1. Statistics Fundamentals

How to calculate p-values

**Important (2): 1. p-values are determined by adding up probabilities 2. A p-value is composed of 3 parts: I. The probability random chance would result in the observation Ii. The probability of observing something else that is equally rare Iii. The probability of observing something rarer or more extreme

Hypothesis Testing and The Null Hypothesis, Clearly, Explained!!!

**Important (5) : 1. "You may remember that the only reason the hypothesis is 13 fewer hours is that it was the first result. " 2. "And if 13 is a reasonable hypothesis, then so is 12 and 13.5; in other words, there are a lot of reasonable hypotheses. How do we know which one to test?" 3. ""and thus, the data does not overwhelmingly convince us to reject the Null Hypothesis" 4. Mnemonic: No Difference in hypothesis = Null Hypothesis (they both start with the letter N) Hypothesis testing is a statistical method used to determine whether a hypothesis about a population is likely to be true or not. 5: Options: you only have 2 options when it comes to Null hypothesis; you Reject or Fail to Reject the Null Hypothesis The null hypothesis is the hypothesis that there is no significant difference between the sample and the population. It's often denoted as H0. The alternative hypothesis is the hypothesis that there is a significant difference between the sample and the population. It's often denoted as Ha.

False Discovery Rates, FDR, clearly explained

**Main Ideas: 1) False Discovery Rates are a tool to weed out bad data that looks good 2) The False Discovery Rate (FDR) can control the number of false positives. Technically, the FDR is not a method to limit false positives, but the term is used interchangeably with the methods. In particular, it is used for the "Benjamini-Hochberg method" Multiple hypothesis testing involves testing many hypotheses simultaneously, which increases the likelihood of false positives (type I errors) even if the individual p-values are small. Imagine you are a detective investigating a crime. Your goal is to identify all of the suspects involved in the crime. You have a list of 100 people who may or may not be suspects, and you decide to interrogate each of them. Now, let's say that only 10 of these people are actually involved in the crime. However, as you interrogate each person, you may mistakenly believe that some innocent people are guilty. These false positives are similar to the idea of Type I errors in statistical hypothesis testing. In this case, false discovery rate (FDR) would be the proportion of people you identified as suspects who are actually innocent. For example, if you identified 20 people as suspects, but half of them were actually innocent, your FDR would be 50%. Similarly, in statistics, FDR is a measure of the proportion of significant results (i.e., results that appear to be statistically significant) that are actually false positives. In other words, it is the proportion of false positives among all positive results. It is an important concept to consider when interpreting statistical results, especially when multiple hypotheses are being tested simultaneously.

p-values: What they are and how to interpret them

*Important: 0. Null = No / p = Yes 1. "p-values are numbers between 0 and 1, that, in this example, quantify how confident we should be that Drug A is different from Drug B 2. "The closer a p-value is to 0, the more confidence we have that Drug A and Drug B are different 3. "So the question is, "how small does a p-value have to be before we are sufficiently confident that Drug A is different from Drug B?In other words, what threshold can we use to make a good decision?" 4. "In practice, a commonly used threshold is 0.05. It means that if there is no difference between Drug A and Drug B, and if we did this exact same experiment a bunch of times, then only 5% of those experiments would result in the wrong decision A p-value is a measure of the evidence against the null hypothesis. It represents the probability of observing a test statistic as extreme as, or more extreme than, the observed value, assuming that the null hypothesis is true. 5. A small p-value (usually less than 0.05) indicates strong evidence against the null hypothesis, while a large p-value (usually greater than 0.05) indicates weak evidence against the null hypothesis. The choice of the significance level (often denoted as alpha) depends on the research question and the consequences of making a Type I error (rejecting the null hypothesis when it is actually true). A commonly used significance level is 0.05. It's important to remember that p-values are not the probability that the null hypothesis is true or false. They only measure the evidence against the null hypothesis based on the observed data. P-values should be interpreted in conjunction with other factors, such as effect size, sample size, and the study design. A small p-value does not necessarily imply a large effect size, and a large sample size can lead to small p-values even if the effect size is small.

Probability is not Likelihood. Find out why!!!

1) Probability vs Likelihood: (I think lol) Probability = Discrete Variables Likelihood = Continuous Variables Probability and likelihood are two different concepts in statistics, but they are often confused with each other. Probability is the measure of how likely an event is to occur given a specific set of conditions. Likelihood is the measure of how likely the observed data is to occur, given a specific parameter value or hypothesis. (This if from chatbot but I think it's on to something. As long as you have distribution parameters, I think you can figure out the likelihood of any data. That's why we have distribution parameters I think!!!!) Probability is calculated by dividing the number of desired outcomes by the total number of possible outcomes. Likelihood is calculated by multiplying the probabilities of each data point given the parameter value or hypothesis. The likelihood function is used to estimate the parameter value or hypothesis that maximizes the probability of observing the given data. Maximum likelihood estimation is a common method for finding the best-fit model or parameters for a given data set. Probability and likelihood are related through Bayes' theorem, which allows for updating prior beliefs based on observed data. In summary, probability is used to calculate the chance of an event happening given certain conditions, while likelihood is used to measure how well a particular model or hypothesis fits the observed data.

