Programming tutorials - page 11
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Tidy data
Tidy data
Hey everyone, today we'll be discussing tidy data, which is a particularly convenient and common format in data science applications. While there are various ways to record information in a spreadsheet, tidy data follows three simple principles to ensure its organization and usefulness.
Firstly, each row in tidy data represents one and only one observation. This means that each row captures all the measurements and details for a single experimental unit.
Secondly, each column represents one and only one variable. Variables are the measured attributes across all the experimental units, and each column focuses on a specific characteristic or aspect.
Lastly, the entire spreadsheet should consist of exactly one type of observation. This ensures that all the data in the spreadsheet relates to the same type of experiment or study.
One significant advantage of tidy data is its ease of expansion. If you obtain new observations or data points, such as new subjects in a medical trial, you can simply add a new row at the bottom of the spreadsheet. Similarly, if you want to include additional variables, you can add new columns to the right of the existing ones.
Let's take a look at a couple of examples. The "mtcars" dataset, available in R, is a tidy data set. Each row represents a single car, and each column represents a specific characteristic of the cars. Ideally, tidy data sets should be accompanied by a data dictionary that explains the meaning of each variable and provides information about the units of measurement. The data dictionary may also include metadata about the data set, such as the recording details.
On the other hand, the "diamonds" data set in the "ggplot2" package is another example of tidy data. Each row corresponds to a single round-cut diamond, and each column represents a characteristic of the diamonds.
However, not all data sets are tidy. For instance, the "construction" data set in the "tidyverse" package is not tidy because two variables, the number of units and the region, are spread across multiple columns.
It's important to note that untidy data is not necessarily bad, as real-world spreadsheets often have their own conventions for specific purposes. However, when it comes to data science and exploring relationships between variables among a large number of observations, tidy data is often more convenient for visualization and modeling.
To wrap up, I want to mention contingency tables, which are a common format for non-tidy data. Contingency tables display counts for different combinations of categorical variables. While they can be useful, transforming them into tidy data with separate columns for each variable and their respective counts can make the data more manageable and easier to analyze.
In summary, tidy data follows the principles of one observation per row, one variable per column, and one type of observation throughout the spreadsheet. By adhering to these principles, tidy data provides a structured and organized format that facilitates data exploration, visualization, and modeling in data science applications.
Experiments and Observational Studies
Experiments and Observational Studies
Hello everyone, today we'll be discussing experiments and observational studies, which are the two fundamental types of research studies in statistics. Understanding the difference between them is crucial. Let's explore each type and their key characteristics.
Experiments: In an experiment, different treatments are applied to different parts of the sample, and the resulting variations are observed. The main objective is to determine cause and effect. If there are distinct outcomes between the treatment groups, we aim to attribute those differences to the specific treatments. Experimental studies involve actively influencing and manipulating the variables.
Observational Studies: On the other hand, observational studies involve researchers measuring characteristics of the population of interest without attempting to influence the responses in any way. The most common type of observational study is a sample survey, where researchers collect data by observing and recording information. The focus is on understanding relationships and patterns within the observed data.
Let's explore a few examples to distinguish between experiments and observational studies:
A group of doctors studies the effect of a new cholesterol-lowering medication by administering it to their patients with high blood pressure. This is an experiment since the doctors are applying a treatment and analyzing the outcomes.
A primatologist observes 10 chimpanzees in their natural habitat, taking detailed notes on their social behavior. This is an observational study as the primatologist is merely observing and recording the behavior without influencing it.
An upholster contacts 500 men and 500 women, asking each individual about their preferred candidate in an upcoming election. This is another example of an observational study. The pollster is collecting data without manipulating the participants or their responses.
Observational studies can be comparative, like the previous example, where men and women are contacted separately for analysis purposes. However, since there is no treatment applied, it remains an observational study.
Certain characteristics define a good experiment. It should be randomized, controlled, and replicable:
In experiments, comparisons are often made between two or more treatment groups, with one group serving as the control. The control group provides a baseline for comparison against the groups receiving specific interventions.
To address the placebo effect, where subjects respond to treatments even if they have no measurable effect, experimenters include a placebo in the control group. Placebos are treatments known to have no real effect, such as a sugar pill or an unrelated lesson for educational studies.
