dropped again in 2015 to about 450,000,000.
Annual subway ridership, on the other hand, rose steadily during the same time period. In
2011, a little over 600,000,000 rides were taken by subway. This number rose to around
650,000,000 in 2012 and continued to climb steadily until it reached around 750,000,000 in
2015.
In every year son on the graph, subway ridership was significantly higher than bus
ridership. In addition, bus ridership went up and down in this time period, while subway
ridership rose consistently.

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Scatterplots
To show the relationship between two numerical variables, the most
useful type of graph is scatterplot. In a scatterplot, the values of one
variable appear on the horizontal axis of a rectangular coordinate
system and the values of one variable appear on the vertical axis. For
each individual or object in the data, an ordered pair of numbers is
collected, one number for each variable, and the pair is represented by
a point in the coordinate system.

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THE LANGUAGE OF CHANGE
Vocabulary used to describe a graph which shows changes over a
period of time. You need to use language expressing change and
appropiate tenses.

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The graph gives information about changes in the birth and death rates in New
Zealand between 1901 and 2101.
In 1901 the birth rate was 20,000 and the death rate was 9,000. In 1861 the birth
rate reached a peak of 66,000 while the death rate was 23,000. In 2001 there
were 55,000 births and 38,000 deaths, and in 2061 there were 60,000 deaths
and 48,000 births.
At the end of the period there were 58,000 deaths and 45,000 births.
Both the birth and death rates changed between 1901 and 2101. Perhaps this
was because a lot of people did not want to have children.

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This is a weak answer which would score a low IELTS band.
Problems:
• Underlength
• Introduction is copied from task
• No comparison between figures
• No focus on general trends
• No reference to the future (see projection on graph)
• Conclusion tries to explain information rathen than summarise it
• Poor linking of ideas (only done by time markers)
• Limited range of gramar and vocabulary

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The graph shows changes in the birth and death rates in New Zealand
since 1901, and forecasts trends up until 2101.
Between 1901 and the present day, the birth rate has been consistently
higher than the death rate. It stood at 20,000 at the start of this period
and increased to a peak of 6,000 in 1961. Since then the rate has
fluctuated between 65 and 50 thousand and it is expected to decline
slowely to around 45,000 births by the end of the century.
In contrast, the death rate started below 10,000 and has increased
steadily until the present time. This increase is expected to be more rapid
between 2021 and 2051 when the rate will probably level off at around
60,000, before dropping slightly in 2101.
Overall, these opposing trends mean that the death rate will probably
overtake the birth rate around 2041 and the large gap between the two
levels will be reversed in the later part of this century.

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This is a strong answer which would score a high IELTS band.
Good points:
• Fulfils criteria for length
• Introduction is paraphrased
• Main sets of data are compared and contrasted
• Clear focus on the different trends
• Important features of the graph, (e.g.cross-over point) included
• Information summarised in conclusion
• Well organised information
• Range of linkers and referencing expressions
• Good range of vocabulary and structures, used accurately

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The line graph shows the percentage of houses using two different types of heating fuel
according to the year of construction of the houses. The information shows that electricity
is more commonly used in older homes, while newer homes use natural gas at a similar
rate as electricity.
Many more older homes are heated with electricity than with natural gas. Sixty percent of
homes built before 1950, as well as an equal percentage of homes builts between 1950
and 1969, use electricity for heat. The percentage falls significantly, to around 45%, for
homes built between 1970 and 1989. This figure stays the same for houses built between
1990 and 2010.
Many fewer older homes are heated with natural gas. Just 15% of homes built before
1950 are heated with this type of fuel. The figure rises steadily, until we see that 45% of
homes built between 1990 and 2010 use natural gas for heat.
While electricity is the most common source of heat in older. In older homes, equal
percentages of the newest homes shown on the graph use electricity and natural gas.

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The owner of a restaurant wants to find out more about where his
patrons are coming from. One day he decided to gather data about the
distance (in miles) that people commuted to get to his restaurant.
People reported the following distances traveled:
14,6,3,2,4,15,11,8,1,7,2,1,3,4,10,22,20
He wants to create a graph that helps him understand the spread of
distances (and the median distance) that people travel. What kind of a
graph should he create?