Conditional Probabilities, Clearly Explained!!!

1. Contingency Table - "in statistics, a contingency table is a type of table in a matrix format that displays the frequency distribution of the variables 2. There's redundancy in the equation so the standard does not include the redundant parts 3. Prior Probability is what we think the probability of an event is, before we have nay evidence or information. It's just a guess or estimate. Conditional Probability, on the other hand, is the probability of an event happening given that we have some additional information or evidence. It's the updated probability after we have taken into account the new information. Think of it like this: if you're trying to guess the color of a ball in a bag without looking, your prior probability of any given color is 1/3 if there are three different colors. But if you peek in the bag and see that the ball is red, your conditional probability of the ball being red is now 1 (or 100%) because you have additional information Conditional probabilities are calculated using the formula P(A|B) = P(A and B) / P(B), where A and B are two events. The vertical line in the notation P(A|B) indicates that we are calculating the probability of A given that B has occurred. The conditional probability of A given B can be used to update our prior beliefs about the probability of A. For example, if we know that someone has tested positive for a disease, we can use the conditional probability of having the disease given a positive test result to update our prior belief about the person's probability of having the disease. The multiplication rule of probability can be used to calculate the probability of two or more events occurring together. For example, P(A and B) = P(A|B) * P(B), or P(A and B) = P(B|A) * P(A). The Bayes' theorem provides a way to calculate conditional probabilities in cases where we know the probability of B given A, but not the probability of A given B. Conditional probabilities are useful in many areas, including medicine, finance, and sports.

Confidence Intervals, Clearly Explained!!!

A confidence interval is a range of values constructed from a sample of data that is likely to contain the true population parameter with a certain degree of confidence. The width of the confidence interval is determined by the level of confidence chosen, the sample size, and the variability of the data. The most common level of confidence used is 95%, which means that if we took many samples and calculated the confidence interval for each one, we would expect 95% of them to contain the true population parameter. The formula for constructing a confidence interval for a mean involves using the sample mean, the standard error, and a critical value from the t-distribution or z-distribution depending on the sample size and whether the population standard deviation is known. The formula for constructing a confidence interval for a proportion involves using the sample proportion, the standard error, and a critical value from the normal distribution. Interpretation of a confidence interval involves recognizing that it provides a plausible range of values for the population parameter and does not imply that the true value is within the interval with any specific probability.

Population and Estimated Parameters, Clearly Explained!!!

A population is a group of individuals or objects that share a common characteristic. For example, the population could be all the people in a city, all the cells in a petri dish, or all the cars produced by a manufacturer. In statistics, parameters are numerical values that describe a population. For example, the mean, median, and standard deviation are all parameters that can be used to describe a population. Because it's often impractical or impossible to measure every member of a population, statisticians often take a sample, which is a smaller group of individuals or objects selected from the population. Statistics that are calculated based on a sample are called estimates. For example, the sample mean and sample standard deviation are estimates of the population mean and standard deviation, respectively. The accuracy of estimates depends on the size and representativeness of the sample. A larger and more representative sample is generally more likely to yield accurate estimates.

StatQuest: Histograms, Clearly Explained

An advantage of using a curve vs a histogram is that the curve is not limited by the width of the bins If we don't have enough time or money to get a ton of measurements... the approximate curve (based on the mean and standard deviation of the data we were able to collect), is usually good enough

Alternative Hypotheses: Main Ideas!!!

An alternative hypothesis is a hypothesis that asserts that there is a significant difference between the sample and the population. It's often denoted as Ha. There are different types of alternative hypotheses, including one-tailed and two-tailed hypotheses. A one-tailed hypothesis predicts the direction of the difference, while a two-tailed hypothesis does not. The choice of alternative hypothesis depends on the research question and the available evidence. A one-tailed hypothesis is appropriate when there is a strong theoretical or empirical basis for predicting the direction of the difference, while a two-tailed hypothesis is appropriate when the direction is not clear. The alternative hypothesis is usually tested against the null hypothesis, which asserts that there is no significant difference between the sample and the population. Hypothesis testing involves calculating a test statistic and comparing it to a critical value or p-value to determine whether the data supports the null hypothesis or the alternative hypothesis. The choice of statistical test depends on the research question and the type of data being analyzed. Common statistical tests include t-tests, ANOVA, and chi-squared tests.