In addition to randomization and control, it's advantageous for the assignment of subjects to treatment groups to be double-blind whenever possible. This means neither the subjects nor the data collectors are aware of who is in which treatment group. Double-blinding helps eliminate bias and ensures unbiased observations and measurements.
There are three important experimental designs to consider:
Understanding these design types helps researchers plan experiments effectively and draw meaningful conclusions from the data. By implementing appropriate experimental designs, researchers can enhance the validity and reliability of their findings.
In summary, experiments and observational studies are two fundamental types of research studies in statistics. Experiments involve applying different treatments and observing their effects to determine cause and effect. On the other hand, observational studies focus on observing and measuring characteristics without actively influencing the responses.
A good experiment should incorporate randomization, control, and replicability. Randomization ensures unbiased assignment of subjects to treatment groups, control minimizes confounding variables, and replication allows for verification of results. Additionally, the inclusion of a control group and the consideration of the placebo effect are important aspects of experimental design.
Different experimental designs, such as completely randomized design, randomized block design, and matched pair design, offer flexibility in addressing specific research questions and accommodating different study scenarios.
By understanding the distinctions between experiments and observational studies and employing appropriate experimental designs, researchers can conduct rigorous studies, draw meaningful conclusions, and contribute to advancing knowledge in their respective fields.
Remember, when planning a research study, carefully consider the research question, the nature of the variables, and the available resources to determine the most suitable approach—whether it be an experiment or an observational study.
Introduction to Statistical Sampling
Introduction to Statistical Sampling
Good day, everyone! Today, we are delving into the fascinating world of statistical sampling. In an ideal scenario, conducting a research study would involve collecting data from the entire population of interest, akin to a census. However, in practice, this is often impractical or impossible. Consider the following research questions: What is the average lifespan of pigeons in New York? Is a new medication effective in reducing LDL cholesterol in patients over 45? What percentage of voters approve of the President's performance? In each case, gathering data from the entire population is not feasible. Therefore, we turn to a more manageable approach: sampling.
Sampling involves selecting a subset, or sample, from the population to represent and draw conclusions about the entire population. However, not all sampling methods are equally reliable. Let's discuss a couple of incorrect approaches to sampling. First, anecdotal evidence, which consists of personal testimonials from individuals known to the researcher, should be met with skepticism. For instance, relying solely on statements like "This pill worked for my whole family" or "I talked to three people today who approve of the President" can lead to biased results. Similarly, convenient sampling, where data is collected from easily accessible sources, such as a political poll conducted in a nearby park or a psychological study using the professor's students, can introduce bias due to the non-random selection of participants.
To ensure the validity of our findings, it is crucial to employ a random sample. In a random sample, a random process determines which individuals from the population are included, with each member having an equal chance of being selected. The goal of a random sample is to avoid sampling bias, which occurs when the statistic derived from the sample systematically overestimates or underestimates the population parameter. It is essential to note that statistics derived from random samples still exhibit variability, as individual samples may differ from the population due to the random selection process. However, on average, the statistic will equal the population parameter.
Let's explore some types of random sampling. The simplest and most intuitive approach is a simple random sample (SRS), where every sample of the same size has an equal chance of being selected. This is typically achieved by obtaining a list of the population members, assigning them numbers, and using a random number generator to select the desired number of individuals. In a stratified sample, the population is divided into groups or strata based on important characteristics like age, sex, or race. Then, a simple random sample is taken from each group, allowing for separate analysis of different subgroups within the population. In a cluster sample, the population is divided into naturally occurring or similar groups or clusters. A random sample of clusters is selected, and every member of the selected clusters is included in the sample. Multi-stage sampling combines these techniques by selecting clusters, then taking random samples within each cluster, repeating the process if necessary.
Now, let's apply these concepts to some examples and identify the sampling methods employed. In the first example, a pollster contacts 400 men and 400 women at random, asking them about their preferred candidate in an upcoming election. This is an instance of stratified sampling, as it gathers information on both men and women while taking a simple random sample within each group. In the second example, researchers randomly select 50 high schools and administer a math proficiency exam to all students within those schools. This represents a cluster sample, where randomization occurs at the school level, and a census is conducted within the selected schools.