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Time Plots
Sometimes data are collected in order to observe changes in a variable over time. For
example, sales for a department store may be collected monthly or yearly. A time plot
of a variable plot each observation corresponding to the time at which it was
measured.
A Dow Jones Timeplot from the Wall Street Journal shows how the stock market
changes over time
Although time plots are commonly used to compare frequencies, as in the example above,
they can be used to compare any numerical data as the data change over time, such as
temperaturas, dollar amounts, percents, heights, and weights.

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NUMERICAL METHODS FOR DESCRIBING DATA
Measures of Central Tendency
Measures of central tendency indicate the “center” of the
data along the number line and are usually reported as
values that represents the data. There are three common
measures of central tendency:
• The mean: The arithmetic mean-usually called the
average.
• The median
• The mode

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A list of numbers, or numerical data, can be described by various
statistical measures. One of the most common of these measures is
the average, or (arithmetic) mean, which locates a type of “center”
for the data. The average of n numbers is defined as the sum of the n
numbers divided by n. For example, the average of 6, 4, 7, 10, and 4 is
6 + 4 + 7 + 10 + 4
5
=
31
5
= 6.2
The median is another type of center for a list of numbers. To
calculate the median of n numbers, first order the numbers from
least to greatest; if n is odd, the median is defined as the middle
number, whereas if n is even, the median is defined as the
average of the two middle numbers. In the example above, the
numbers, in order, are 4,4,6,7,10, and the median is 6, the
middle number.
DESCRIPTIVE STATISTICS

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Meausures of Position
QUARTILES
The quartiles must divide the numbers into four groups with the same
amount of numbers in each group, in this case groups of three.
To order a set of numbers into quartiles, we first of all have to put the
numbers in order from the lowest to the highest.
The median splits the numbers into two equal parts and is the second
Quartile, Q2
To calculate what the other two quartiles, Q1 and Q3, are, you
calculate the median of the upper and lower halves.
The median of the lower half is called Q1.
The median of the upper half is called Q3.

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The quartiles must divide the numbers into four groups with the same
amount of numbers in each group, in this case groups of three.
Q2 =M divides the data into two equal parts-trthe lesser numbers and the
greater numbers-and then Q1 is the median of the lesser numbers and Q3 is
the median of the greater numbers.

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Measures of Dispersion
Measures of dispersión indicate the degree of “spread” of the
data. The most common statistics used as measures of
dispersión are the range, the interquartile range, and the
standard deviation. These stadistics measure the spread of the
data in different ways.
The range of the numbers in group of data is the difference
between the greatest number G in the data and the least
number L in the data, that is, G-L. For example, given the list 11,
10, 5, 13, 21, the range of the numbers is 21 – 5 =16.

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A measure of dispersión that is not affected by outliers is the interquartile range.
It is defined as the difference between the third quartile and the first quartile,
that is, Q3 – Q1. Thus, the interquartile range measures the spread of the middle
half of the data.
Example
1)Boxplots or box and whisker plots as a summary

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The data below represents the number of essays that students in Mr. Ji’s class
wrote
2,3,5,5,6,7,8,8,11
Which box plot correctly summarizes the data?

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The data below represents the number of pages each student in Ashwin’s class
read during Reading time.
16,16,16,20,21,21,23,25,26,26,28,28
Which box plot correctly summarizes the data?

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Unlike the range and the interquartile range, the standard deviation is a measure
of spread that depends on each number in the list. Using the mean as the center
of the data, the standar deviation takes into account how much each value
differs from the mean and then takes a type of average of these differences. As a
result, the more the data are spread away from the mean, the greater the
estándar deviation; and the more the data are clustered around the mean, the
lesser the estándar deviation.
The standard deviation of a group of n numerical data is computed by (1)
calculating the mean of the n values. (2) finding the difference between the
mean and each n values. (3) squaring each of the differences, (4) finding the
average of the n squared differences, and (5) taking the nonnegative square root
of the average squared difference.

