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Statistics Chapter 1 overview
Citation preview
Terminology
1
Introduction
2
Statistics (as opposed to statistic) is the science
of gathering, organizing, analyzing, and
interpreting numerical and categorical
information.
“Statistics is the art of decision making in the
presence of uncertainty”
CPT
3
Descriptive Statistics involve methods
of organizing, picturing, and
summarizing information from samples
or populations (Chapters 1-3)
Inferential Statistics involve methods of
using information form a sample to draw
conclusions regarding the population
(Chapters 7-11)
4
5
Descriptive Statistics
Probability
Inferential
Statistics
1. Choose a topic and identify the problem to be addressed.
2. Background research –past research including historical descriptive statistics and references.
3. Develop a conjecture or hypothesis. 4. Design an experiment 5. Gather additional information through
experimentation and observation. 6. Analyze the results and interpret the results –if
hypothesis is rejected, then repeat Step 3 thru 6. 7. Formulate conclusions and draw inferences
6
A population, N, is a group of individual persons, objects, or items that one wishes to better understand certain characteristics about and from which samples are taken for statistical measurement.
The population data is the complete collection of information from all of the individuals or subjects of interest in a given study.
7
A census is a survey of every individual in the
population and the information gathered is called
the population data.
Recall: the population data is the complete
collection of information from all of the
individuals or subjects of interest in a given
study.
8
A sample, n, is a partial collection of
information for only some of the individuals or
subjects of interest in a given study
The sample size, n, of a sample is the number of
observations that constitute the sample. Whereas
the size of a population is denoted by the capital
letter N, the size of the sample is denoted by the
lowercase letter n.
9
Descriptive Statistics involve methods of
organizing and summarizing information (data)
and presenting it numerically or visually
(graphically).
Stem-and-leaf, Frequency Tables, and
Contingency Tables
Bar Charts, Pie Charts, and Histograms
Scatter Plots
Among others…
10
The distribution of the data is a list of all the
values recorded in a sample; that is, the observed
outcomes and their frequency.
Distributions can be given in tabular form as a
frequency or contingency table or illustrated
graphically in the form of a bar chart or
histogram.
11
Characteristics of
Distribution
The mean, median, and mode
are central tendencies
Uniform distribution is
symmetric, not skewed
The range of information
between minimum and
maximum
Location
Shape
Spread
12
Location within a distribution is a specific value
within the domain of the data such as the
extremes: minimum and maximum, central
tendencies, etc.
13
Outliers are data within a distribution of the
data, but outside the overall pattern (cluster) of
the graph; that is, extreme values that can distort
the interpretation of the data by creating
misleading statistics.
14
Common
Locations
Minimum and Maximum
Mean, Median, and Mode (among other weighted or trimmed means)
A value, xp%, such that p% of the observed values are to the left
p%=0%, 25%, 50%, 75%, 100%
Extremes
Central
Tendencies
Percentiles
Quartile
15
Shape of a distribution describes the symmetry
or lack thereof (skewness).
Data that is symmetric exhibits balance and self-
similarity whereas skewness is a measure of the
asymmetry.
16
Common Shapes
Equal frequencies – no mode
The mean equals the median equals the mode - looks like a
The above are symmetric
Left: mean < median Right: mean > median
Uniform
Bell
Shaped
Symmetric
Skewed
17
Spread of a distribution is a measure which
indicates how the data values are distributed, a
measure of the dispersion or variability within a
group of values. Some appropriate measures
include:
Range (from minimum to maximum)
Variance (mean square error)
Deviation (square root of variance)
Where error=observed value – expected value
18
Common Measures
Minimum and Maximum
Mean, Median, and Mode (among other weighted or trimmed means)
Range, Variance, and Standard Deviation
Count, Relative Frequency, Rate
Extremes
Central
Tendencies
Deviations
Frequency &
Proportions
19
An extreme is a characteristic farthest removed
from the ordinary; common extremes are the
minimum, which is the least observed value in a
sample, and the maximum, the greatest observed
value in a sample.
20
Measure of a characteristic which clusters around
a central value, using the data to estimate the
central tendency, called averages; there are
three common measures: the mode, median, and
mean.
