{"id":34973,"date":"2024-10-16T01:07:57","date_gmt":"2024-10-16T05:07:57","guid":{"rendered":"https:\/\/statanalytica.com\/blog\/?p=34973"},"modified":"2024-10-16T01:15:50","modified_gmt":"2024-10-16T05:15:50","slug":"characteristics-of-binomial-distribution","status":"publish","type":"post","link":"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/","title":{"rendered":"Key Characteristics of Binomial Distribution: A Comprehensive Guide"},"content":{"rendered":"\n<p>Probability distributions are important in statistics because they explain the likelihood of different outcomes in a series of experiments or events. The binomial distribution is a popular probability distribution that plays an important role in statistical analysis and decision-making. Its simplicity and effectiveness make it a useful tool in a variety of fields. From quality control in manufacturing to predicting outcomes in medical testing, the binomial distribution is essential for analyzing situations with two possible outcomes: success or failure. This blog will dive deep into the characteristics of binomial distribution, formulas, and real-life applications.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"what-is-a-binomial-distribution\"><\/span>What is a Binomial Distribution?<span class=\"ez-toc-section-end\"><\/span><\/h2><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-6a09a2845c03b\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #ff5104;color:#ff5104\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #ff5104;color:#ff5104\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-6a09a2845c03b\" checked aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#what-is-a-binomial-distribution\" >What is a Binomial Distribution?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#key-characteristics-of-binomial-distribution\" >Key Characteristics of Binomial Distribution<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#1-fixed-number-of-trials-n\" >1. Fixed Number of Trials (n)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#2-two-possible-outcomes-success-or-failure\" >2. Two Possible Outcomes (Success or Failure)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#3-constant-probability-of-success-p\" >3. Constant Probability of Success (p)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#4-independence-of-trials\" >4. Independence of Trials<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#mean-variance-and-standard-deviation-of-binomial-distribution\" >Mean, Variance, and Standard Deviation of Binomial Distribution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#real-life-examples-of-binomial-distribution\" >Real-Life Examples of Binomial Distribution<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#coin-toss-experiments\" >Coin Toss Experiments<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#quality-control-in-manufacturing\" >Quality Control in Manufacturing<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#medical-testing\" >Medical Testing<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#binomial-distribution-vs-other-distributions\" >Binomial Distribution vs Other Distributions<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#binomial-distribution-vs-normal-distribution\" >Binomial Distribution vs. Normal Distribution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#binomial-distribution-vs-poisson-distribution\" >Binomial Distribution vs. Poisson Distribution<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#conclusion\" >Conclusion<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#faqs\" >FAQs<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#what-are-the-main-conditions-required-for-a-binomial-distribution\" >What are the main conditions required for a binomial distribution?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/statanalytica.com\/blog\/characteristics-of-binomial-distribution\/#how-is-the-binomial-distribution-used-in-real-life-applications\" >How is the binomial distribution used in real-life applications?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n\n\n\n\n<p>One discrete probability distribution that summarizes the chances of getting a fixed number of successes in a series of independent and identical trials is the binomial distribution. It would be best if you had a constant success probability (p), a fixed number of trials (n), and two possible outcomes (success or failure). This is the binomial distribution formula:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"830\" height=\"299\" src=\"https:\/\/statanalytica.com\/blog\/wp-content\/uploads\/2024\/10\/Binomial-Distribution.png\" alt=\"\" class=\"wp-image-34974\" srcset=\"https:\/\/statanalytica.com\/blog\/wp-content\/uploads\/2024\/10\/Binomial-Distribution.png 830w, https:\/\/statanalytica.com\/blog\/wp-content\/uploads\/2024\/10\/Binomial-Distribution-300x108.png 300w, https:\/\/statanalytica.com\/blog\/wp-content\/uploads\/2024\/10\/Binomial-Distribution-768x277.png 768w, https:\/\/statanalytica.com\/blog\/wp-content\/uploads\/2024\/10\/Binomial-Distribution-150x54.png 150w\" sizes=\"(max-width: 830px) 100vw, 830px\" \/><\/figure>\n\n\n\n<p>The distribution must meet certain conditions to be classified as binomial. These include a fixed number of trials, independent trials, and two possible outcomes for each trial, with the probability of success remaining constant across all trials.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"key-characteristics-of-binomial-distribution\"><\/span>Key Characteristics of Binomial Distribution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"1-fixed-number-of-trials-n\"><\/span>1. Fixed Number of Trials (n)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>One of the unique characteristics of a binomial distribution is that the number of trials, denoted by n, is constant throughout the experiment. This means that the total number of attempts to observe a specific outcome (success or failure) is predetermined and remains constant throughout the process. For example, if an experiment requires flipping a coin ten times, the number of trials is set at ten. This consistency enables precise probability calculations because the number of opportunities for success is clearly defined.