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For the five data 0, 7, 8, 10 and 10, the standard deviation can be computed as
follows. First, the mean of the data is 7, and the squared differences from the mean
are
(7-0)2, (7-7)2, (7-8)2, (7-10)2, (7-10)2 or 49, 0, 1, 9, 9. The average of the five squared
differences is 68/5, or 13.6, and the positive square root of 13.6 is approximately
3.7.
The table shows the numbers of packages shipped daily by each of five
companies during a 4 day period. The standard deviation of the numbers of
packages shipped daily during the period was greatest for which of the five
companies?
A)A B)B C)C D)D E)E

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Note on terminology: the term “standard deviation” defined above is
slightly different from another measure of dispersión, the sample standard
deviation. The latter term is qualified with the word “sample” and is
computed by dividing the sum of the squared differences by n-1 instead of
n. The sample standard deviation is only slightly different from the
standard deviation but it is preffered for technical reasons for a sample
data that is taken from a larger population of data. Sometimes the
standard deviation is called the population standard deviation to help
distinguish it from the sample standard deviation.

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Example: Six hundred applicants for several post office Jobs were rated on a scale from
1 to 50 points. The ratings had a mean of 32.5 points and a standard deviation of 7.1
points. How many standard deviations above or below the mean is a rating of 48
points? A rating of 30 points? A rating of 20 points?
Let d be the standard deviation, so d = 7.1 points. Note that 1 standard deviation
above the mean is
32.5 + 7.1 = 39.6
And 2 standard deviations above the mean is
39.6 + 7.1 = 46.7
So 48 is a Little more than 2 standard deviations above the mean. Since 48 is
actually 15.5 points above the mean, the number of standard deviations that 48
is above the mean is 15.5/7.1 ≈ 2.2. Thus, to answer the question, we first
found the difference from the mean and then we divided by the standard
deviation. The number of standard deviations that a rating of 30 is away from
the mean is
30 − 32.5
7.1
=
−2.5
7.1
= −0.4

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Where the negative sign indicates that the rating is 0.4 standard deviation below
the mean.
The number of standard deviations that a rating of 20 is away from the mean is
20 − 32.5
7.1
=
−12.5
7.1
= −1.8
Where the negative sign indicates that the rating is 1.8 standard deviations below
the mean.
In the example above, each value can be located with respect to the mean by using
the standard deviation as a ruler. The process of substructing the mean from each
value and then dividing the result by the standard deviation is called
standarization. Standarization is a useful tool because for each data value, it
provides a measure of position relative to the rest of the data independently of the
variable for which the datas was collected and the units of the variable.
Graphic of the standard deviation and the bell curve

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IDENTIFY INDEPENDENT AND DEPENDENT EVENTS
Patrick picks a marble at random. Without putting the first marble back,
he picks a second marble at random.
Are these two events dependent or independent?
Two events are dependent if the outcome of the first event can affect
the outcome of the second event.
The two events are dependent. Patrick does not put the first marble
back, so his first pick affects which marbles are left for his second pick.

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THE BASIC COUNTING PRINCIPLE
When there are m ways to do one thing, and n ways to do another,
then there are mxn ways of doing both.
Example: you have 3 shirts and 4 pants.
That means 3.4=12 different outfits
Example there are 6 flavors of ice-cream, and 3 different cones.
That means 6.3=18 different single scoop ice-creams you could
order.

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You spin the spinner and pick a marble. How many outcomes are
posible?
Make a tree diagram, then count the branches.

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The first event has 4 outcomes: E, F, G, and H.
The second event has 5 outcomes: Green (G), orange (O), White (W),
Yellow (Y), and purple (P).
Make a tree diagram:
Count the number of branches. There are 20 branches, so there are 20
possible outcomes.

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Example: You are buying a new car.
There are 2 body styles:
Five colors available:
There are 3 models:
GL (Standard model),
SS (Sports model with bigger engine)
SL (Luxury model with leather seats)

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Independent or Dependent?
But it only works when all choices are independent of each other.
If one choice affects another choice (i.e.depends on another choice), then
a simple multiplication is not right.

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COUNTING METHODS
Uncertainty is part of the process of making decisions and
predicting outcomes. Uncertainty is adressed with the ideas and
methods of probability theory. Since elementary probability
requieres an understanding of counting methods, we now turn to a
discussion of counting objects in a systematic way before reviewing
probability.

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ADDITION AND MULTIPLICATION PRINCIPLE
ADDITION Key word: OR
Event A can happen instead of B
MULTIPLICATION Key word: AND
Events A and B happen together
Examples:
You walk into a store and there are 5 types of pens and 3 types of
pencils
1. If you want to buy 1 pen and 1 pencil, how many ways are possible?
2. If you want to buy only 1 pencil or 1 pen, how many ways are
possible?