Bayes' Theorem, Clearly Explained!!!

Bayes' theorem is a formula that relates conditional probabilities. It tells us how to update our beliefs about the probability of an event given new information. Conditional probability, on the other hand, is simply the probability of an event happening given that we have some additional information or evidence. It doesn't necessarily involve updating our beliefs based on new information. To put it more specifically, conditional probability is a concept that describes the likelihood of an event given a certain condition, whereas Bayes' theorem is a mathematical formula that uses conditional probability to calculate the probability of an event after new information is taken into account. In other words, Bayes' theorem is a tool we can use to calculate conditional probabilities when we have additional information or evidence. It's a way of updating our beliefs about the probability of an event in light of new information. So, while conditional probability and Bayes' theorem are related, Bayes' theorem is a specific mathematical formula that involves conditional probabilities and is used to update our beliefs based on new evidence.

Using Bootstrapping to Calculate p-values!!!

Bootstrapping is a resampling technique that involves repeatedly sampling from the data to estimate the sampling distribution of a statistic. One way to use bootstrapping to calculate p-values is to generate many bootstrap samples under the null hypothesis, and then calculate the test statistic for each bootstrap sample. The p-value is then estimated as the proportion of bootstrap samples in which the test statistic is equal to or more extreme than the observed test statistic. This approach is particularly useful when it is difficult to calculate the null distribution of the test statistic analytically, or when the assumptions underlying traditional parametric tests are not met. When using bootstrapping to calculate p-values, it is important to consider the number of bootstrap samples used, as well as the specific method for generating the bootstrap samples (e.g., whether or not to use stratified sampling or antithetic sampling). While bootstrapping can be computationally intensive, it can provide a powerful and flexible tool for hypothesis testing when traditional parametric methods are not appropriate.

Bootstrapping Main Ideas!!!

Bootstrapping is a resampling technique used to estimate the sampling distribution of a statistic. The bootstrap method involves creating many "bootstrap samples" by randomly sampling with replacement from the original sample. By repeatedly resampling from the original sample, we can create a distribution of sample statistics that approximates the sampling distribution of the statistic of interest. Bootstrapping can be used to estimate standard errors, confidence intervals, and p-values for a variety of statistics, including means, medians, correlations, and regression coefficients. Bootstrapping does not assume any specific distribution of the data, making it a powerful and flexible method for estimating the properties of the sampling distribution of a statistic. Bootstrapping can be computationally intensive, especially for large sample sizes, but it is becoming increasingly feasible with advances in computing power.

Covariance, Clearly Explained!!!

Covariance vs Correlation: Covariance is an indicator of the extent to which 2 random variables are dependent on each other. A higher number denotes higher dependency. Correlation is a statistical measure that indicates how strongly two variables are related. Covariance = measure of dependence / Correlation = measure of relationship Covariance is a measure of the direction and strength of the relationship between two variables. It indicates how much two variables change together. Covariance can be positive, negative, or zero. A positive covariance indicates that the variables tend to increase or decrease together, while a negative covariance indicates that one variable tends to increase while the other decreases. A covariance of zero indicates that there is no relationship between the variables. Covariance is affected by the units of measurement of the variables. It is important to standardize the variables before calculating covariance, which can be done by subtracting the mean and dividing by the standard deviation. Covariance is related to correlation, which is a standardized measure of the relationship between two variables. Correlation ranges from -1 to 1, and is unaffected by the units of measurement. Covariance can be used to calculate the slope of a regression line, which is used to predict the value of one variable based on the value of another variable. Covariance has limitations, and is affected by outliers and non-linear relationships between variables.

Calculating the Mean, Variance and Standard Deviation, Clearly Explained!!!

Important: " I'm making a big deal about calculating vs estimating variance because it makes a big difference that we'll talk about later." The mean is a measure of central tendency that represents the average value of a set of numerical data. It's calculated by adding up all the values and dividing by the number of values. The variance is a measure of how spread out the data is. It's calculated by subtracting the mean from each value, squaring the differences, adding them up, and dividing by the number of values minus one. The standard deviation is another measure of how spread out the data is. It's simply the square root of the variance. The mean, variance, and standard deviation are all affected by outliers, or extreme values in the data that can skew the results. Therefore, it's important to check for and handle outliers appropriately. These measures can provide useful insights into the distribution of the data, and can be used to compare different datasets or to identify trends and patterns.

Expected Values for Continuous Variables Part 1!!!

