Calculo Diferencial - Module 1
"Probability and Statistics"(2)
1 Alejandro Baruch Saucedo Esparza - A01400284
1.1 Lic. Saúl Garcia
2 Probability and Statistics
2.1 Statistics: The science of collecting, describe, analyze, and interpret data.
2.1.1 Descriptive Statistics
220.127.116.11 Collection, organization,
overview and presentation of
sample data. Main tools:
Tables of Numbers, Graphs
and calculated quantities.
2.1.2 Inferential Statistics
18.104.22.168 Obtaining inferences or
conclusions (via conjectures)
about populations based on
information taken from the
22.214.171.124.1 Population or Universe: The
totality of the elements or
things under consideration
126.96.36.199.1.1 Parameter: Measure of
overview that is
calculated to describe
a characteristic of a
188.8.131.52.2 Sample: The
portion of the
Population that is
selected for the
184.108.40.206.2.1 Statistic: Measure of
overview that is calculated
to describe a characteristic
of a single sample of the
220.127.116.11.3 Random Variable:
characteristics or values
that is subject to variations
due to chance
18.104.22.168.3.1 Qualitative random variables
22.214.171.124.3.1.1 Produce categorical responses
to describe an element of a
126.96.36.199.3.1.2 Quantitative random variables
188.8.131.52.184.108.40.206 Discrete: Numerical
responses arising from a
process of counting
from a process of
2.2 Measurement Scales
220.127.116.11.1 Each value on the
measurement scale has a
18.104.22.168.1 Values on the measurement scale
have an ordered relationship to one
another. That means some values are
larger and some are smaller.
22.214.171.124 Equal Intervals
126.96.36.199.1 Scale units along the scale are
equal to one another. This means
that the difference between 1 and 2
would be equal to the difference
between 19 and 20.
188.8.131.52 Minimum value of zero
184.108.40.206.1 The scale has a true zero
point, below which no
2.2.2 Nominal Scale of Measurement
220.127.116.11 Satisfies the identity property of
measurement. Values assigned to
variables represent a descriptive
category, but have no inherent
numerical value with respect to
Individuals may be classified
as "male" or "female", but
neither value represents
more or less "gender" than
2.2.3 Ordinal Scale of Measurement
18.104.22.168 Scale has the property of
both identity and magnitude.
Each value on the ordinal
scale has a unique meaning,
and it has an ordered
relationship to every other
value on the scale.
22.214.171.124.1 Example:The results of a
horse race, reported as
"win", "place", and "show".
We know the rank order in
which horses finished the
2.2.4 Interval Scale of Measurement
126.96.36.199 Has the properties of identity,
magnitude, and equal intervals. With
an interval scale, you know not only
whether different values are bigger or
smaller, you also know how much
bigger or smaller they are.
188.8.131.52.1 Example: The Fahrenheit scale to measure
temperature. The scale is made up of equal
temperature units, so that the difference
between 40 and 50 degrees Fahrenheit is
equal to the difference between 50 and 60
2.2.5 Ratio Scale of Measurement
184.108.40.206 Satisfies all four of the properties
of measurement: identity,
magnitude, equal intervals, and a
minimum value of zero.
220.127.116.11.1 Example: The weight of an object. Each value
on the weight scale has a unique meaning,
weights can be rank ordered, units along the
weight scale are equal to one another, and the
scale has a minimum value of zero.
2.3 Sampling methods
2.3.1 The way that observations are
selected from a population to be in
the sample for a sample survey.
18.104.22.168 Population parameter. A population
parameter is the true value of a population
22.214.171.124 Sample statistic. A sample statistic is
an estimate, based on sample data, of
a population parameter. Is strongly
affected by the way that sample
observations are chosen; that is by the
2.3.2 Sample survey: Estimate
the value of some
attribute of a population.
2.3.3 • Probability samples.
126.96.36.199 Each population element has a
known (non-zero) chance of being
chosen for the sample.
2.3.4 Non-probability samples
188.8.131.52 We do not know the probability that
each population element will be chosen,
and/or we cannot be sure that each
population element has a non-zero
chance of being chosen.
184.108.40.206.1 Advantages:convenience and cost.
Disadvantages: Do not allow you to
estimate the extent to which sample
statistics are likely to differ from
220.127.116.11 • Voluntary sample
18.104.22.168.1 A voluntary sample is
made up of people who
self-select into the survey.
22.214.171.124 • Convenience sample
126.96.36.199.1 A convenience sample is
made up of people who are
easy to reach.