<\/p>\n\n\n\n<p>The fixed nature of trials is essential for ensuring that the probability distribution accurately reflects the situation. Without this condition, the probabilities would become dynamic, making it impossible to calculate the outcomes with the binomial formula. Fixed trials are critical in scenarios such as quality testing, where a set number of products are inspected for flaws, and clinical trials, where a fixed number of patients are assessed for treatment efficacy. Without this fixed structure, the statistical model couldn&#8217;t make reliable predictions.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"2-two-possible-outcomes-success-or-failure\"><\/span>2. Two Possible Outcomes (Success or Failure)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Another key Characteristics of Binomial Distribution is that each trial has only two possible outcomes: success or failure. These outcomes are mutually exclusive and collectively exhaustive, which means that for any given trial, one of them will be successful or unsuccessful, but not both. For example, in the context of a coin toss, the two outcomes are heads (success) and tails (failure).<\/p>\n\n\n\n<p>This binary nature is a fundamental feature of the binomial distribution, which reduces complex events to manageable success\/failure conditions. In real-world applications, success does not always imply something positive; it can simply refer to the outcome of interest. In medical testing, for example, success may refer to a patient responding to a treatment, whereas failure refers to no response.&nbsp;<\/p>\n\n\n\n<p>Similarly, in quality control, success could mean producing a defect-free product, whereas failure would mean discovering a defect. This clear division of outcomes allows for precise probability calculations, making the binomial distribution widely applicable in binary decision-making situations.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"3-constant-probability-of-success-p\"><\/span>3. Constant Probability of Success (p)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>To apply a binomial distribution, the probability of success (p) must be constant for each trial, regardless of previous successes or failures. For example, in a fair coin toss, the probability of getting heads (success) is 0.5, and this probability remains constant regardless of how many heads or tails occurred in previous tosses.<\/p>\n\n\n\n<p>This condition of constant probability guarantees that the distribution&#8217;s outcomes are predictable and that each trial has the same chance of success. In practice, this may apply to scenarios such as a manufacturing line where the probability of producing a defective product (failure) remains constant for each item, assuming the process does not change.&nbsp;<\/p>\n\n\n\n<p>The constant probability is important because it keeps the trials separate from one another, preventing previous results from influencing subsequent ones. Without this condition, the binomial formula for calculating probabilities would be invalid because the probability of each trial would vary depending on the outcomes of previous trials.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"4-independence-of-trials\"><\/span>4. Independence of Trials<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The independence of trials is one of the key Characteristics of Binomial Distribution, which means that the outcome of one trial does not affect the outcome of another. In other words, previous trial results have no bearing on future trials, and each trial is conducted independently. For example, in a series of ten coin tosses, the outcome of any one toss (heads or tails) has no bearing on the outcome of the next toss. Regardless of how many heads or tails have been tossed previously, the probability of getting heads remains 0.5.<\/p>\n\n\n\n<p>Independence is an important condition because it guarantees that each trial is statistically independent, which is required when using the binomial distribution. With this independence, the results would fit the binomial model, necessitating the use of a different distribution. Independence is important in experiments where external factors do not interfere with the trials.&nbsp;<\/p>\n\n\n\n<p>For example, when testing the success rate of a vaccine in different patients, each patient&#8217;s response is independent of the others, allowing the binomial distribution to predict success rates. Similarly, in product testing, each product can be evaluated independently because the quality of one item has no bearing on the next.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"mean-variance-and-standard-deviation-of-binomial-distribution\"><\/span>Mean, Variance, and Standard Deviation of Binomial Distribution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>To better understand the distribution, it&#8217;s important to know its key metrics\u2014mean, variance, and standard deviation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mean (\u03bc = np):<br><\/strong>To get the average or predicted value, we multiply the number of trials (n) by the success probability (p). If you know the average number of trials, you may estimate the average number of successes.<\/li>\n\n\n\n<li><strong>Variance (\u03c3\u00b2 = npq):<br><\/strong>With q = 1 &#8211; p representing the failure probability, the variance can be expressed as npq, which quantifies the data&#8217;s dispersion. Here, we can see how much the success rate varies from trial to trial.<\/li>\n\n\n\n<li><strong>Standard Deviation (\u03c3 = \u221anpq):<br><\/strong>A measure of the dispersion of results around the mean, the standard deviation is calculated by taking the square root of the variance. A lower <a href=\"https:\/\/statanalytica.com\/blog\/standard-deviation-in-excel\/\" target=\"_blank\" rel=\"noreferrer noopener\">standard deviation<\/a> indicates that the outcomes are close to the expected value, while a higher standard deviation suggests more variability.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"real-life-examples-of-binomial-distribution\"><\/span>Real-Life Examples of Binomial Distribution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>After knowing the characteristics of binomial distribution, you should also learn the real life examples of binomial distribution. Mentioned below are some Real-Life Examples of Binomial Distribution:-<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"coin-toss-experiments\"><\/span>Coin Toss Experiments<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Repeatedly tossing a coin is one of the most basic applications of binomial distribution. For every trial, a head indicates success and a tail indicates failure. Every flip has its own unique probability and is completely separate from the others.