In the case of a continuous variable, the probability of any specific value is zero, so we can't simply use the formula for expected value that we use for discrete variables. Instead, we can use a formula that involves integrating the product of the continuous variable and its probability density function (PDF) over the entire range of possible values. The resulting value represents the center of mass of the distribution, or the point at which the distribution would balance if it were a physical object. The expected value for a continuous variable has important applications in statistics, such as in calculating the mean, variance, and standard deviation of a distribution. The expected value for a continuous variable can be influenced by various factors, such as the shape and spread of the distribution, the probability density function, and any biases or constraints in the experiment. In practice, calculating the expected value for a continuous variable may require using specialized software or numerical integration techniques.

One or Two Tailed P-Values

In the context of hypothesis testing, a one-tailed test is when we are only interested in detecting an effect in one direction, either positive or negative. A two-tailed test is when we are interested in detecting an effect in either direction, positive or negative. The choice between one-tailed and two-tailed test depends on the research question and hypothesis. If we have a strong reason to believe that an effect will only go in one direction, we can use a one-tailed test, which will give us more power to detect the effect. However, if we are uncertain about the direction of the effect, or we want to be able to detect effects in both directions, we should use a two-tailed test. The p-value obtained from a one-tailed test is the probability of observing a result as extreme as the one we obtained, assuming the null hypothesis is true, in the direction specified by the alternative hypothesis. For a two-tailed test, the p-value is the probability of observing a result as extreme or more extreme than the one we obtained, assuming the null hypothesis is true, in either direction. In general, a smaller p-value provides stronger evidence against the null hypothesis. When interpreting p-values, it's important to consider the research question and the choice between a one-tailed or two-tailed test, as well as the significance level and effect size.

Logs (logarithms), Clearly Explained!!!

Logs are the inverse operation of exponentiation, meaning they undo the effect of exponentiation. The logarithm of a number tells you what power you need to raise the base to, in order to get the number. Logs can be useful when working with data that has a wide range of values, because they can help to compress the data and make it easier to analyze. Logs can also be used to transform data that has a skewed distribution into a more normal distribution, which can make it easier to apply statistical tests. Logs have a variety of applications in fields such as finance, science, and engineering, where they are used to express relationships between variables that may not be easily understood in their raw form. Common logarithms (base 10) and natural logarithms (base e) are the two most commonly used types of logarithms.

Maximum Likelihood for the Normal Distribution, step-by-step!!!

In the video "Maximum Likelihood for the Normal Distribution, step-by-step!!!", the StatQuest host explains how to calculate the maximum likelihood estimate (MLE) for the parameters of the normal distribution, which is used to model continuous data that are approximately normally distributed. The video starts by reviewing the probability density function (PDF) of the normal distribution, which has two parameters, the mean (µ) and the standard deviation (σ). The host explains that the MLE for these parameters is the set of values that maximizes the likelihood function, which is the PDF evaluated at the observed data points. The host then demonstrates how to calculate the log-likelihood function for a given data set, which simplifies the calculations and allows for easy maximization. He explains that the log-likelihood function is the sum of the logarithms of the PDF evaluated at each data point. To find the MLE, the host then takes the derivative of the log-likelihood function with respect to each parameter and sets them equal to zero. He solves for µ and σ separately and shows how to use calculus and algebra to arrive at the maximum values. Finally, the host provides examples of how to use the MLE for hypothesis testing, such as comparing the means of two normal distributions or testing for normality of a data set. Overall, the video provides a step-by-step explanation of how to calculate the MLE for the parameters of the normal distribution, which is a fundamental concept in statistics and data analysis.

Odds and Log(Odds), Clearly Explained!!! Part 1

In this StatQuest video, "Odds and Log(Odds), Clearly Explained!!!" it is explained that odds are used to represent the probability of an event happening divided by the probability of it not happening. Odds are frequently used in the context of binary outcomes, where there are only two possible outcomes such as heads or tails in a coin flip. Log odds, or logit, is the logarithm of the odds and is used in logistic regression to model the probability of a binary outcome as a function of one or more predictor variables. By taking the logarithm of the odds, we can convert the range of the odds from (0, infinity) to (-infinity, infinity), making it easier to work with in statistical models. In addition, the video goes on to explain how to calculate the odds and log odds using contingency tables, and how to interpret them in the context of a logistic regression model.