In the third example, a car dealership uses a customer list to randomly select 200 previous car buyers and contacts each one for a satisfaction survey. This is a typical example of a simple random sample, as each group of 200 customers has an equal chance of being selected. Lastly, a medical group randomly chooses 35 US hospitals and then takes a random sample of 50 patients from each hospital to examine the cost of their care. This scenario demonstrates a multistage sample. Initially, clusters (hospitals) are randomly selected, followed by a simple random sample within each chosen hospital.
Before concluding, it's worth mentioning another sampling method, known as a systematic sample. While not a form of random sampling, it can be used as a substitute under specific circumstances. In a systematic sample, members of the population are selected using a predetermined pattern. For example, a grocery store could survey every 20th person exiting the store to gauge customer satisfaction. A systematic sample can be as effective as a random sample when the population is homogeneous, meaning there are no relevant patterns within it. However, caution must be exercised to ensure that the sampling pattern does not align with any existing patterns in the population, as this could introduce bias.
To summarize, statistical sampling is a vital tool when it is impractical or impossible to collect data from an entire population. Random sampling methods, such as simple random samples, stratified samples, cluster samples, and multistage samples, help mitigate sampling bias and increase the likelihood of obtaining representative and unbiased results. While random samples introduce variability, statistics derived from them, on average, align with the population parameters. Understanding the strengths and limitations of different sampling methods is crucial for conducting reliable and accurate research studies.
Bias and Variability in Statistics
Bias and Variability in Statistics
Hello everyone! Today, we're diving into the concepts of bias and variability in statistics. The overarching goal of statistical inference is to draw conclusions about populations based on sample data. To achieve this, we often use statistics, which are numerical descriptions of samples, to estimate the corresponding parameters, which are numerical descriptions of populations.
To illustrate this, let's consider an example. Suppose a survey of 1,200 voters reveals that Candidate A is leading Candidate B by 8 percentage points. We can view this 8-point difference as a statistic, an estimate of how much Candidate A is expected to win by. On the other hand, the actual outcome of the election, which is the true difference in support between the candidates, represents the parameter.
In some cases, the statistic and the parameter will align perfectly. However, more often than not, they will differ to some extent. For instance, the actual outcome of the election might show that Candidate A wins by 7.8 percentage points. While such deviations can occur due to random chance, they can pose a problem when assessing the quality of a statistic.
This leads us to the concept of bias. A statistic, represented as P-hat, is considered unbiased if, on average, it is equal to the corresponding parameter, denoted as P. In other words, a good statistic should not systematically overestimate or underestimate the parameter. It is important to note that we are using the term "bias" in a technical sense here, unrelated to prejudice or discrimination.
Several common sources of bias can affect surveys. Sampling bias occurs when not all members of the population have an equal chance of being selected in a random sample. For example, if a telephone poll excludes cell phones, it may skew the results towards older individuals, potentially differing from the overall population's views. Non-response bias arises when those who refuse to participate in a survey differ from those who do, leading to potential biases in the collected data.
Asymmetric questions or biased wording can influence respondents to answer in a certain way, introducing bias into the results. Social desirability bias occurs when respondents are inclined to provide answers that are socially acceptable or viewed positively. For instance, if individuals are asked about their dental hygiene practices, they might overestimate the number of times they brushed their teeth due to social desirability bias.
In experimental studies, bias can stem from factors such as lack of control or blinding. If experimental groups differ beyond the treatment being applied, it can introduce bias into the results. Randomization is crucial to ensure uniformity and reduce bias.
While an unbiased statistic aims to estimate the parameter accurately, variability accounts for the tendency of statistics to vary across different random samples. Even with an unbiased sampling method, each random sample is likely to yield a different statistic due to chance alone. It's important to note that variability is not a form of bias. Just because a poll did not precisely predict an election outcome does not necessarily imply it was flawed.
To help visualize the difference between bias and variability, imagine throwing darts at a bull's-eye. Low variability and low bias would mean that your darts consistently hit the target, tightly clustered around the bull's-eye. High variability but low bias would result in scattered darts, still centered around the bull's-eye. Conversely, high variability and high bias would lead to widely scattered darts, missing the bull's-eye consistently. However, even in the worst-case scenario, it's possible for a study to hit the bull's-eye once, indicating that occasional correct outcomes can occur despite high bias and variability.