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SPACES METHOD
1. Draw spaces equal to the number of elements to be taken at
the same time.
2. Write the number of options for each space.
3. Multiply
Factorial
𝑛! = 1𝑥2𝑥3𝑥 … 𝑥 𝑛 − 1 𝑥𝑛

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ARRANGEMENTS
In how many ways can n elements be arranged?
1. When all n elements are different: Arrangements = n!
2. When some elements are repeated:
Arrangements =
𝑛!
𝑟𝑒𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛𝑠!
Examples:
1. In how many ways can you arrange the letters ABCD?
2. In how many ways can you arrange the letters AABBC?

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CODES
1. Look for key phrases like “different digits” or “the elements can be
repeated” to know if the number of options decrease in each space. If
it’s not explicit, asume that they can be repeated.
2. Always start with the spaces that have restrictions.
Examples:
1. How many 4-digit codes can be created using the digits 3 to 8?
2. How many 6-digit odd integers can be formed using all of the digits:
0,2,4,5,7, and 8?

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SUBGROUPS
Order matters: You want all possible combinations. AB and BA are different.
Only multiply.
Order doesn’t matter: You want only one combination. AB and BA are the
same. Multiply and divide by n! (n is the number of spaces)
Subgroups of people:
When doing subgroups of people, the order doesn’t matter by default. The
exception is when the persons are differentiated by position or role.
Examples:
In a room there are 6 men and 5 women
1. How many pairs of 1 man and 1 woman can be formed?
2. How many groups of 4 men and 3 women can be formed?
3. How many 3 person committees can be formed in which one person is
president, another is secretary and the other is treasurer?

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SPACES METHOD AND COMBINATORICS FORMULA
Example: making a subgroup of 4 elements out of 9 elements
9𝑥8𝑥7𝑥6
4!
=
9𝑥8𝑥7𝑥6
4!
𝑥
5!
5!
=
9!
4! 𝑥5!
= 𝐶4
9

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Sets and Lists
A set is a collection of objects that have some property, whether it is the
collection of all positive integers, all points in a circular región, or all
students in a school that have studied French. The objects of a set are
called members or elements. Some sets are finite, which means that their
members can be completely counted. Finite sets can, in principle, have all
their members listed, using curly brackets, such as the set of even digits
0,2,4,6,8 . Sets that are not finite are called infinite sets, such as the set of
all integers. A set that has no members is called an empty set and its
symbol is ɸ.
If A and B are sets and all the elements of A are also members of B, then A
is a subset of B. Also by convention ɸ is a subset of every set.

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If all the alements of a set S are also elements of a set T, then S is a subset of T; for
example, 𝑆 = −5,0,1 is a subset of T𝑇 = −5,0,1,4,10 .
SETS
In mathematics a set is a collection of numbers or other objects.
The order in which the elements are listed in a set ……….. matter; ;
thus −5,0,1 = 0,1, −5

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The diagram illustrates a fact about any two finite sets S and T; the number of
elements in their unión equals the sum of their individual numbers of elements
minus the number of elements in their intersection (because the latter are
counted twice in the sum); more concisely,
𝑆 ∪ 𝑇 = 𝑆 + 𝑇 − 𝑆 ∩ 𝑇 .
This counting method is called the general addition rule for two sets. As a
special case, if S and T are disjoint, then
𝑆 ∪ 𝑇 = 𝑆 + 𝑇
Since 𝑆 ∩ 𝑇 = ɸ

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Example 1
Each of 25 people is enrolled in history, mathematics, or both. If 20 are
enrolled in history and 18 are enrolled in mathematics, how many are
enrolled in both history and mathematics?