21
Common Central
Tendencies
Most frequently observed value A value such that 50% of the
observed values fall to the left Sum of all data divided by
number of data points (equal weights)
Average such that the weights are not equal
Average such that some of the weights are zero
Mode Median Mean Weighted Mean Trimmed Mean
22
Common
Deviations
Maximum minus minimum
Q3 – Q1
Expected value of the
differences (error)-squared
Square root of the variance
Range IQR (Interquartile Range)
Variance Standard Deviation
23
Common
Frequency &
Proportions
1,2,3,…
The number of times a given value
occurs (a count)
The ratio of the frequency of one value to the sample size
Part in ratio to the whole – the relative frequency
Count
Frequency
Relative
Frequency
Rate
24
A statistical measure is sensitive if the
computed value changes readily even if a
single observed data value is different. Also, a
statistical method is sensitive if the decision
changes radically based on the assumptions
we made to develop it.
A statistical method is robust if the decision
is not strongly dependent on the assumptions.
Not formally defined in the text
25
The degree of confidence represents the
proportion of times the statistical methodology
used captures the true state of nature.
This value will be denoted (1-)%, where is
the level of significance.
26
An event is considered statistically significant if
its occurrence is unlikely to happen by chance.
This value will be denoted by %.
27
Inferential Statistics involves methods of
analyzing and interpreting descriptive statistics
to draw conclusions regarding a particular
characteristic in the population with a certain
degree of assurance based on a preset level of
significance and specified assumptions.
28
Hypothesis Testing is the use of a statistical
method in arguing for or against a hypothesized
value based on observed information and using
this information to make a decision regarding an
initial hypothesis and an alternative hypothesis.
Not formally defined in chapter 1
29
An experiment is the method (procedure) that
we follow to obtain data or information.
An experiment design is the art of planning
and executing experiments designed to gather
information (data) from the population, N, in
such a way as to ensure the sample, n, is
representative of the population, N.
30
Individuals are the people, places, or things
included in a study and for which information is
gathered. In medical research studies,
individuals are referred as the subjects in the
study.
31
A variable is a distinct characteristic of an individual to be observed and measured. These observed data can be qualitative or quantitative. A qualitative (categorical) variable is a variable
that describes the individual by placing the individual into a category or group
A quantitative (numerical) variable is a variable that takes on a real value or numerical measurement for which sums, differences, and ratios have meaning.
32
Regression is a statistical procedure used to
estimate the relationship among variables,
specifically between the primary (response)
variable of interest and all other variables.
33
In a study, the response variable is the primary
variable of interest; that is, the objective in the
given study.
This variable is also referred to as the dependent
variable (although the relationship of
dependence or cause-and-effect is yet to be
determined).
34
The explanatory variables are the extraneous variables that have been measured but are not the primary variable of interest; they are used to understand the behavior of the response (primary) variable. This variable is also referred to as the independent variable (although the relationship of dependent/independent has not yet been established).
35
Lurking variable(s) are the unknown variables
that have not been measured; however, they do
contribute to the response (primary) variable and
are not included as an explanatory variable.
36
Correlation is a measure of association between
a response variable and an explanatory variable.
Correlation measures the strength and direction
of a simple linear relationship, that is, a straight
line.
37
Causation is more than a measure of association
between a response variable and an explanatory
variable. It also implies direct cause or
dependence.
Correlation between two events may be a
common response to a lurking variable.
38
Confounding variables are the variables that
have been measured and are significantly
contributing variables; however, their
independent contributions to the subject response
are indistinguishable and are not deemed
significantly contributing in the larger model.
39
Discrete/Continuous
40
An instrument is any means by which
information is gathered or measured such as an
exam, survey, or other rulers such as a barometer,
thermometer, etc.
41
A parameter is a numerical measure that describes the outlined characteristic of the population such as central tendencies (mean, median, mode, and proportion), spread (range, variance, and standard deviation), and shape (symmetric and skewed). In general, when a specific parameter is not specified, the lowercase Greek letter Theta () is used to denote a population parameter.
42
Common
Parameters
Mean
Variance
Standard Deviation
Proportions
Correlation
43
p
2
A sample survey is a survey of only some of the
individuals in the population and the information
gathered is called the sample data.
The number of individuals included in the
sample survey is called the sample size, n.