Understanding bias and variability is essential for evaluating the quality of statistics and interpreting research findings accurately.
Constructing Frequency Distributions
Constructing Frequency Distributions
Hello everyone! Today, we're going to delve into constructing frequency distributions to summarize and analyze quantitative data. When we have a set of numerical observations, it's essential to understand the shape, center, and spread of the data. To achieve this, simply staring at the data won't be sufficient. We need to summarize it in a meaningful way, and that's where frequency distributions come into play.
A frequency distribution involves dividing the data into several classes or intervals and then determining how many observations fall into each class. Let's consider an example where we have a range of values from 11 to 25. To create a frequency distribution, we can divide this range into five classes and count the number of observations in each class.
In the notation used for interval notation, a hard bracket on the left [ indicates that the left endpoint is included in each interval, while a soft bracket on the right ) indicates that the right endpoint is not included. It means that the boundary values, such as 14, 17, 20, and 23, always go into the next higher class. Additionally, the class widths are all equal, in this case, three units each.
By examining the frequency distribution, we can already gain some insights into the data. The center of the data appears to be around 18, falling within the 17 to 20 class, which has a higher frequency. The rest of the data shows relative symmetry around this central spike.
Now, let's go through a step-by-step process for constructing a frequency distribution. Firstly, we need to decide on the number of classes to use. While there isn't a strict rule, a good starting point is typically between 5 and 20 classes. If we use too few classes, we won't capture enough detail in the distribution, hindering our ability to understand the data. On the other hand, using too many classes results in low counts per class, making it challenging to discern the shape of the data.
Once we determine the number of classes, we proceed to calculate the class width. To do this, we compute the range of the data by subtracting the minimum value from the maximum value. Then, we divide the range by the number of classes. It's crucial to round up the class width to ensure that all observations fall into one of the classes. Rounding down may cause some data points to be excluded from the distribution.
Next, we find the lower boundaries for each class. We start with the minimum value as the lower boundary of the first class. Then, we add the class width to obtain the lower boundary of the second class, and so on. Each class's upper boundary is just below the lower boundary of the next class.
Finally, we count how many observations fall into each class by examining the data set. For example, let's consider a scenario where we construct a frequency distribution using eight classes for a given data set. We calculate the range of the data, which is 115.5 - 52.0 = 63.5. Dividing this range by eight, we get a class width of 7.9, which we round up to 8.0. Starting from the minimum value of 52, we add 8.0 to obtain the lower boundaries for each class: 52, 60, 68, and so on.
By going through the data set and counting the observations falling into each class, we obtain the frequencies. It's important to note that the classes should not overlap, and their widths should remain the same. This ensures that each observation is assigned to a single class.
To enhance our understanding of the frequency distribution, we can expand the table by adding columns for class midpoints, relative frequencies, and cumulative frequencies. Class midpoints represent the average value within each interval. We compute them by taking the average of the lower and upper boundaries of each class. For example, the midpoint for the class from 52 to 60 is (52 + 60) / 2 = 56, and for the class from 60 to 68, it is (60 + 68) / 2 = 64, and so on.
Relative frequencies provide insights into the proportion of observations within each class relative to the total size of the data set. To calculate relative frequencies, we divide the frequency of each class by the total size of the data set. For instance, dividing the frequency 11 by the data set size of 50 gives us a relative frequency of 0.22. Similarly, dividing 8 by 50 yields a relative frequency of 0.16.
Cumulative frequencies are obtained by summing up the frequencies for each interval and all the intervals that came before it. The cumulative frequency for the first interval, from 52 to 60, remains the same as its frequency, which is 11. To find the cumulative frequency for the next interval, we add its frequency (8) to the cumulative frequency of the previous interval. For example, the cumulative frequency for the second interval, from 60 to 68, is 11 + 8 = 19. We continue this process for each interval, summing the frequencies and previous cumulative frequencies to obtain the cumulative frequencies for subsequent intervals.
It's important to note that the sum of all the frequencies should be equal to the total size of the data set (in this case, 50). The sum of the relative frequencies should always be 1, indicating the entirety of the data set. Finally, the last value in the column of cumulative frequencies should match the size of the data set.
Expanding the frequency distribution table with columns for class midpoints, relative frequencies, and cumulative frequencies helps provide a more comprehensive understanding of the data distribution. It allows us to observe the central tendencies, proportions, and cumulative proportions of the data in a more organized and insightful manner.