2.3.5 Probability sampling methods:
They guarantee that the sample
chosen is representative of the
population. This ensures that
the statistical conclusions will be
188.8.131.52 Simple random sampling
184.108.40.206.1 Any sampling method that has the
following properties. o The population
consists of N objects. o The sample
consists of n objects. o If all possible
samples of n objects are equally likely
to occur, the sampling method is called
simple random sampling.
220.127.116.11.1.1 A good example would be the
lottery method. Each of the N
population members is assigned a
unique number. The numbers are
placed in a bowl and thoroughly mixed.
Then, a blind-folded researcher selects
18.104.22.168 Stratified sampling
22.214.171.124.1 The population is divided into groups, based on some
characteristic. Then, within each group, a probability sample
(often a simple random sample) is selected. In stratified
sampling, the groups are called strata.
126.96.36.199.1.1 Ex: a national survey. Divide the population
into groups or strata, based on geography -
north, east, south, and west. Then, within
each stratum, we might randomly select
188.8.131.52 Cluster sampling
184.108.40.206.1 Every member of the population is assigned to one,
and only one, group. Each group is called a cluster. A
sample of clusters is chosen, using a probability
method (often simple random sampling). Only
individuals within sampled clusters are surveyed.
220.127.116.11.1.1 The difference between cluster sampling and
stratified sampling. With stratified sampling, the
sample includes elements from each stratum.
With cluster sampling, in contrast, the sample
includes elements only from sampled clusters.
18.104.22.168 Multistage sampling
22.214.171.124.1 We select a sample by
using combinations of
126.96.36.199.1 We create a list of every member of the population. From the list,
we randomly select the first sample element from the first k
elements on the population list. Thereafter, we select every kth
element on the list.
188.8.131.52.1.1 This method is different from simple random sampling since
every possible sample of n elements is not equally likely.
2.4 Frequency Distribution
2.4.1 Data set with large numbers
184.108.40.206 Grouped Frequency Distribution
220.127.116.11.1.1 lower class limits, upper class limits, class width, class mark
18.104.22.168.2.1 Make sure each data item will fit into one
22.214.171.124.2.2 Try to make all class the same width
126.96.36.199.2.3 Make sure the classes do not overlap
188.8.131.52.2.4 Use from 5 to 12 classes(too few or to many classes can
obscure the tendencies in the data
184.108.40.206 How many times an event happens
2.4.3 Relative Frequency
220.127.116.11 Frequency over total
elements given in percentage
2.4.4 Visual displays of data
18.104.22.168.1 To demonstrate how a
respect to something.
22.214.171.124 Circle Graphs
126.96.36.199.1 Uses a circle to represent
the total of all the categories
and divides circle into
sections which sizes show
the relative magnitudes of
the categories. 360º=100%
188.8.131.52 Bar Graphs
184.108.40.206.1 Frequency distribution of non-numerical
observation bars are not touching one
another and sometimes are arranged
horizontally rather than vertically.
220.127.116.11 Stem and leaf graphs
18.104.22.168.1 Numbers Grouped(min. to solve an exam)
22.214.171.124.1 Aseries of rectangles whose
lengths represent the
frequencies, are placed next to
2.5 Measures of central tendencies
126.96.36.199 Average: Most common measure.
Addition of all items and then diving
the sum by the number of items.
2.5.2 Weightened mean
188.8.131.52 Sum of all products of items
weighting factors, divided by
the sum of all weighting
184.108.40.206 Is not so sensitive to extreme
values. Divides a groups of
numbers into two parts, with
half the numbers below the
median and half above it
220.127.116.11.1 Position of the median
18.104.22.168.1.1 Frequency Distribution
22.214.171.124.2 Steps to find it
126.96.36.199.2.1 1. Rank items(airing them in numerical order from
least to greatest) 2.if number odd, median is the
middle list. 3. Every median is mean of the 2 middle
188.8.131.52.1 value that occurs more often
2.6 Measures of dispersion
184.108.40.206 A straight forward measure of dispersion.
Range=(the greatest value in the set- the
least value in the set.
2.6.2 Standard deviation
220.127.116.11 Based on deviations from
the mean at data value
2.6.3 Coefficient of variation
2.6.4 Chevyshev´s theorem
18.104.22.168 For any set of numbers regardless of how they are
distributed, the fraction of them that lie within K standard
deviations of their mean (where k>1) is at least
22.214.171.124.1 k= numbers of
2.7 Measures of position
126.96.36.199 If approximately n percent of the items in a distribution
re less than the number x, then x is the nth percentile
of the distribution, denoted Pn
2.7.3 Deciles and Quartiles
188.8.131.52 Deciles are the nine values(denoted D1-D9) along the scale that divide
a data set into ten(approximately) equal-sized parts, and quartiles are
the thress values(Q1-Q3)that divide a data set into 4 (approximately)