statistics.calpoly.edu Open in urlscan Pro
13.57.72.33  Public Scan

Form analysis
3 forms found in the DOM

GET https://www.calpoly.edu/search/?q=

<form method="get" title="Search Form" action="https://www.calpoly.edu/search/?q=">
  <div>
    <input id="searchField" type="text" maxlength="256" placeholder="Search" title="Search this site" name="q" alt="Search Text" tabindex="1">
    <input id="searchSubmit" type="image" name="submit" title="Submit Search Query" alt="Go" src="/sites/all/themes/cpstandard_r/common/images_html/header/search-arrow.png" tabindex="1">
  </div>
</form>

GET https://www.calpoly.edu/search/?q=

<form method="get" title="Search Form" action="https://www.calpoly.edu/search/?q=">
  <div id="mobileSearchForm" style="min-width: 200px; max-width: 300px; margin: 5px auto 8px;">
    <input type="text" maxlength="256" placeholder="Search" title="Search this site" name="q" alt="Search Text" style="width: 85%;">
    <input type="submit" name="submit" title="Submit Search Query" alt="Go" src="" tabindex="2" value="Go" style="height: 25px; margin: 0px 0px 0px -4px; border: none; color: white; background: rgb(111, 99, 69);">
  </div>
</form>

GET https://www.calpoly.edu/search/?q=

<form method="get" title="Search Form" action="https://www.calpoly.edu/search/?q=">
  <div id="mobileSearchForm" style="min-width: 200px; max-width: 300px; margin: 5px auto 8px;">
    <input id="searchField" type="text" maxlength="256" placeholder="Search" title="Search this site" name="q" alt="Search Text" style="width: 85%;">
    <input id="searchSubmit" type="submit" name="submit" title="Submit Search Query" alt="Go" src="" tabindex="2" value="Go" style="height: 25px; margin: 0px 0px 0px -4px; border: none; color: white; background: rgb(111, 99, 69);">
  </div>
</form>

Text Content

 * Skip to Content
 * ?

 * Current Students
 * Prospective Students
 * Parents
 * Business Community
 * Faculty & Staff
 * Alumni

--------------------------------------------------------------------------------

my CalPoly login
 * Quick Links
   * Tutoring
   * Stat Club
   * Contact Us
 * Contact
 * 

Menu ContactSearch
 * 

 * Home
 * About the Department
 * Curriculum
 * Current Students
 * Prospective Students
 * Directory & Office Hours
 * Giving
 * 
 * More About the Department
   * Mission & Objective
   * Statistics Department History
   * Directory
   * Consulting
   * Current Newsletter
   * Past Newsletters
 * 
 * undefined
   * Tutoring
   * Stat Club
   * Contact Us
 * Information for...
   * Current Students
   * Prospective Students
   * Parents
   * Business Community
   * Faculty & Staff
   * Alumni

 * Home
 * About the Department
 * Curriculum
 * Current Students
 * Prospective Students
 * Directory & Office Hours
 * Giving
 * 
 * More About the Department
   * Mission & Objective
   * Statistics Department History
   * Directory
   * Consulting
   * Current Newsletter
   * Past Newsletters
 * 
 * undefined
   * Tutoring
   * Stat Club
   * Contact Us
 * Information for...
   * Current Students
   * Prospective Students
   * Parents
   * Business Community
   * Faculty & Staff
   * Alumni

 * 


STATISTICS DEPARTMENT


BAILEY COLLEGE OF SCIENCE AND MATHEMATICS

 * Home
 * About the Department
 * Curriculum
 * Current Students
 * Prospective Students
 * Directory & Office Hours
 * Giving

Home : Shiny


MORE ABOUT THE DEPARTMENT

 * Mission & Objective
 * Statistics Department History
 * Directory
 * Consulting
 * Current Newsletter
 * Past Newsletters




WEBSITE UPDATE

 * Submit Update Request


SHINY


OUR COLLECTION OF APPS

Regression

Correlation and Regression Game

Multiple Regression Visualization

Inference

Benford's Law: Data Examples (Census and Stock Exchange)