In summary, constructing a frequency distribution involves dividing data into classes, determining class widths, calculating lower boundaries, counting observations in each class, and analyzing the resulting frequencies. Expanding the table with additional information, such as class midpoints, relative frequencies, and cumulative frequencies, can further enhance our understanding of the data set's characteristics.
Histograms, Frequency Polygons, and Ogives
Histograms, Frequency Polygons, and Ogives
Hey everyone, today we're diving into the world of graphing data. We'll be exploring histograms, frequency polygons, and ogives, which are all visual representations of single-variable distributions. As we explore these different types of displays, we'll use the expanded frequency distribution we created in the previous video as an example. To refresh your memory, we started with a dataset consisting of 50 values ranging from approximately 52 to 116. We divided the dataset into eight classes of equal width and determined the number of values in each class to construct the frequency distribution.
Let's begin with the most important and commonly used visual representation of a single-variable dataset: the frequency histogram. In a histogram, we plot the data values on the horizontal axis and the frequencies on the vertical axis. Specifically, we label the class midpoints, such as 56, 64, 72, and so on, on the horizontal axis. Above each midpoint, we draw a bar whose height corresponds to the frequency of that class. For example, if the frequencies for the first few classes are 11, 8, 9, and so on, the bars will have those respective heights.
It's important to note that histograms represent frequency using area. More area indicates a larger amount of data. When we look at the plot, our eyes are naturally drawn to areas with more data, giving us an intuitive understanding of the shape, center, and spread of the dataset. For instance, in this histogram, we can see that the data is more likely to cluster around 56 rather than 112. Additionally, it's worth mentioning that when drawing a histogram, we don't leave gaps between adjacent classes, unlike in a bar chart where gaps are typically present between bars representing categorical variables.
Sometimes histograms are drawn with the horizontal axis labeled with the endpoints of the classes instead of the midpoints, and that's perfectly acceptable. The graph conveys the same information regardless of which labeling approach is used. Another option is to plot relative frequency instead of frequency on the histogram, which should yield a similar shape. The only difference would be a change in the scaling of the horizontal axis to accommodate the relative frequency values.
Another visual display method similar to the histogram is the frequency polygon. Here, we still plot the data values on the horizontal axis and represent frequencies on the vertical axis. However, instead of drawing bars, we plot a point for each class. These points correspond to the midpoints on the horizontal axis and their respective frequencies on the vertical axis. We then connect these points with lines. To ensure that the polygon appears complete, we add an extra point below the first midpoint and another above the last midpoint, each extending by one class width.
Lastly, we can represent the data using an ogive, which displays cumulative frequencies. When constructing an ogive, we plot the upper class boundaries on the horizontal axis and the cumulative frequencies on the vertical axis. We start with a point on the horizontal axis corresponding to the first lower class boundary. The purpose of the ogive is to show, for any given x-value, how many data points in our distribution fall below that value.
I hope this clarifies the concepts of graphing data using histograms, frequency polygons, and ogives. These visual displays provide valuable insights into the distribution of single-variable datasets.
Your First RStudio Session
Your First RStudio Session
Hello everyone, in today's session, we are excited to open up our studio for the first time. Our main focus will be on exploring the basic functionality and getting comfortable working in this environment. When you first open up our studio, you'll notice three different panes, but in this video, we will primarily concentrate on the console tab in the leftmost pane. However, we will briefly mention the other panes as we progress, saving a more detailed discussion for future videos.
To begin, let's explore the console tab, which acts as a scientific calculator in R. You can perform basic arithmetic operations, such as addition, subtraction, multiplication, and division. For instance, if we calculate 8 plus 12, the answer is 20. It's important to note that the answer is displayed without the square brackets, which we will explain later in this video. Additionally, you can add spaces for readability, as R ignores spaces when entered on the command line.
R provides a wide range of built-in functions, such as the square root function. For example, the square root of 9 is 3. Similarly, you can perform trigonometric operations, absolute value calculations, and more. The function names are usually intuitive, but in case you're unsure, a quick Google search will help you find the correct syntax.