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"quality-control-in-manufacturing\"><\/span>Quality Control in Manufacturing<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The binomial distribution is a useful tool for quality control in manufacturing. If a manufacturing facility estimates a 2% defect rate for a batch of 1000 items, for instance, the binomial distribution can be used to forecast the number of defective products in that batch.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"medical-testing\"><\/span>Medical Testing<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The binomial distribution is often used to analyze medical test results, such as determining a treatment&#8217;s success rate. For example, if a treatment has a 70% success rate, the distribution can predict how many patients out of a given number are likely to recover.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"binomial-distribution-vs-other-distributions\"><\/span>Binomial Distribution vs Other Distributions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>It is crucial to recognize when a binomial distribution is more appropriate than, say, a normal or Poisson distribution.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"binomial-distribution-vs-normal-distribution\"><\/span>Binomial Distribution vs. Normal Distribution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>In contrast to continuous data, which is often distributed using the normal distribution, discrete outcomes (success or failure) are best handled by the binomial distribution. In theory, the binomial distribution can be improved to a normal distribution with enough trials.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"binomial-distribution-vs-poisson-distribution\"><\/span>Binomial Distribution vs. Poisson Distribution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>A defined number of separate trials is utilized with the binomial distribution, whereas events that happen independently across a fixed duration or region are used with the Poisson distribution. Whenever there is a steady rate of irregular events, such as calls to a customer service center, use Poisson.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"conclusion\"><\/span>Conclusion<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Understanding the key characteristics of binomial distribution is essential for statistical analysis in various fields. Whether you&#8217;re managing quality control in a manufacturing process or analyzing success rates in medical trials, the binomial distribution provides a reliable way to predict outcomes based on probability.&nbsp;<\/p>\n\n\n\n<p>By mastering the mean, variance, and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Standard_deviation\" target=\"_blank\" rel=\"noreferrer noopener\">standard deviation<\/a> formulas, you can make more informed decisions and improve the accuracy of your data analysis. This distribution remains a fundamental tool for decision-making in the modern world, helping businesses and researchers optimize their operations and predictions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"faqs\"><\/span>FAQs<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1729054997844\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"what-are-the-main-conditions-required-for-a-binomial-distribution\"><\/span>What are the main conditions required for a binomial distribution?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>For a binomial distribution to be applicable, four key conditions must be met: (1) A fixed number of trials (n), (2) each trial must have only two possible outcomes (success or failure), (3) the probability of success (p) must remain constant across all trials, and (4) each trial must be independent, meaning the outcome of one trial does not influence the outcome of others.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1729055007971\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"how-is-the-binomial-distribution-used-in-real-life-applications\"><\/span>How is the binomial distribution used in real-life applications?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>The binomial distribution is commonly used in various fields, such as quality control (e.g., determining the probability of defective products in manufacturing), medical testing (e.g., evaluating the success rate of a treatment), and even simple experiments like flipping a coin. It helps analyze situations where there are repeated independent trials with two possible outcomes and a constant probability of success.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Probability distributions are important in statistics because they explain the likelihood of different outcomes in a series of experiments or events. The binomial distribution is a popular probability distribution that plays an important role in statistical analysis and decision-making. Its simplicity and effectiveness make it a useful tool in a variety of fields. From quality [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":34975,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[76],"tags":[4364],"class_list":["post-34973","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-statistics","tag-characteristics-of-binomial-distribution"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/34973","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/comments?post=34973"}],"version-history":[{"count":2,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/34973\/revisions"}],"predecessor-version":[{"id":34978,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/34973\/revisions\/34978"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/media\/34975"}],"wp:attachment":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/media?parent=34973"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/categories?post=34973"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/tags?post=34973"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}