Benford's Law: Sequences

Hot Hand Phenomenon: Randomization-based Analysis

Length/Coverage Optimal Confidence Intervals

Performance of the Wilcoxon-Mann-Whitney Test vs. t-test

t-test with diagnostics

Testing Violation of the Constant Variance Condition for ANOVA

Probability

and

Randomness

Chaos Game: Two Dimensions

Chaos Game: Three Dimensions

Gambler's Ruin

Longest Run of Heads or Tails

Distribution Theory

and

Estimation

Maximum Likelihood Estimation for the Binomial Distribution

Probability Distribution Viewer

Random Variable Generation

Sampling Distributions of Various Statistics

Special Topics

Heaped Distribution Estimation

Hierarchical Models

Population Genetics

Reference: Doi, J., Potter, G., Wong, J., Alcaraz, I., and Chi, P. (2016) “Web
Application Teaching Tools for Statistics Using R and Shiny.” Technology
Innovations in Statistics Education 9(1). Available at
http://escholarship.org/uc/item/00d4q8cp. Corresponding Author: Jimmy Doi


REGRESSION

 





Correlation and Regression Game
Link 1 Link 2

Correlation and regression are two closely related topics. Often they are taught
to students at about the same time, thus giving them the opportunity to explore
both topics in the same application is an intuitive choice. The goals of this
application are then two. The first is to give students with an experience where
they can easily visualize different graphical representations of the full range
of correlation values. The second is to give students an interactive experience
where they can learn to recognize and apply the different features that
determine the solutions to both correlation and regression problems.

The app provides both topics separated by different tabs. In the correlation
portion of the app, a random set of data, with a certain correlation, is
generated and user must attempt to correctly guess the correlation of the data.
The user can mouse over the individual data points if they wish to manually
calculate the correlation. Once the user submits an answer they are notified
whether they guess correctly on not through a popup alert, this feature appears
in both portions of the application. In the regression portion of the app, again
random data is generated without a superimposed regression line. The user must
input values for the slope and intercept, after which a line and residuals are
drawn. After the user has already submitted an estimate once, they may edit the
parameter estimates interactively to see how the changes effect the residuals.
The app then notifies the user whether they were correct or not in guessing the
least squares regression line.

Back to top

 





Multiple Regression Visualization
Link 1 Link 2

Scatterplots are often useful to visualize the relationship between two
quantitative variables. However, with Multiple Regression, there are more than
one predictor variables used to model one response variable. Thus, a simple
scatterplot is no longer adequate to graphically represent all of the variables.
In the case of two predictor variables, we can illustrate this in three
dimensions, or also in two dimensions with appropriate color schemes. This
applet shows us these illustrations for a variety of datasets.

Back to top

--------------------------------------------------------------------------------


INFERENCE

 





Benford's Law: Data Examples (Census I, Census II, US and World Stock Exchange)
Link 1 Link 2

Census I: The first-digit distribution of many US Census variables is known to
closely follow Benford's Law. We will consider several census variables
available from County Totals Dataset: Population, Population Change and
Estimated Components of Population Change. The app will apply a goodness of fit
test of the observed frequencies of first-digits for the selected variable. The
variables under consideration are: Annual Resident Total Population Estimate
(2013 to 2016), Annual Births (2013 to 2016), Annual Deaths (2013 to 2016).

Census II: We also consider several census variables available from US Census
State & County QuickFacts. The app will apply a goodness of fit test of the
observed frequencies of first-digits for the selected variable. The variables
under consideration are: Housing Units (2013), Households (2008-12), Veterans
(2008-12), Nonemployer Establishments (2012), Private Nonfarm Establishments
(2012), Private Nonfarm Employment (2012), Retail Sales (2007).

US Stock Markets: For an analysis of US stock market data the app will download
information from the Wall Street Journal website from the most recent end of day
market data. The data will be based on various market variables for all
companies listed in one of four stock markets. The app will apply a goodness of
fit test of the observed frequencies of first-digits for the selected variable
in the specified stock market.

World Stock Markets: A similar goodness of fit analysis is done for market
variables from various world stock markets based on data accessed from
investing.com. For the selected stock market, if trading is active at the point
of data access, the results will be based on the most current market data. If
the market is closed at point of access, then all information will be based on
the most recent end of day market data. 

Back to top

 





Benford's Law: Sequences (Additive, Power, Prime Number)
Link 1 Link 2

This app examines the first-digit distribution of various sequences: Additive,
Power, and Prime Number

Additive Sequence: Consider a sequence of numbers where we fix the initial two
numbers and then the value of each subsequent number is the sum of the previous
two. We will call this an additive sequence. When the initial two numbers are
both 1 then this yields the famous Fibonacci sequence. If you only consider the
first digit of each number in an additive sequence and examine its distribution,
is it the case that it closely follows Benford's Law? This app generates an
additive sequence, for a given length and initial sequence numbers, and applies
a goodness of fit test of the observed frequencies of first digits to Benford's
Law.