One helpful feature in RStudio is the ability to recall previous commands using the up arrow key. This allows you to retrieve a previous command and make edits if needed. For instance, if you want to calculate the square root of 10 instead of 9, you can press the up arrow key, delete the 9, and enter 10 to get approximately 3.162278.
By default, R displays six digits of accuracy to the right of the decimal point. However, you can adjust this setting under the preferences menu according to your needs.
Now, let's move on to defining variables. In R, you can assign values to variables using the assignment operator, which is a left arrow ( <- ) or an equal sign ( = ). It is recommended to use the left arrow for assignments. For example, let's define a variable named "x" and set it equal to 3. After the assignment, the environment tab in the upper right pane will display "x = 3" to remind us of the assignment. If we simply type the variable name "x" in the console and press enter, R will print its value, which is 3 in this case.
You can perform arithmetic operations using variables, just like with numeric values. For instance, if we calculate 3 plus x, the result is 6. R respects the order of operations, so expressions like 1 plus 2 times x will evaluate to 7 rather than 9.
R becomes more powerful when we assign variables as vectors. To create a vector, we use the concatenate function (c) followed by parentheses and the values we want to include. For example, let's assign the vector "y" to the values 1, 5, 6, and 9. After defining the vector, typing "y" and pressing enter will display its values: 1, 5, 6, and 9. Now we can perform arithmetic operations on the vector, such as adding 2 to each element (y + 2) or applying mathematical functions like the square root (sqrt(y)).
In addition to arithmetic operations, we can also summarize vectors. For instance, we can calculate the median (median(y)) or the sum (sum(y)) of the vector. R provides numerous functions to manipulate vectors, and if you're unsure about a specific function, a quick Google search will provide the necessary information.There are two additional features in RStudio that I'd like to mention before we move on. The first one is the
History tab located at the top of the console. By clicking on it, you can access a list of your most recent commands. You can scroll through the history to review and reuse previous commands, which can be a time-saving feature. Even if you quit RStudio and come back later, the command history will still be available.
To reuse a command from the history, simply double-click on it, and it will appear in the console. You can then make any necessary edits and reevaluate the command. This feature allows you to easily revisit and modify your previous commands.
The second feature I want to highlight is the ability to give variables names consisting of more than one letter. For example, let's say we want to create a variable named "numbers" and assign it the values 1, 2, 3, 4, 5, and 6. We can do this by entering "numbers <- c(1, 2, 3, 4, 5, 6)" in the console. Once the assignment is made, we can perform various operations on the variable, such as calculating the square root of "numbers" (sqrt(numbers)).
Now, let's move on to loading a data set and exploring some of the actions we can take with loaded data. In the lower right-hand pane of RStudio, you'll find a file browser. Navigate to the location of your data set and select it. For example, let's choose the "body" data set. Click on the "Import Dataset" button to import the data set into RStudio.
During the import process, you'll see a preview of the data set's spreadsheet format. In the upper right pane, the environment tab will display a new object called "body_data." This object represents a data frame with 300 observations and 15 variables. Essentially, it's a table with 300 rows and 15 columns. You can interact with the data set by sorting columns, scrolling horizontally to view more columns, and treating it similarly to an Excel file.
To work with specific variables in the data frame, we need to specify them using the dollar sign ($) notation. For example, if we're interested in the "age" variable, we can type "body_data$age" in the console. RStudio will provide a list of available variables as you start typing. By pressing enter, you'll see a list of all the ages in the data set in the order they appear.
Once we have isolated a specific variable, such as "body_data$age," we can perform operations on it just like any other variable. For instance, we can calculate the mean age of all individuals in the data set by typing "mean(body_data$age)" in the console. In this case, the average age is determined to be 47.0.
In addition to the mean, you can explore other statistics such as the standard deviation, median, sum, minimum, maximum, and more using the appropriate functions. We will delve deeper into these data manipulation techniques in future videos, exploring the power of R for statistical analysis.
That concludes our overview of opening up our studio, basic functionality, and working with variables and data sets. Stay tuned for future videos where we will explore more advanced features and techniques in RStudio.
Histograms and Frequency Polygons in R
Histograms and Frequency Polygons in R
Hello everyone, in today's video, we will be creating visually appealing histograms and frequency polygons in R using the qplot command. There are various ways to create graphics in R, but I personally believe that the ggplot2 package produces the best-looking displays. To get started, we will be using the qplot command in ggplot2.