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Last year 26 members of a certain club traveled to England, 26 members
traveled to France, and 32 members traveled to Italy. Last year no
members of the club traveled to both England and France, 6 members
traveled to both England and Italy, and 11 members traveled to both
France and Italy. How many members of the club traveled to at least one
of these three countries last year?
Rpta:67
GMAT 2018 Test Guide

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A bicicly trainer studies 50 bicyclists to examine how the finishing time for
bicycle race was related to the amount of physical training in the three months
before the race. To measure the amount of training, the trainer developed a
training index, measured in “units” and base don the intensity of each bicyclist’s
training. The data and the trend of the data, represented by a line, are
displayed in the scatterplot above.
(a) How many of the 50 bicyclists had both a training index less than 50 units
and a finishing time less than 4.5 hours?
(b) What percent of the 10 fastest bicyclists in the race had a training index less
than 90 units?
Source: GRE Test Guide

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In a survey of 250 European travelers, 93 have traveled to Africa, 155 have
traveled to Asia, and of these two groups, 70 have traveled to both continents, as
illustrated in the Venn diagram above.
(a) How many of the travelers surveyed have traveled to Africa but not to Asia?
(b) How many of the travelers surveyed have traveled to at least one of the two
continents of Africa and Asia?
(c) How many of the travelers surveyed have traveled neither to Africa nor to
Asia?

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Example 2:
In a certain production lot, 40 percent of the toys are red and the remaining
toys are Green. Half of the toys are small and half are large. If 10 percent of the
toys are red and small, and 40 toys are green and large, how many of the toys
are red and large?
GMAT Test guide 2018

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GMAT 2018 Test Guide
In a certain production lot, 40 percent of the toys are red and the
remaining toys are green. Half of the toys are small and half are large. If
10 percent of the toys are red and small, and 40 toys are green and large,
how many of the toys are red and large.

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PROBABILITY
Theoretical vs Experimental Probability
When asked about the probability of a coin landing on heads,
you would probably answer that the chance is ½ or 50%.
Tails and heads
Imagine that you toss that same coin 20 times. How many times would
you expect it tol and on heads. You might say, 50% of the time, or half
of the 20 times. So you would expect it tol and on heads 10 times. This
is the theoretical probability.

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The theoretical probability is what you expect to happen, but it isn’t always
what actually happens. The table below shows the results after Sunil tosed
the coin 20 times.
This shows the experimental probability. You can think of it as the
probability determined from the results of an experiment. It is what
actually happens instead of what you were expecting to happen.
The experimental probability of landing on heads is
13
20
=
65
100
= 65%.
It actually landed on heads more times than we expected.
Outcomes Frequency
Heads 13
Tails 7
Total 20

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Now, Sunil continues to toss the same coin for 50 total tosses. The results are
shown below.
Now the experimental probability of landing on head is
26
50
=
52
100
= 52%.
The probability is still slightly higher than expected, but as more trials
were conducted, the experimental probability became closer to the
theoretical probability.
Outcomes Frequency
Heads 26
Tails 24
Total 50

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Use the table below to determine the probability of each number on a number
cube.
Rolling a 3 (use the table)
22
100
= 0.22 = 22%
What is the theoretical probability of rolling a 3?
1
6
= 0.166666 … ≈ 17%
Rolling a number less than 3 (use the table)
(Rolling a 1 or 2)
36
100
= 0.36 = 36%
Rolling a 3 or a 5 (use the table)
40
100
= 0.40 = 40%
Outcome Frequency
1 16
2 20
3 22
4 10
5 18
6 14
Total 100

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GEOMETRIC PROBABILITY
Geometric probability is a tool to deal with the problem of infinite
outcomes by measuring the number of outcomes geometrically, in terms
of length, area, or volumen. In basic probability, we usually encounter
problems that are “discrete”. However, some of the most interesting
problems involve “continuous” variables, for instance the arrival time of
your bus. Random events that take place in continuous sample space may
invoke geometric imagery for at least two reasons; due to the nature of
the problema or due to the nature of the solution.
Some problems such as:
• Buffon’s needle
• Birds on a wire
• Bertrand’s Paradox
• Stick Broken into three pieces

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DISTRIBUTIONS OF DATA, RANDOM VARIABLES, AND PROBABILITY DISTRIBUTIONS
Distributions of data
Relative frequency distribution or histogram are a common way to show
how numerical data are distributed. In a histogram, the áreas of the bars
indicate where the data are concentrated.

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Random variables
When analyzing data, it is common to choose a value of the data at
random and consider that choice as a random experiment.
A random variable, usually written X, is a variable whose posible
values are numerical outcomes of a random phenomenon.
There are two types of random variables, discrete and continuous.