Power Sequence: Consider a sequence of the form b1, b2, …, bn, where b is called
the base. We will call this a power sequence. If you only consider the first
digit of each number in a power sequence and examine its distribution, is it the
case that it closely follows Benford's Law? This app generates the power
sequence, for a given b and n, and applies a goodness of fit test of the
observed frequencies of first digits to Benford's Law.

Prime Number Sequence: Consider the sequence of prime numbers less than or equal
to some power of 10. An article from 2009 shows that the distribution of the
first digit of these prime numbers is well described by what's known as
Generalized Benford's Law (GBL) . This app generates the prime numbers less than
or equal to 103, 104, 105, or 106 and applies a goodness of fit test of the
observed frequencies of first digits to GBL.

Back to top

 





Hot Hand Phenomenon: Randomization-based Analysis
Link

Many basketball players and fans alike believe in the "hot hand" phenomemon: the
idea that making several shots in a row increases a player's chance of making
the next shot. Does the hot hand in basketball really exist? This app can be
used to perform a statistical test for "hot hand" type behavior in sequences of
success/failure trials, such as the shot attempts of a basketball player. A data
set containing the results of each shot attempt for players in the NBA Three
Point Contest from 2013 through 2017 is included.

Back to top

 





Length/Coverage Optimal (LCO) Confidence Intervals
Link 1 Link 2

In 2014, Schilling and Doi developed a binomial confidence interval procedure
that produces coverage probabilities always at least equal to the stated
confidence level (e.g., a strict method), and which, among all procedures that
have this property, give confidence intervals having the minimum possible
average length and the highest possible coverages. They called this the LCO
method (for length/coverage optimal). This Shiny app generates LCO confidence
intervals for any n = 1, 2, ..., 200 and any confidence level between 80% and
99%. The user may select the accuracy of the intervals to be at the 2nd, 3rd, or
4th decimal place.

Back to top

 





Performance of Wilcoxon-Mann-Whitney Test vs. t-test
Link 1 Link 2

The goal of this app is to compare the performance of a nonparametric to a
parametric test for the difference in two population means. Specifically,
performance is measured in the app either by Type I error rate or power, and the
two respective tests for comparison are the Wilcoxon-Mann-Whitney (WMW) test and
the two-sample t-test. Recall that for the test conditions to be satisfied, the
two-sample t-test requires either the two population distributions to be normal
or large enough sample sizes while the WMW test requires the two population
distributions to have the same shape. Users have the option to produce different
scenarios and conclude the better test either through a lower Type I error rate
(if the two population means are the same) or a higher power (if they are not).

When users first launch the app, they are presented with the goal of the study.
Then, a game demonstrates to users the difficulty of identifying the population
distributions of sample data. Following the first two introductory tabs, users
can proceed to comparing performance. They have the option to choose a tab
corresponding to their choice of the population distributions. Within each tab,
either a single comparison or comparisons over a range can be conducted. The
settings available for users to adjust are sample sizes, population means,
significance level, number of simulations, and range of comparison values. In
addition, visualizations are implemented to communicate results to users. For a
single comparison, the outputs are distributions of the test statistics and
gauges. For comparisons over a range, the output illustrates the performance of
the two tests in each comparison.

Back to top

 





t-test with diagnostics
Link 1 Link 2

This app focuses on conducting a t-test and checking the normality condition.
Both the one-sample and two-sample t-tests are implemented in this app. Recall
that for the t-test to be valid either sample size(s) need to be large enough or
the population distribution(s) needs to be a Normal distribution. To begin the
app, data configuration is required. Users have the choice to either use sample
data or upload their own data when first launching the app. Customization is
needed in respect to the uploaded data. After selecting their option, users can
proceed to visualizing the data. A histogram is presented for one sample while
comparative boxplots are presented for two samples. In addition, summary
statistics are also available for display.

The hypothesis test tab displays the null and alternative hypotheses. The
settings available for users to adjust are the hypothesized value, the direction
for the alternative hypothesis, and the significance level. For users who are
not familiar with the concept of the hypothesis test, they can click on a link
that shows information in a popover. Additional information on the one-sample
and two-sample t-tests is also available. When users have run the t-test, the
output includes items such as the shaded t-distribution, t-statistic, and the
p-value. The point estimate(s) and confidence interval can also be outputted by
users’ request. In the normality condition tab, the Shapiro-Wilk normality test
is performed and a Q-Q plot is displayed. In all relevant outputs throughout the
app, sample interpretations from popovers are included for users to understand
the results of the hypothesis test.