For our demonstration, we will be working with the "faithful" dataset, which is built-in with R. This dataset consists of 272 observations of eruption time and waiting time between eruptions in minutes from the Old Faithful geyser in Yellowstone National Park, USA.
To plot histograms and frequency polygons for the "waiting" variable, we will need to install the ggplot2 package first. If you haven't installed it yet, you can do so by typing "install.packages('ggplot2')". Once installed, you need to load the package every time you start a new session by typing "library(ggplot2)".
Now let's focus on the plotting. To create a histogram, we specify the variable on the x-axis using the "x" argument, like this: "qplot(x = waiting, data = faithful, geom = 'histogram')". This will generate a histogram that looks better than the one produced by base R's hist command.
However, there are a few improvements we can make. Let's start by adding labels and a main title to the graph. We can use the arguments "xlab" for the x-axis label, "ylab" for the y-axis label, and "main" for the main title. For example: "qplot(x = waiting, data = faithful, geom = 'histogram', xlab = 'Waiting Time', ylab = 'Frequency', main = 'Old Faithful')".
Next, let's address the appearance of the bars. By default, the bars might appear to run together. To differentiate them, we can add a border color using the "color" argument, such as "color = 'darkblue'". Additionally, we can change the fill color of the bars using the "fill" argument, like "fill = 'lightblue'".
Now, if we want to create a frequency polygon instead of a histogram, we can change the "geom" argument to "geom = 'freqpoly'". This will plot the frequency polygon using the same variable on the x-axis. Remember to remove the "fill" argument since it's not applicable in this case.
You might also want to adjust the number of bins in the histogram using the "bins" argument. By default, R uses 30 bins, but you can change it to a different value, such as "bins = 20", to have more or fewer bins.
Finally, I want to mention an alternative way to specify the data. Instead of using the "$" notation, you can directly specify the dataset using the "data" argument, like "qplot(x = waiting, data = faithful, geom = 'histogram')". This can be useful when working with multiple variables.
That wraps up our tutorial on creating histograms and frequency polygons in R using the qplot command. Feel free to explore and experiment with different settings to create visually appealing and informative graphics.
Stem-and-Leaf Plots
Stem-and-Leaf Plots
Hello everyone, in today's discussion, we will explore the concept of stem-and-leaf plots. Stem-and-leaf plots offer a simple and informative way to visualize the distribution of a single variable. They are especially effective for small data sets as they retain all the information without any loss during visualization. To better understand them, let's dive into some examples.
A typical stem plot consists of a vertical bar, where each digit to the right of the bar represents a data point. These digits represent the last significant digit of each observation, while the values to the left of the bar represent the higher place value digits. For instance, in the given distribution, the initial values are 27, 29, and 32.
Note the key at the top, where the decimal point is one digit to the right of the slash. Stem-and-leaf plots do not incorporate decimals directly; instead, the key indicates the place value. This way, we can differentiate between 27, 2.7, or 0.27.
Now, let's construct a stem-and-leaf plot for the following data set. Here, the tenths place will serve as the leaves, and the two digits to the left of the decimal point will be the stems. So, the first few entries will be 34.3, 34.9, and then proceeding to the next stem, 35/1 (the decimal point aligns with the slash).
The complete plot is as follows: 34.3 34/9 and so on.
It is important to note that every stem between the first and last is included, even if there are no corresponding leaves. This allows us to observe the shape of the data in an unbiased manner. For instance, the values 39.0 and 39.1 are not immediately next to 37.5, leaving some space in between.
However, two potential difficulties can arise when constructing a stem-and-leaf plot. Firstly, if the data contains too many significant figures, such as in the given example, using the last digit as the leaf would result in over 400 stems. To avoid this, rounding the data is recommended. In this case, rounding to the nearest hundred provides a reasonable number of stems.
The second problem occurs when there are too many data points per stem, as shown in another example. To address this, using the thousandths place for leaves and the tenths and hundredths for stems seems appropriate. However, this would only result in three stems (2.1, 2.2, and 2.3). Although technically accurate, this plot fails to depict the desired distribution shape.