The sample data is a subset of the population
data often denoted: x1, x2,…. xn.
44
A statistic is a numerical measure that yields an estimate of a population parameter. That is, a numerical measure that uses the data from the sample to estimate the outlined characteristic of the population. As opposed to Statistics - the study of how to gather, organize, analyze and interpret information.
45
POPULATION IS TO SAMPLE AS CENSUS IS TO SAMPLE SURVEY
POPULATION IS TO SAMPLE AS PARAMETER IS TO STATISTIC
46
Measure involves any standard of comparison, estimation, or judgment; property of an individual given a numerical value; a quantity, a count, a degree, a rate, or a proportion. In terms of data collections, the measured values are referred to as the outcomes or observed values. Two types of measure are discrete and continuous.
47
A discrete measure is such that the set of
possible observed outcomes are separate,
distinct, and finite such as a count.
Discrete measures are such that the outcomes can
be enumerated: one, two, three, etc.
48
Examples of
Discrete Measures
Number of children in a family tree –
depending on the number of generations
included in the tree, there can be either 1,
2, 3, …, but not 1.5 – nothing between 1
and 2, or 2 and 3, etc.
Count of whole beans – depending on the
number of pods included, there can be
either 1, 2, 3, …, but not 1.2 since the
count is restricted to the whole number
Frequency of blue-eyed men and green-
eyed women.
Number
Count
Frequency
49
A continuous measure is such that the set of
possible observed outcomes are infinite and
uncountable.
Continuous measures are dense; that is, between
any two values (outcomes) there exist another
value (outcome) such as a mean or rate.
50
Examples of
Continuous
Measures
Length of a road – it can measure 1 mile or 2
miles, and between these possible measures
exists 1.5 miles, 1.24 miles; in fact, between
any two values there exist other possible values
Height of a man – a man can be 5 feet tall or 6
feet tall, and between these potential values
exist 5.5 feet, 5.14 feet, etc. While we might
not have an instrument precise enough to
measure the 1/100th of a foot, this measure
exists
Age of a woman – between this moment and
the next, there is a continuous existence.
Between 1 yrs and 2 yrs, 1.8 yrs exist, etc.
Length
Height
Age
51
Samplings
52
53
Validity refers to the degree of accuracy to which a study reflects the specific concept or characteristic that the analyst or researcher is attempting to measure. Internal validity is the degree to which one can draw valid conclusions about the causal effect between variables. External validity is the degree to which one can extend the findings that are relevant to subjects and settings outside those included in the experimental design.
54
For example, when evaluating a class of 180 students from a single mass lecture of STA 2023, can this information be used to evaluate all students taking STA 2023 given there is more than one section taught by different instructors? Internal Validity – drawing conclusions about this specific subjects inside the study. External Validity – the ability to extend conclusions to subjects outside of the study.
55
56
57
Bias is a consistent deviation of the statistics to
one side of the parameter.
LOW BIAS HIGH BIAS
58
For example, when weighing out coffee to be
ground and brewed at a coffee shop, the
employee forgets to zero-out the scale with the
cup used to measure the coffee. This leads to the
coffee measured in the cup to be off by the
weight of the cup.
Solution: add the weight of the cup in coffee to
each cup.
59
60
61
Variability measures the degree of dispersion within a given data set. Some common measures of dispersion include range, mean (average) deviation, standard deviation, variance, inter-quartile range, and mean difference. Variability can appear as gaps in the data when illustrated graphically.
62
Reliable refers to the accuracy and precision of
the actual measuring instrument or procedure.
A reliable measure is a (precise) measurement
such that the random error is small.
63
Valid (Accurate)
Reliable (Precise)
We like samples to
represent the population
and the measures taken to
represent the parameters
estimated. These statistics
need to be a valid measure,
accurately estimating the
parameter with low bias as
well as be reliable,
measured with such
precision as to have low
variability when estimating
the parameter using
statistics.