Back to top

 





Testing Violation of the Constant Variance Condition for ANOVA
Link 1 Link 2

The ANOVA F-test is used to test for difference in means between groups, and
requires the conditions of normality (or large sample size), independence, and
constant variance.  A common rule of thumb for the constant variance condition
is that the ratio of largest to smallest standard deviation is less than or
equal to two.  This application implements a user-guided simulation study to
assess the consequences of non-constant variance on the Type I error rate of the
ANOVA F-test.  The application enables the user to visualize data with different
standard deviations, reinforces the concepts of sampling distribution, null
distribution, and Type I error, and allows the user to uncover a rule of thumb
for the constant variance condition. 

At left, the user specifies standard deviations for three hypothetical
populations and sample sizes to be drawn from each of the populations.  When the
user presses the “Draw samples” button, data will be simulated from normal
distributions with mean zero and the specified standard deviations and sample
sizes and displayed in dot plots in the left graph.  The ANOVA F-statistic for
the simulated data is plotted in the graph at right, and the critical value for
a 0.05-level hypothesis test is shown in red.  As more samples are drawn (with
the option to draw up to 1,000 samples at a time), more F-statistics are plotted
in the sampling distribution on the right.  The Type I error rate is estimated
as the proportion of samples for which the null hypothesis was rejected, and is
displayed below the graphs.  Below the graphs (not included in the picture
above) is guidance for a suggested series of simulation studies allowing the
user to compare different specifications systematically and uncover the rule of
thumb for the constant variance condition.  

Back to top

 

--------------------------------------------------------------------------------


PROBABILITY AND RANDOMNESS

 





Chaos Game: Two Dimensions
Link 1 Link 2

In the two dimensional version of the Chaos Game we start with a regular polygon
and mark selected points which will typically be the vertices. These points will
be called endpoints and will be marked in red. The game begins by randomly
choosing a starting point and one of the endpoints. Mark a new point at a fixed
distance ratio from the starting point to the endpoint (e.g., halfway to the
endpoint). Select another endpoint at random and, with the most recently created
point, repeat the process to generate the next point and continue. By applying
the right distance ratio the resulting set of points can converge to a beautiful
image known as a fractal. For each polygon the required distance ratio to yield
a fractal will be provided, but try different settings to see what other
patterns may arise!

Back to top

 





Chaos Game: Three Dimensions
Link 1 Link 2

In the three dimensional version of the Chaos Game we start with a regular
polyhedron and mark selected points which will typically be the vertices. These
points will be called endpoints and will be marked with red squares. The game
begins by randomly choosing a starting point and one of the endpoints. Mark a
new point at a fixed distance ratio from the starting point to the endpoint
(e.g., halfway to the endpoint). Select another endpoint at random and, with the
most recently created point, repeat the process to generate the next point and
continue. By applying the right distance ratio the resulting set of points can
converge to a beautiful image known as a fractal. For each polyhedron the
required distance ratio to yield a fractal will be provided, but try different
settings to see what other patterns may arise!

Back to top

 





Gambler's Ruin
Link 1 Link 2

The Gambler’s Ruin is a well-known problem that can be used to illustrate a
variety of probability concepts.

Two players are playing a game against each other, betting the same amount on
each turn (here, we use $1). On each turn of the game, Player A has a fixed
probability p of winning $1 from Player B, where 0<p<1. The probability that
Player B will win $1 from Player A is 1-p. Player A and Player B each start with
some initial fortune (which may or may not be equal to each other), and the game
continues until one player has all of the money.

The Gambler’s Ruin problem is useful for teaching conditional probability,
Markov chains, and for simply visualizing a stochastic process. This app shows a
graphical representation of one iteration of the Gambler’s Ruin, and also can
simulate many runs under a variety of settings that may be manipulated, to
obtain simulated estimates of the average length of a game, and the probability
that Player A will win under those settings. In a mathematical statistics class,
the simulated estimates from this app could be used to corroborate analytic
solutions.