Maximum Likelihood for the Exponential Distribution, Clearly Explained

In this Statquest video, the concept of maximum likelihood estimation (MLE) is explained using the exponential distribution as an example. The exponential distribution is commonly used to model the time between events occurring at a constant rate. MLE is a statistical method that can be used to estimate the parameters of the exponential distribution, namely the rate parameter λ. The video explains that MLE involves finding the value of λ that maximizes the likelihood function, which is the probability of observing the data given a specific value of λ. This involves taking the product of the probabilities of observing each data point, assuming they are independent and identically distributed. The video uses an example of observing the times between a series of events to demonstrate how to calculate the likelihood function. The concept of the log-likelihood function is introduced as a way to simplify the calculations involved in MLE. The video explains that taking the logarithm of the likelihood function does not change the location of the maximum value, but it does change the shape of the function to make it easier to differentiate and calculate. Finally, the concept of bias in MLE is briefly discussed, which refers to the tendency of the estimated value to systematically differ from the true value. The video concludes by emphasizing that MLE is a powerful and flexible tool for estimating parameters of statistical models, but it is important to be aware of its limitations and potential biases.

Why Dividing By N Underestimates the Variance

In this video, the Statquest host explains why dividing by N (the sample size) underestimates the variance of a sample. The main idea is that dividing by N instead of N-1 produces a biased estimate of the variance that can lead to incorrect inferences in statistical analysis. The video starts with an example of two samples with the same mean but different variances. Dividing by N results in a smaller estimate of the variance for the larger sample, even though the true variance is the same for both samples. The host then explains that the problem is caused by the fact that dividing by N uses information from the sample mean to estimate the variance, which leads to a bias. The video then goes on to explain why dividing by N-1 produces an unbiased estimate of the variance. It is because using N-1 degrees of freedom corrects for the bias introduced by using the sample mean to estimate the variance. The video concludes with a brief discussion of the practical implications of using N-1 instead of N, emphasizing the importance of using the correct formula to obtain accurate estimates of the variance and make correct statistical inferences.

Lowess and Loess, Clearly Explained

Lowess (LOcally WEighted Scatterplot Smoothing) and Loess (LOcally Estimated Scatterplot Smoothing) are non-parametric methods for fitting a smooth curve to a scatterplot of data points. Both methods involve dividing the data into windows, fitting a polynomial model within each window, and weighting the contribution of each data point to the local fit according to its distance from the center of the window. The smoothing parameter, which determines the degree of smoothing, can be adjusted to balance between overfitting (too much smoothing) and underfitting (not enough smoothing). Lowess and Loess can be particularly useful for identifying trends and patterns in noisy data, as they can capture non-linear relationships that might be missed by linear regression. Lowess and Loess can be implemented in various statistical software packages, such as R, Python, and MATLAB.

Maximum Likelihood for the Binomial Distribution, Clearly Explained!!!

Maximum Likelihood Estimation (MLE) is a method for finding the values of the parameters of a distribution that maximize the likelihood of observing the data that we have. The Binomial distribution is a discrete probability distribution that models the probability of getting a certain number of successes (or failures) in a fixed number of independent trials. The MLE for the probability of success in a binomial distribution is the proportion of successes observed in the sample. The likelihood function is the probability of observing the data given a set of parameter values. The likelihood function for the binomial distribution is given by the product of the probability of success to the power of the number of successes and the probability of failure to the power of the number of failures. The log-likelihood function is the natural logarithm of the likelihood function. Taking the logarithm makes the calculations easier because we can turn products into sums. To find the MLE, we take the derivative of the log-likelihood function with respect to the parameter of interest, set it equal to zero, and solve for the parameter. The MLE provides us with a point estimate of the parameter, but we also need to assess the uncertainty around the estimate. One way to do this is to find the standard error of the estimate and use it to construct a confidence interval. We can use the MLE to perform hypothesis tests. For example, we can test whether the probability of success is equal to a certain value by comparing the log-likelihood under the null hypothesis to the log-likelihood under the alternative hypothesis. We can use the likelihood ratio test to perform this test.

p-hacking: What it is and how to avoid it!

P-hacking is the practice of exploiting flexibility in data analysis to obtain a p-value below a certain threshold (usually 0.05), regardless of whether the result is meaningful or not. P-hacking can lead to false positives and can undermine the validity and reliability of scientific research. P-hacking can take many forms, such as selecting the best-performing statistical test out of many options, excluding or including certain data points, or transforming the data to achieve a desired result. To avoid p-hacking, it's important to plan the data analysis in advance, define the hypothesis and analysis plan, and stick to it. It's also important to be transparent and report all results, even if they are not statistically significant or contradict the hypothesis. Replication and collaboration can also help avoid p-hacking, as independent researchers can verify the results and identify any inconsistencies or errors. P-values should be interpreted in conjunction with other factors, such as effect size, sample size, and the study design. A small p-value does not necessarily imply a large effect size, and a large sample size can lead to small p-values even if the effect size is small.

Pearson's Correlation, Clearly Explained!!!