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Discrete Random Variables
A discrete random variable is one which may take on only a countable number of
distinct values such as 0,1,2,3,4,........ Discrete random variables are usually (but not
necessarily) counts. If a random variable can take only a finite number of distinct
values, then it must be discrete. Examples of discrete random variables include the
number of children in a family, the Friday night attendance at a cinema, the number
of patients in a doctor's surgery, the number of defective light bulbs in a box of
ten.The probability distribution of a discrete random variable is a list of probabilities
associated with each of its possible values. It is also sometimes called the probability
function or the probability mass function.

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Suppose a variable X can take the values 1, 2, 3, or 4.
The probabilities associated with each outcome are described by the
following table:
The probability that X is equal to 2 or 3 is the sum of the two
probabilities: P(X = 2 or X = 3) = P(X = 2) + P(X = 3) = 0.3 + 0.4 = 0.7.
Similarly, the probability that X is greater than 1 is equal to 1 - P(X = 1) = 1
- 0.1 = 0.9, by the complement rule.
Outcome 1 2 3 4
Probability 0.1 0.3 0.4 0.2

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This distribution may also be described by the probability
histogram shown to the right:

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Continuous Random Variables
A continuous random variable is one which takes an infinite number of
possible values. Continuous random variables are usually measurements.
Examples include height, weight, the amount of sugar in an orange, the time
required to run a mile.
A continuous random variable is not defined at specific values. Instead, it is
defined over an interval of values, and is represented by the area under a
curve (in advanced mathematics, this is known as an ……….). The
probability of observing any single value is equal to 0, since the number of
values which may be assumed by the random variable is infinite.
Suppose a random variable X may take all values over an interval of real
numbers. Then the probability that X is in the set of outcomes A, P(A), is
defined to be the area above A and under a curve. The curve, which
represents a function p(x), must satisfy the following:
1: The curve has no negative values (p(x) > 0 for all x)
2: The total area under the curve is equal to 1.
A curve meeting these requirements is known as a density curve
.

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The Normal Distribution

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7.51 8.22 7.86 8.36
8.09 7.83 8.30 8.01
7.73 8.25 7.96 8.53
A vending machine is designed to dispense 8 ounces of coffee into a cup. After a test
that recorded the number of ounces of coffee in each of 1,000 cups dispensed by the
vending machine, the 12 listed amounts, in ounces, were selected from the data. If the
1,000 recorded amounts have a mean of 8.1 ounces and a standard deviation of 0.3
ounces, how many of the 12 listed amounts are within 1.5 standard deviations of the
mean?
A)Four B)Six C)Nine D)Ten E)Eleven
Source: Dominate the GMAT

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The figure above shows a normal distribution with mean m and standar deviation d,
including approximate percents of the distribution corresponding to the six regions
shown.
Suppose the heights of a population of 3,000 adult penguins are approximately
normally distributed with a mean of 65 centimeters and standard deviation of 5
centimeters.
(a) Approximately how many of the adult penguins are between 65 centimeters and
75 centimeters and 75 centimeters tall?
(b) If an adult penguin is chosen at random from the population, approximately
what is the probability that the penguin’s height will be less than 60 centimeters?
Give your answer to the nearest 0.05

81
[1] Soft CPU Cores para FPGA’s
[Online] Available:
http://www.1-core.com/library/digital/soft-cpu-cores/
[2] MITx 6.002x: Circuits & Electronics
[Online] Available:
https://6002x.mitx.mit.edu/
[3] Definición de FPGA dada por Xilinx
[Online] Available:
http://www.xilinx.com/products/silicon-devices/fpga/index.htm
[4] Sistema de Visualización de Imágenes a 8 Colores empleando la tarjeta de desarrollo Digilent Spartan 3
[Online] Available:
http://jupiter.utm.mx/~tesis_dig/11608.pdf
[5] Circuit Design with VHDL
Volnei A. Pedroni
[6] A Brief History of VHDL
[Online] Available:
https://www.doulos.com/knowhow/vhdl_designers_guide/a_brief_history_of_vhdl/
[7] Rapid Prototyping Of Digital Systems Sopc Edition
-James O. Hamblen
-Tyson S. Hall
-Michael D. Furman
[8] Dominate the GMAT
Standard Deviation Example - GMAT Statistics - GMAT Quant
https://www.youtube.com/watch?v=xBKZ-TsGkKw 23/10/2018 Taller 2018

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