Back to top

 





Longest Run of Heads or Tails
Link 1 Link 2

One popular class activity to help students understand chance behavior is to
observe the runs of consecutive heads or tails in a sequence of coin flips. When
asked to write down a simulated sequence of 100 tosses of a fair coin, most
students are hesitant to create runs of heads or tails exceeding 4. Students are
often surprised to find that the longest run of heads or tails turns out to be
much higher based on 100 tosses of an actual coin.

This Shiny app allows the user to simulate the outcomes of a fair coin flipped n
times (n = 10, 20, ..., 400). In an accompanying plot of outcomes any runs of at
least a specified length are marked in color, and the length of the longest run
is displayed. The user can easily re-randomize the sequence of coin flips and
quickly get a sense of typical longest run values. From the plot students may
also be quite surprised to see how many long runs occur in the sequence.

The user may choose to display the predicted approximate length of the longest
run and an approximate 95% prediction interval for the length of the longest
run. Details on these two estimators can be found in Schilling (1990). See
Schilling (2012) for a more recent and related article.

Back to top

 

--------------------------------------------------------------------------------


DISTRIBUTION THEORY AND ESTIMATION

 





Maximum Likelihood Estimation for the Binomial Distribution
Link 1 Link 2

This app provides an introduction to the concept of maximum likelihood
estimation by working through the example of the binomial distribution. The
first tab shows the probability mass function (pmf) of the binomial
distribution. The user specifies the parameters to see various pmfs, and is
guided to understand that this function takes the number of successes (x) as
input and provides a probability as output.

The pmf is then contrasted with the likelihood function in the second tab. Here
the user specifies the fixed "parameters" (x and n) for the likelihood function,
and the likelihood curve is graphed. Here the user sees that the input to the
function is p rather than x, and the text explains that inputs and parameters
have effectively been switched. The user is guided to input various values of x
and discover that the likelihood function is always maximized at p=x/n. The
third tab displays the likelihood and log likelihood side-by-side so the user
understands they achieve the maximum in the same location.

Back to top

 





Probability Distribution Viewer
Link 1 Link 2

Probability distributions, p-values, and percentiles are fundamental topics
taught to introductory statistics students. Often, students are presented these
topics with static images in textbooks, but frequently do not have access to a
dynamic and interactive tool they can use for exploration. For example, in the
case of p-values, introductory students are frequently shown how to use tables
to obtain a range of possible p-values associated with their test statistic
along with a guiding image, but this is often difficult for students to
understand. The goal of this application is to provide the student with an
intuitive, simple, and comprehensive visualization of the three aforementioned
topics. At the moment, many but not all continuous distributions (Beta, Cauchy,
Chi-Squared, Exponential, F, Gamma, Logistic, Log Normal, Normal, Student’s t,
Uniform, and Weibull) are available in the application. Support for discrete may
be added in the future.

When the app first renders, the user is shown by default the standard normal
distribution. The student may vary the both the distribution and parameters
corresponding to the distribution of their choice from the options at the top
under "Distribution." This enables the student to see how the shape of their
selected distribution changes as these values change. For students that are
interested in visualizing probabilities and percentiles, the probability viewer
app easily allows the student to select between two types of inputs. The student
can also select between different shaded tail visualizations for the inputted
percentiles or probabilities. Whether the student chooses to input a percentile
or probability, the app will automatically calculate the value that the student
did not input, corresponding to the one was. After inputting all required
values, a graph appears with the appropriate distribution, the percentile and
probability pairs, and the appropriate shading.

Back to top

 





Random Variable Generation
Link 1 Link 2

The Probability Integral Transform and the Accept-Reject Algorithm are two
methods for generating a random variable with some desired distribution. This
Shiny app demonstrates how they work, through two examples of each method.

For the Accept-Reject Algorithm (shown above), the examples demonstrated in this
app are the Beta distribution and the truncated Normal distribution. A
side-by-side plot shows each point that has been generated. Users have the
option to generate one replicate at a time, to examine and understand the
mechanics of how the algorithm is accomplishing its task, with details of each
replicate given below the plots. Additionally, up to 500 replicates can be
generated at once, to build towards a greater representation of points and
confirm that the algorithm does in fact result in the desired distribution.

The Probability Integral Transform (not shown) is demonstrated with the
Exponential distribution, and an arbitrary, unnamed distribution. In this
demonstration, users again have the option to generate one replicate at a time,
with side-by-side plots showing each point, and details of each replicate given
below the plots. Users can also generate up to 500 replicates at once to view
the overall distribution that is produced.