Pearson's correlation coefficient (r) is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with 0 indicating no linear relationship. Pearson's correlation is affected by outliers, and may not accurately reflect the strength of the relationship if there are extreme values in the data. Pearson's correlation is only appropriate for assessing the linear relationship between two variables. If the relationship is non-linear, other correlation measures such as Spearman's rank correlation may be more appropriate. Pearson's correlation can be used to test hypotheses about the strength of the relationship between two variables. The null hypothesis is that the correlation coefficient is zero, indicating no linear relationship. Pearson's correlation can be interpreted using a correlation matrix or scatterplot, which can help visualize the relationship between multiple variables.

Power Analysis, Clearly Explained!!!

Power analysis involves calculating statistical power, which is the probability of rejecting the null hypothesis when it is false. High statistical power is desirable, as it increases the likelihood of detecting a true effect. The key factors that influence statistical power include the effect size, significance level, sample size, and variability in the data. Power analysis is useful for determining the sample size needed to achieve a desired level of power for a given effect size and significance level. Power analysis can be conducted using statistical software or online calculators, which typically require inputs such as the effect size, significance level, sample size, and variability in the data. Power analysis can help researchers design studies that are more likely to detect significant effects and avoid false negatives. Power analysis is also important for evaluating the quality of existing studies and determining the reliability of their results.

The Main Ideas behind Probability Distributions

Probability distributions can be either discrete or continuous. Discrete distributions describe outcomes that can only take on certain values, such as the number of heads in a series of coin tosses. Continuous distributions describe outcomes that can take on any value within a certain range, such as the height of a randomly selected person. The shape of a probability distribution can provide insights into the likelihood of different outcomes. For example, a bell-shaped distribution, known as a normal distribution, is common in many natural phenomena, while a skewed distribution can indicate the presence of outliers or a non-normal underlying process. Probability distributions are characterized by certain parameters, such as the mean and standard deviation, which can be used to calculate probabilities and other descriptive statistics. Probability distributions can be used to model real-world phenomena and make predictions about future events.

Quantile-Quantile Plots (QQ plots), Clearly Explained (IMPORTANT-shows how to use distributions)

QQ plots are a way to compare the distribution of a sample to a theoretical distribution, such as the normal distribution. The QQ plot plots the quantiles of the sample against the quantiles of the theoretical distribution. If the sample is normally distributed, the points on the QQ plot should form a straight line. The y-axis of the QQ plot represents the values of the sample, while the x-axis represents the corresponding quantiles of the theoretical distribution. If the sample is not normally distributed, the QQ plot will show a deviation from a straight line. For example, if the sample has heavy tails or is skewed, the points on the QQ plot will deviate from the straight line. QQ plots can be used to compare the distribution of different samples to each other or to a theoretical distribution. QQ plots can be used in combination with hypothesis testing to assess whether a sample is normally distributed.

Quantile Normalization, Clearly Explained!!!

Quantile normalization is a method for normalizing two or more data sets with different distributions so that they have the same distribution. This is important when combining data sets for analysis. Quantile normalization involves the following steps: Rank the values in each data set from smallest to largest Calculate the mean of the ranked values across all data sets Replace the original values in each data set with the mean of the corresponding ranks across all data sets The resulting normalized data sets will have the same distribution, which is the average of the distributions of the original data sets. Quantile normalization is useful in many applications, such as gene expression analysis and microarray data analysis. One potential downside of quantile normalization is that it assumes that most of the genes are not differentially expressed. If this assumption is violated, the results of the normalization may be inaccurate.

R-squared, Clearly Explained!!!

R-squared measures how well the regression line fits the data by calculating the proportion of the total variance in the dependent variable that is explained by the independent variable(s) included in the model. The R-squared value ranges from 0 to 1, with 1 indicating that all the variance in the dependent variable is explained by the independent variable(s) included in the model, and 0 indicating that none of the variance is explained. The video uses a simple example with data points scattered around a regression line to explain the concept of R-squared. It also illustrates how R-squared can be misleading if the model is overfitted or if there are outliers in the data. The video concludes with a discussion on the limitations of R-squared, including its inability to determine causality and its reliance on the choice of independent variable(s) included in the model.

Sample Size and Effective Sample Size, Clearly Explained

Sample size refers to the number of observations or individuals in a sample. A larger sample size generally results in a more precise estimate of a population parameter, such as the mean or standard deviation. The effective sample size takes into account the correlation between observations in a sample, which can affect the precision of estimates. When observations are independent, the effective sample size is equal to the actual sample size. When observations are correlated, the effective sample size will be smaller than the actual sample size, reducing the precision of estimates. The effective sample size can be calculated using the formula N_eff = N(1 - ρ), where N is the actual sample size and ρ is the average correlation between observations. The effective sample size is important to consider when designing experiments or studies and when interpreting statistical results. In some cases, increasing the sample size may not improve the precision of estimates if the observations are highly correlated, and alternative methods such as adjusting for correlation or using more sophisticated statistical models may be necessary.

