Start with your data matrix \(X =\begin{bmatrix}2 & 0 \\0 & 2 \\3 & 3 \\\end{bmatrix} \) Step 1: Centering (standard for PCA): \( X_{centered} =X – \bar{X} \) Sometimes you also scale (normalize) each feature to have unit variance (standard deviation = 1). This is called standardization. \( X_{scaled} = \frac{X – \bar{X}}{\sigma} \) … Read more
1. Definition A Z-score, also known as a standard score, measures how many standard deviations a data point is from the mean of a dataset. It is used in statistics to standardize data, compare different distributions, and detect outliers. A positive Z-score means the data point is above the mean, while a negative Z-score indicates … Read more
Chi-Square Critical Value Table (Distribution Table) df p = 0.995 p = 0.99 p = 0.975 p = 0.95 p = 0.9 p = 0.75 p = 0.5 p = 0.25 p = 0.1 p = 0.05 p = 0.025 p = 0.01 p = 0.005 1 0.000 0.000 0.001 0.004 0.016 0.102 0.455 1.323 … Read more
T-Distribution Table Student’s T-Distribution Table This table shows critical values of the t-distribution for both one-tailed and two-tailed tests. df α = 0.1 α = 0.05 α = 0.01 One-Tail Two-Tail One-Tail Two-Tail One-Tail Two-Tail 1 3.0777 6.3138 6.3138 12.7062 31.8205 63.6567 2 1.8856 2.9200 2.9200 4.3027 6.9646 9.9248 3 1.6377 2.3534 2.3534 3.1824 4.5407 … Read more
Central Limit Theorem (CLT) states that when you take a sufficiently large number of independent random samples from a population (regardless of the population’s original distribution), the sampling distribution of the sample mean will approach a normal distribution. This tutorial uses two examples showing while the original data do not follow normal distribution (e.g., uniform … Read more
Two-tail Standard Normal Z-table Two-tail Standard Normal Z-Table |Z| 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 1.0000 0.9920 0.9840 0.9761 0.9681 0.9601 0.9522 0.9442 0.9362 0.9283 0.1 0.9203 0.9124 0.9045 0.8966 0.8887 0.8808 0.8729 0.8650 0.8572 0.8493 0.2 0.8415 0.8337 0.8259 0.8181 0.8103 0.8026 0.7949 0.7872 0.7795 0.7718 0.3 0.7642 0.7566 … Read more
The following is the F Distribution Critical Value Table. The numbers in the table are critical values for F distribution with probably p in right tail of the distribution (see the figure illustration below). F Distribution Critical Values df2 p-value df1 (Degrees of freedom in numerator) 1 2 3 4 5 6 7 8 12 … Read more
This tutorial shows how you can calcuate sample size for one variable chi-square test here. The example in this tutorial is that a marketing agency tests two versions of packaging (A version vs. B Version). The null hypothesis is that there is no difference between these two versions, and the alternative hypothesis is that there … Read more
This tutorial is to compare OLS (Ordinary Least Square) and Maximum Likelihood Estimate (MLE) in linear regression. We are going to use simple linear regression as examples here. Most of the conclusions can be directly extended into general linear regressions. OLS in Linear Regression Coefficients The principle of ordinary least squares is to minimize the … Read more
This tutorial is going to explain what Maximum Likelihood Estimation (MLE) is and how Maximum Likelihood Estimation (MLE) can be used in linear regression. Basics of Maximum Likelihood Estimation Before discussing about linear regression, we need to have a basic idea of MLE. In particular, assume that \( y_i \) are all independently and identically … Read more