64
ACCURACY (VALID MEASURE)
HITS THE TARGET’S “BULL’S EYE”
PRECISION (RELIABLE MEASURE) HITS THE SAME LOCATION REPEATEDLY
65
ACCURATE INACCURATE
66
PRECISE IMPRECISE
67
PRECISE IMPRECISE
68
Nominal, Ordinal, Interval, & Ratio
69
Common
Levels of Measure
Data that consist of names, labels, or categories
Data that can be arranged in order; however, differences between data values cannot be determined or are meaningless
Data that can be ordered and differences have meaning, but ratios do not (equal distances, but no fixed zero)
Ordinal and interval, but ratios have meaning (equal distances and fixed zero)
Nominal
Ordinal
Interval
Ratio
70
A nominal measure is one that measures a
characteristic of an individual by name only;
information in the form of categorical data where
the order of the categories is not relevant.
Names only – no calculations can be preformed.
71
Examples of
Nominal Measures
Can be made ordinal if considered alphabetically, but
otherwise, this is a name only
There are relations among the digits that make up such
numbers, but there is not a true ordering, difference, or
ratio
While these codes can be “ordered” numerically, the
order is arbitrary and therefore not meaningful – the zip
code 33617 is not “less than” the zip code 33620 – the
only difference is geographical
Male/Female: these are clearly labels for which there is
no order other than “alphabetically”; however, it is
meaningless to argue “less than” or “greater than” in
general
Surnames
SSN
Zip Code
Gender
72
An ordinal measure is one that measures a
characteristic of an individual by the rank order
(1st, 2nd, 3rd, etc.) of the entities measured or by
implied ordering such as worst, bad, good, great.
Ordering the measured outcomes.
73
A simple ranking imposes an order on the
measured characteristic of an individual and the
set of natural numbers by defining a relationship
that establishes the position within a sequence of
outcomes "ranked higher than," "ranked lower
than," or "ranked equal to.“
Imposing an ordinal scale.
74
A Likert scale establishes the hierarchy within a
sequence of outcomes.
For example, “how attractive is a person on a
scale from 1 to 10,” 1 meaning not very
attractive to a 10 which represents perfect
attraction.
75
Examples of
Ordinal Measures
What is the best-selling flavor of ice cream?
A five-point scale by which to evaluate an
instructor: poor, unsatisfactory, satisfactory,
good, great
Due to inconsistencies found in “sizes”
between designers – a size “0” is smaller than a
size “2,” which is smaller than a “4,” but this does
not mean the difference between a “2” and a “4” is
the same as the difference between a “0” and a “2.”
Furthermore, a “4” is not twice as large as a “2”;
this ratio has no meaning.
Ranking
Likert
Scale
Dress Size
&
Shoe Size
76
An interval measure is one that measures a
characteristic of an individual where differences
between measures have meaning; that is, the
distance between two adjacent units is the same
but there is not a meaning zero point. An interval
measure is such that sums and averages have
meaning; however, ratios do not have meaning.
Sums (differences) but not ratios.
77
Examples of
Interval Measures
If your watch reads 12:05 and mine reads 12:07, then my watch reads a later time than yours; hence the measure is at least ordinal. However there is a 2-minute difference, therefore this measure is interval. It is not ratio since 12:07 in ratio to 12:05 has no meaning.
If the daytime temperature is 50°F in New York and 100°F in Miami, then it is 50°F hotter in Miami than it is in New York. While the ratio of 100°F to 50°F is 2, this measure has no meaning and is therefore an invalid measure. You can not say 100°F is “twice as hot” as 50°F.
Some may argue that degrees Kelvin, which has an “absolute zero,” is ratio; however, in general, temperature is interval.
Time of Day
Temperature
78
A ratio measure is one that measures a
characteristic of an individual where not only do
differences between measures have meaning, but
ratios also have meaning. That is, a measure in
which any two adjoining values are the same
distance apart and there is a true zero point. Ratio
measures have fixed zeros; that is, an interval
measure with a true zero.
79
Examples of Ratio
Measures
At 2:00, the measure is 2 hours past noon and at 4:00, the measure is 4 hours past noon, 4 hours is greater than 2 hours; hence at least ordinal. The difference between 4 hours and 2 hours is 2 hours, which has meaning; hence at least interval. Moreover, the ratio of 4 hours to 2 hours is 2, that is 4 hours is twice as much time as 2 hours; thus this measure is ratio.
If you are 6 feet tall and your child is 3 feet tall, then you are taller than your child (ordinal), you are 3 feet taller than your child (interval), and you are twice as tall as your child (ratio). Therefore, this measure is Ratio.