Sampling from a Distribution, Clearly Explained

Sampling from a distribution involves randomly selecting values from a population or a probability distribution. The goal of sampling is to obtain a representative subset of the population that can be used to make inferences or predictions about the population as a whole. There are different types of sampling techniques, including simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Each technique has its own advantages and disadvantages, and the choice of technique depends on the specific research question and available resources. Sampling can introduce uncertainty and variability into the results, which can be quantified using statistical measures such as confidence intervals and standard errors. Sampling is widely used in many areas of science, social science, and engineering, and is essential for making reliable inferences and predictions based on data.

Statistical Power, Clearly Explained!!!

Statistical power is the probability of rejecting the null hypothesis when it is false. In other words, it measures the ability of a statistical test to detect a true effect. Statistical power depends on several factors, such as the effect size, sample size, significance level, and variability in the data. Increasing any of these factors can increase statistical power. Statistical power is typically calculated before conducting a study, as it can help determine the sample size needed to achieve a desired level of power. Low statistical power can lead to false negatives (type II errors), which occur when a true effect is not detected due to inadequate sample size or other factors. A common way to increase statistical power is to increase the sample size, as this reduces the variability in the data and increases the likelihood of detecting a true effect. Statistical power is an important concept in experimental design and is often used to evaluate the quality of a study and its results.

Quantiles and Percentiles, Clearly Explained

The Statquest video "Quantiles and Percentiles, Clearly Explained" explains the concepts of quantiles and percentiles, which are statistical measures used to describe the distribution of a dataset. Quantiles divide the data into equal portions, while percentiles divide the data into 100 equal portions. The video provides an example of a dataset and shows how to calculate various quantiles and percentiles. It also explains the differences between quartiles, deciles, and percentiles, and how they can be used to summarize and compare different datasets. Additionally, the video discusses how to interpret and use percentile ranks, which measure the relative position of a value within a dataset. Overall, the video provides a clear and accessible introduction to the concepts of quantiles and percentiles in statistics.

The Binomial Distribution and Test, Clearly Explained

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. The parameters of the binomial distribution are the number of trials (n) and the probability of success (p), which together determine the mean and variance of the distribution. The binomial distribution can be used to calculate the probability of observing a specific number of successes in a given number of trials. The binomial test is a statistical test that can be used to test the null hypothesis that the probability of success is equal to a specific value, against an alternative hypothesis that the probability of success is different from that value. The binomial test involves calculating the probability of observing the number of successes that were actually observed, or a more extreme value, under the null hypothesis. The p-value is the probability of observing the test statistic or a more extreme value under the null hypothesis, and a small p-value indicates evidence against the null hypothesis. The significance level, or alpha, is the threshold at which we reject the null hypothesis based on the p-value. The binomial distribution and test have important applications in fields such as biology, medicine, and social sciences, where we often want to test whether a certain outcome is due to chance or some other factor.

The Central Limit Theorem, Clearly Explained

The central limit theorem states that if we take repeated samples of size n from any population with a finite mean and variance, the sampling distribution of the means will be approximately normal, regardless of the shape of the original population. The central limit theorem is important because it allows us to make statistical inferences about a population based on a sample, even when we don't know the distribution of the population. The sampling distribution of the mean has a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. As the sample size increases, the sampling distribution of the mean becomes more and more normal, regardless of the shape of the original population. The central limit theorem has important applications in fields such as quality control, finance, and social sciences, where we often want to make inferences about a population based on a sample. The central limit theorem can be used to estimate the mean of a population by calculating the mean of a sample and using the sampling distribution of the mean to construct a confidence interval.

Expected Values, Main Ideas!!!

The expected value is a measure of the central tendency of a random variable, and represents the average value we would expect to observe over many trials of an experiment. The expected value can be calculated by multiplying each possible value of the random variable by its corresponding probability, and then summing these products. The expected value is not necessarily equal to any of the actual values that the random variable can take, but rather represents a hypothetical average value. The expected value has important applications in statistics, such as in calculating the mean, variance, and standard deviation of a distribution. ---------------------------------------------------------------------- Let's say you are planning a picnic with your friends, and you want to bring some snacks. You have two options: bringing a bag of apples or a bag of oranges. You know that your friends like both fruits, but you are not sure which one they would prefer. Now, let's assign some values to each option. You know that a bag of apples costs $5 and has a 70% chance of being enjoyed by your friends, while a bag of oranges costs $6 and has a 90% chance of being enjoyed by your friends. To calculate the expected value of each option, you would multiply the probability of each outcome by its corresponding value and then add them up. So, for the bag of apples: Expected value of the bag of apples = (0.7 x $5) + (0.3 x $0) = $3.50 This means that, on average, you can expect to get $3.50 worth of enjoyment from bringing the bag of apples. For the bag of oranges: Expected value of the bag of oranges = (0.9 x $6) + (0.1 x $0) = $5.40 This means that, on average, you can expect to get $5.40 worth of enjoyment from bringing the bag of oranges. Therefore, based on the expected values, it would be better to bring the bag of oranges, as it has a higher expected value. In summary, expected value is a way to calculate the average outcome of a situation based on probabilities and values assigned to each outcome, similar to how you would calculate the expected value of the snacks you bring to a picnic based on the probability of your friends enjoying them and their cost.