To overcome this issue, we can split the stems. By duplicating each stem and assigning the first half to the final digits (leaves) from 0 to 4 and the second half to the digits from 5 to 9, we can obtain a better representation. For example, stem 2.1 would be split into 2.10 to 2.14 (first half) and 2.15 to 2.18 (second half). This resolves the previous difficulty and provides a more informative view of the data.
This additional detail can be revealing, as seen in this example where the split stems highlight a symmetric distribution, contrary to the previous display that appeared right-skewed. Stem-and-leaf plots offer valuable insights into data distributions while preserving all the essential information.
Stem-and-Leaf Plots in R
Stem-and-Leaf Plots in R
Hello, everyone! Today, we will explore the fascinating world of stem-and-leaf plots. A stem-and-leaf plot, also known as a stem plot, is a visual representation of data for a single variable. It is particularly well-suited for small data sets, as it provides insights into the shape, center, and spread of the data. To enhance our understanding, we will work through two examples.
Firstly, let's dive into the built-in "faithful" data set. This data set consists of 272 observations of eruption length and waiting time for the famous Old Faithful geyser in the United States. All measurements are recorded in seconds. In R, the basic command to create a stem plot is conveniently named "stem." We need to specify the name of the variable we want to analyze from the "faithful" data set. Let's begin with the waiting time variable.
Observe the key located at the top of the stem plot. The decimal point is positioned one digit to the right of the slash. By looking at the stem plot, we can identify the first couple of values in the data set, which are 43 and 45. Notably, R automatically splits the stems to accommodate a range of values. For instance, the 40s are divided into the range of 40-44 in the first stem and 45-49 in the second stem, and so forth.
If we wish to override the automatic stem splitting, we can utilize the "scale" argument. This argument allows us to adjust the height of the stem plot by specifying a scaling factor. In this case, to prevent stem splitting, we can halve the height of the stems by setting "scale = 0.5." Although it may not enhance the visual appeal, it serves as a valuable illustration of using the "scale" argument.
Now, let's move on to the second example. We have a data set comprising 20 observations of reaction times in milliseconds to a visual stimulus by participants in a research study. As before, we will begin with a basic stem plot. In this case, the decimal point is two digits to the right of the slash. For instance, "3/1" represents "310."
Please note that some rounding occurs in this plot. The minimum value in the data set is actually 309, resulting in a slight loss of information. As with the previous example, we can modify the default settings using the "scale" command. Let's experiment with that by adjusting the scaling factor. For instance, setting "scale = 0.5" may provide even less intuition about the shape of the data set compared to our original stem plot. However, if we double the length of the stem plot, we can gain a better understanding of the data's distribution.
In this modified plot, you will notice that the stems have transitioned from single digits to two digits. For instance, when we read the first few values represented in the data set, we observe 307 and 309. Additionally, the next listed stem is "32" instead of "31." This occurrence arises because the data starting with "30" and "31" is combined into a single stem. Consequently, there is a potential loss of information. However, the leaves continue to increase in order.
To avoid skipping values in the stems and capture all the data without omissions, we need to further adjust the scaling factor. In this case, we can make the stem plot five times longer than the original version. This allows us to achieve a stem plot that includes all the data without any stem skipping, aligning with our desired representation.
While this final display encompasses the complete data set, it may not be the most optimal choice due to its excessive length. It becomes challenging to perceive the shape, patterns, and underlying trends in the data set. Considering the alternatives, the best options for a clear and informative stem plot are either the one without overriding the stem splitting or the original stem plot we started with.
By selecting either of these options, we strike a balance between capturing the data's essence and maintaining a concise and visually interpretable representation. It's important to remember that the purpose of a stem-and-leaf plot is to provide intuition and insight into the distribution of data, allowing us to identify central tendencies, variations, and outliers.
So, in conclusion, stem-and-leaf plots are valuable tools for analyzing small data sets. They offer a straightforward and visual means to grasp the shape, center, and spread of the data. By experimenting with the scaling factor and stem splitting, we can adjust the plot to meet our specific requirements. However, it's crucial to strike a balance between capturing the complete data set and maintaining a clear representation that facilitates data analysis and interpretation.
Now that we have explored stem-and-leaf plots through two examples, we have gained valuable insights into their usage and customization. Armed with this knowledge, we can apply stem-and-leaf plots to other data sets to unravel their hidden stories and make informed decisions based on data analysis.