If you are 36 years old and your child is 12 years old, then you are older than your child (ordinal), you are 24 years older than your child (interval), and you are three times as old as your child (ratio). Therefore, this measure is Ratio.
Time Past
Noon
Height
Age
80
Changing Level of
Measure
What is your yearly salary? (a continuous scale)
Interval–what is your income bracket? (a discrete scale) 0-9,999, 10,000-19,999, 20,000-29,999, 30,000-39,999, 40,000-49,999, 50,000-59,999, etc.?
Where the difference between intervals is 10,000
Ordinal–what is your tax bracket? (a discrete scale) 0-9,999, 10,000-39,999, 40,000-59,999, 60,000or more?
Where difference are not well-defined
Nominal–in what currency are you paid?
Dollar, Yen, Euro, etc. (ordinal if you consider exchange rates)
Ratio
Interval
Ordinal
Nominal
81
SRS, Systematic, Cluster, Stratified, etc.
82
Samples
Simple Random
Samples
Systematic
Cluster Samples
Stratified
Samples
Convenience
Samples
83
A random sample is a sample of size n taken
from a population of size N in such a way that
each individual observed has an equally likely
chance of being selected.
84
A simple random sample (SRS) is such that
(1) each individual has an equally likely chance
of being selected as well as
(2) all groups of size n have an equally likely
chance of being selected.
85
Common Sampling
Schemes
Using a system to select
Using clusters of individuals
that are pre-existing
Using “clusters” of individuals
selected by a specified strata
Using individuals who are
conveniently surveyed
More than one stage of
sampling done in succession
Systematic
Cluster
Stratified
Convenience
(Volunteer
Response)
Multi-stage
86
Systematic sampling is a sample such that every
kth individual or item is measured.
Every 3rd: 1, not 2, not 3, 4, not 5, not 6, 7
Every 5th: 1, 6, 11,16,… or 2, 7, 12, 17,…
or 5,10,15,20…. Etc.
87
Cluster sampling is such that groups are
selected based on pre-existing groups that is
arbitrary to the individual and not based on any
characteristic of the individual.
In the country, by region
In the state, by zip code
In the state or nation, by area code
For example, in a state, randomly selecting five
counties and surveying 100 individual from each
88
Stratified sampling is such that individuals are first grouped by specific characteristics such as gender and then samples are taken from each group or strata. Individuals grouped by gender Individuals grouped by age Individuals grouped by race For example, grouping individuals by gender, male/female, then selecting 100 individuals from each group
89
Convenience sampling is such that individuals
are selected based upon ease of access. Such
sampling techniques are prone to bias. An
example of a convenience sampling is a
volunteer response.
Individuals as they passed by
Individuals willing to call in on a talk show
Individuals who agree to take online surveys
90
Multistage sampling is such that more than one
sampling technique is employed in the gathering
of information.
First stratify by gender, then systematically
take every other individual in each group.
First cluster individuals by state, then poll
these regions using mailers which individuals
have the option to fill out at their convenience
91
Too Regular
Implausible Numbers Inconsistencies
Missing Information
Non-Adherers
Non-sampling Error
Hidden Agenda
Hidden Bias
Survey Error
Under-coverage
Incorrect Arithmetic
92
Control, Randomization, Replication, & Enough Information
93
An observational study is an experiment
designed to observe without interference from
the observer in that every effort is made not to
sway the subject response or lead a subject in
their response.
Do not sway individuals!
94
Common
Observational
Studies
Historical data (past)
Single point in time (present)
Data gathered over an extended
period of time (future)
Retrospective
studies
Cross
Sectional
Prospective
studies
(Longitudinal)
95
An experimental study is an experiment designed to be observed with interference from the observer in that specific treatments are applied to the individuals, in an effort to measure differences in the subject response. Note: the treatments used in an experiment are intended to sway the outcome of the subject response. Subject Treatment Response (Outcome)
96
A treatment is any condition set forth that is
applied to the individual or subject in an effort to
determine differences among a variety of
treatments as compared to each other or a control
group.
97
A control group is a group created for sake of
comparison. This group can be one of the
treatment groups or a group that receives a false
treatment called a placebo.