Maximum Likelihood, clearly explained!!!

The flash card "Probability is not likelihood. Find out Why" actually has the pic of the StatQuest slides. The video "Maximum Likelihood, Clearly Explained!!!" provides an overview of the concept of maximum likelihood and how it can be used in statistical modeling. Maximum likelihood is a method for estimating the parameters of a statistical model that maximizes the likelihood function, which is the probability of observing the data given a set of parameter values. The video explains how to use the likelihood function to estimate the values of the parameters that make the observed data most likely. The video also illustrates how to calculate the likelihood function for a simple example using a coin toss experiment. Finally, the video emphasizes the importance of checking the assumptions of the statistical model and the limitations of the maximum likelihood approach.

The Normal Distribution, Clearly Explained

The normal distribution is characterized by its bell-shaped curve, which is symmetric around the mean. The mean and standard deviation of a normal distribution can be used to calculate the probability of certain outcomes, and can be used to make predictions about future events. The empirical rule, also known as the 68-95-99.7 rule, states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. The normal distribution is widely used in statistical inference, hypothesis testing, and modeling real-world phenomena. There are several properties of the normal distribution that make it useful for many applications, including its symmetry, its mathematical tractability, and its prevalence in many natural phenomena.

The Standard error, Clearly Explained!!!

The standard error (SE) is an estimate of the variability of the sample mean from the population mean. SE is calculated by dividing the sample standard deviation by the square root of the sample size. SE decreases as the sample size increases, indicating that larger samples produce more accurate estimates of the population mean. The SE can be used to calculate confidence intervals, which provide a range of values in which the true population mean is likely to fall with a certain level of confidence. The SE is not the same as the standard deviation, which measures the variability of the data points in the sample. The SE is a measure of how well the sample mean estimates the population mean, while the standard deviation is a measure of how spread out the individual data points are.

The Main Ideas of Fitting a Line to Data (The Main Ideas of Least Squares and Linear Regression)

The video begins by introducing the concept of linear regression and its main goal, which is to fit a line to the data in a way that minimizes the distance between the line and the data points. The video then discusses the concept of residual sum of squares, which is the sum of the squares of the distances between the predicted values of the line and the actual data points. The video then explains how the least squares method is used to find the line that minimizes the residual sum of squares. This involves finding the slope and intercept of the line that minimize the sum of the squared residuals. The video demonstrates this process using an example data set. The video also discusses the concept of correlation coefficient (r), which measures the strength and direction of the linear relationship between two variables. It is shown that the square of the correlation coefficient (r-squared) is equal to the proportion of the variation in the dependent variable that is explained by the independent variable. Finally, the video discusses some limitations and assumptions of linear regression, such as linearity, independence, homoscedasticity, and normality of residuals. The importance of checking these assumptions is emphasized to ensure the validity of the linear regression model.

Odds Ratios and Log(Odds Ratios), Clearly Explained!!!

The video starts with a brief recap of odds and how they can be calculated for binary outcomes. The host then goes on to define the odds ratio, which compares the odds of an event occurring in one group relative to another group. The OR can be calculated by dividing the odds in one group by the odds in the other group. Next, the video discusses how to interpret the OR. If the OR is 1, it means that the odds of the event occurring in both groups are equal. If the OR is greater than 1, it means that the odds of the event occurring are higher in the first group compared to the second group. If the OR is less than 1, it means that the odds of the event occurring are higher in the second group compared to the first group. The host then explains how to calculate the logOR, which is the natural logarithm of the OR. The logOR is preferred in statistical analysis because it has desirable properties, such as being symmetric around 0 and having an approximately normal distribution. The video also explains how to interpret the logOR. If the logOR is 0, it means that the odds of the event occurring are equal in both groups. If the logOR is positive, it means that the odds of the event occurring are higher in the first group compared to the second group. If the logOR is negative, it means that the odds of the event occurring are higher in the second group compared to the first group. Finally, the video gives an example of how to calculate and interpret the OR and logOR using a hypothetical dataset.