Back to top

 





Sampling Distributions of Various Statistics
Link 1 Link 2

This app allows the user to draw repeated samples from a specified population
shape (normal, left-skewed, right-skewed, uniform, or bimodal). The user also
specifies a statistic from the pull-down menu in the left panel. When a sample
is generated by pressing the "Draw samples" button, a histogram of that sample
is plotted in the graph at left, and the sample statistic is added to the
sampling distribution histogram at right. The total number of samples is tracked
at the bottom of the page, and the user may also elect to display the mean and
standard deviation of the sampling distribution by checking the box. Above these
two graphs, the user may also click to display the population curve and
parameter of interest.

Back to top

 

--------------------------------------------------------------------------------


SPECIAL TOPICS

 





Heaped Distribution Estimation
Link 1 Link 2

Data often exhibit a heaped distribution in situations when there are either
rounding or recall issues. Then, heaping is observed in the distribution when
there are unusual spikes at certain values. In this app, the focus is heaping
present at multiples of 5. Two rounding behaviors are assumed and they are
accounted for in the form of two rounding probabilities. The first rounding
probability describes the tendency to round with smaller values, while the
second rounding probability describes the tendency to round with larger values.
Therefore, a mixture model is constructed with a specified distribution and the
two rounding probabilities. Throughout the app, interpretations in popovers are
provided for users to understand the different stages of the demonstration.

Users have the option to either simulate data or upload data to begin the app.
There are five distributions for users to choose and the parameters can be
adjusted. The proceeding tab describes the rounding process to users; the actual
and rounded/heaped distributions are visually displayed for users to compare.
With the heaped distribution, the goal for users is to estimate the actual
distribution with maximum likelihood. After obtaining the estimates, confidence
intervals can be produced either based on the inverse Fisher information matrix
or bootstrapping. For users to validate the method, a simulation study can be
performed in the last tab of the app. They can compare the means of the MLE
distributions to the specified underlying parameters.

Back to top

 





Hierarchical Models
Link 1 Link 2

Hierarchical models are used when there is nesting of observational units in the
data and variables are observed on multiple levels of the hierarchy. Failure to
account for the hierarchy in the data may result in invalid conclusions.
However, hierarchical models are not always needed for nested data as the
intraclass correlation coefficient determines the requirement. This app focuses
on illustrating the concept of hierarchical models by comparing the method to
the two others at the extremes: the pooled and unpooled methods. Users are shown
mathematically and visually how the hierarchical estimates are weighted averages
and how they serve as a balance between the pooled and unpooled estimates; the
two related ideas of shrinkage and borrowing strength are illustrated in this
process.

Users have the capability to either use sample data sets or upload their own
data to learn about hierarchical models through case studies. The three
different scenarios for learning are varying-intercept, varying-intercept and
varying-slope, and varying-intercept and varying-slope with level 2 predictor.
In each scenario, users are first presented outputs and graphs of the pooled and
unpooled method. Then they proceed to the hierarchical model and different
concepts of this method are explained in compartments. Interpretations are
included throughout the outputs for users to comprehend the ideas. Additionally,
each scenario contains a comparison of the three modelling methods with
visualizations. For those who are familiar with Bayesian methods, a tab is
available to run a Bayesian hierarchical model. After grasping the concept of
hierarchical models, users can analyze their own data with their own specified
model.

Back to top

 





Population Genetics
Link

Genetic drift, or the variation in the relative frequency of particular
genotypes within a population, can lead to a higher prevalence or disappearance
of certain alleles within a population. Genetic drift is often more visible
within smaller populations when compared to larger populations. By randomly
pairing observations to represent mating couples, this app simulates the genetic
drift of an allele based on a starting population size and the fitness levels of
each genotype. A second population may also be generated to make comparisons
based on population size or fitness levels of certain genotypes.

Back to top

 

--------------------------------------------------------------------------------


RELATED CONTENT






FACULTY RESOURCES

Faculty Issued Permissions

Common Final Request

Alternate Final/Event Request

BCSM Documents


DEI IN THE BAILEY COLLEGE



Learn More

CONNECT WITH CAL POLY

 

 * CP Home
 * Directory
 * Campus Maps & Directions
 * University Store
 * Calendar
 * Employment
 * Campus Policies
 * Contact Us

 * Privacy Notice
 * Web Accessibility Statement
 * Title IX

© 2024 California Polytechnic State University    San Luis Obispo, California
93407
Phone: 805-756-1111