Experimental Group: Subject Treatment Response (Outcome)
Control Group: Subject No Treatment (placebo) or Secondary Treatment Response (Outcome)
98
The placebo effect occurs when a subject
receives a false treatment (such as a sugar pill) or
no treatment, but (incorrectly) believes he or she
is in fact receiving treatment and responds
favorably.
99
In an experimental design, a block is a group of
individuals stratified based on a similar
characteristic and given treatments.
A block design is an experimental design in
which individuals or subjects are grouped into
categories or blocks and then test blocks are
treated as experimental units given different
treatments.
100
A randomized-block design is an experimental
design in which individual subjects are matched
based on a specific variable. The subjects are
then put into blocks of the same size as the
number of treatments and then each block is
assigned to different treatment groups randomly.
101
A (single) blind experiment is an experiment in
which individual subjects do not know the
treatment they receive; however, the researcher is
aware.
A double blind experiment is an experiment in
which neither the individual subjects nor the
researcher are aware of who received what
treatment.
102
Principles of
Experimental
Design
A comparative or control group
Selected at random
To verify validity and reliability
More important in inferential
statistics and not so much in
descriptive statistics
Control
Randomization
Replication
Enough
Information
103
Stages of Sampling
Define population of concern
The set of variables to be measured
Systematic, Cluster, Stratified, etc.
Large Enough n (compared to N)
Implement sampling plan (ED)
Action of data collection
Population Sampling Frame Sampling Method Sampling Size (n) Experimental Design (ED) Sampling
104
Medical Trials and Simulations
105
Medical Trials
Internal Review Board
Independent Ethics Committee
Ethical Review board
Requires that the individual (1) be
informed and (2) give consent
IRB
IEC
ERB
Informed
Consent
106
Anonymity is when no personal information is taken, a coding system is in place to allow the subject to get the information regarding a survey without giving out any personal information; that is, the information is not personally identifiable. Confidentiality is when personal information is given, but not shared. Only the statistical summaries are made available to other organizations or persons involved in the study.
107
Informed consent is when the individual
person is both informed of the ramifications
involved in the study and gives consent to
participate in the knowledge of such things as
side effects.
108
Simulation is the imitation of a natural
process using general characteristics or
behaviors in an effort to mimic or model the
natural system.
“A simulation is only as good as the underlying
analytical model"
CPT
Can be used to verify statistical methods.
109
Examples of
Simulation
ONE POSSIBILITY:
Let evens represent a head and
odds represent a tail.
Hence the sequence
1,5,4,6,5
would represent
T,T,H,H,T
Use a fair dice to simulate
the tossing of a fair coin.
110
Random digit chart is the table of digits
selected at random and placed in a table in
Appendix B which can be used to simulate or
sample data.
07892632401926795457
111
Examples of
Simulation using
Random Digits
Let 0-5 represent a boy and
6-9 represent a girl; hence,
the sequence of random numbers
078
would simulate the sequence of
children: boy, girl, girl.
A man has a 60%
chance of having a
boy and a 40%
change of having a
girl, use the random
digit chart to
simulate the birth
order of three
children
Random digits: 07892632401926795457
112
Randomization or random charts can be used to sample or re-sample the data.
For example, if there are 100 data points available and we only need 30, then we can randomly select this sample by enumerating the data and using the random chart to select the required number with or without replacement. With replacement, we can resample 200 times even though there are only half this many data points to start – this technique is called bootstrapping.
113
Examples of
Sampling using
Random Digits
Let: 0 represent A, 1 -B, 2 -C, 3 -D, 4 -E, 5 –F, 6 -G, 7 -H, 8 -I and 9 -J. Using the random set of digits 9263 generate a random committee as follows:
9 J 2 C 6 G 3 D
A committee of four
is to be selected from
a group of ten
individuals: A, B, C,
D, E, F, G, H, I, and
J. Using the random
set of digits
07892632401926795457
generate a random
committee. Explain.
114
115
Descriptive Statistics vs.
Inferential Statistics
Population vs. Sample
N vs. n
Census vs. Sample Survey
Representative Samples
Sampling Techniques
Simulations
Re-sampling
Statistical
Perspective
Biologist have
microscopes
Physicist have
telescopes
Statisticians have
kaleidoscopes
116