{"id":226256,"date":"2024-06-28T11:04:30","date_gmt":"2024-06-28T11:04:30","guid":{"rendered":"https:\/\/namso-gen.co\/blog\/?p=226256"},"modified":"2024-06-28T11:04:30","modified_gmt":"2024-06-28T11:04:30","slug":"what-is-probability-normal-distribution-below-a-certain-value-in-matlab","status":"publish","type":"post","link":"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/","title":{"rendered":"What is probability normal distribution below a certain value in MATLAB?"},"content":{"rendered":"<p>The probability below a certain value in a normal distribution can be determined using MATLAB, a powerful programming language widely used for scientific computing and data analysis. In MATLAB, the probability of observing a value below a certain threshold in a normal distribution can be calculated using the cumulative distribution function (CDF) or the normcdf function.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_62 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title \" >Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" 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: #999;color:#999\" 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><\/span><\/a><\/span><\/div>\n<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:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#Calculating_the_probability_using_the_cumulative_distribution_function_CDF\" title=\"Calculating the probability using the cumulative distribution function (CDF)\">Calculating the probability using the cumulative distribution function (CDF)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#What_do_the_mean_and_standard_deviation_represent_in_a_normal_distribution\" title=\"What do the mean and standard deviation represent in a normal distribution?\">What do the mean and standard deviation represent in a normal distribution?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#Can_the_value_x_be_any_real_number\" title=\"Can the value x be any real number?\">Can the value x be any real number?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#What_is_the_range_of_the_probability_calculated\" title=\"What is the range of the probability calculated?\">What is the range of the probability calculated?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#Example_Usage\" title=\"Example Usage\">Example Usage<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#What_is_the_difference_between_using_normcdf_and_using_CDF_directly\" title=\"What is the difference between using normcdf and using CDF directly?\">What is the difference between using normcdf and using CDF directly?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#Can_the_normcdf_function_handle_vector_input\" title=\"Can the normcdf function handle vector input?\">Can the normcdf function handle vector input?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#Using_the_inverse_cumulative_distribution_function_invCDF\" title=\"Using the inverse cumulative distribution function (invCDF)\">Using the inverse cumulative distribution function (invCDF)<\/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:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#Can_the_norminv_function_handle_vector_input\" title=\"Can the norminv function handle vector input?\">Can the norminv function handle vector input?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#How_can_I_verify_the_calculated_probabilities\" title=\"How can I verify the calculated probabilities?\">How can I verify the calculated probabilities?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#1_Can_MATLAB_be_used_for_other_probability_distributions\" title=\"1. Can MATLAB be used for other probability distributions?\">1. Can MATLAB be used for other probability distributions?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#2_What_if_I_want_to_calculate_the_probability_between_two_threshold_values\" title=\"2. What if I want to calculate the probability between two threshold values?\">2. What if I want to calculate the probability between two threshold values?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#3_Is_MATLAB_the_only_programming_language_that_can_calculate_probabilities_in_a_normal_distribution\" title=\"3. Is MATLAB the only programming language that can calculate probabilities in a normal distribution?\">3. Is MATLAB the only programming language that can calculate probabilities in a 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:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#4_What_if_the_threshold_value_is_outside_the_range_of_observed_data\" title=\"4. What if the threshold value is outside the range of observed data?\">4. What if the threshold value is outside the range of observed data?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#5_Can_MATLAB_handle_non-normal_distributions\" title=\"5. Can MATLAB handle non-normal distributions?\">5. Can MATLAB handle non-normal distributions?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#6_How_are_probabilities_in_a_normal_distribution_related_to_the_percentile_ranks\" title=\"6. How are probabilities in a normal distribution related to the percentile ranks?\">6. How are probabilities in a normal distribution related to the percentile ranks?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#7_Is_it_possible_to_calculate_probabilities_for_multi-dimensional_normal_distributions_in_MATLAB\" title=\"7. Is it possible to calculate probabilities for multi-dimensional normal distributions in MATLAB?\">7. Is it possible to calculate probabilities for multi-dimensional normal distributions in MATLAB?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#8_Can_I_use_normcdf_to_calculate_the_probability_of_a_value_above_a_certain_threshold\" title=\"8. Can I use normcdf to calculate the probability of a value above a certain threshold?\">8. Can I use normcdf to calculate the probability of a value above a certain threshold?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#9_How_can_I_plot_the_normal_distribution_with_the_calculated_probability\" title=\"9. How can I plot the normal distribution with the calculated probability?\">9. How can I plot the normal distribution with the calculated probability?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#10_Is_MATLAB_suitable_for_large-scale_data_analysis\" title=\"10. Is MATLAB suitable for large-scale data analysis?\">10. Is MATLAB suitable for large-scale data analysis?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#11_Can_I_calculate_probabilities_for_non-symmetric_normal_distributions\" title=\"11. Can I calculate probabilities for non-symmetric normal distributions?\">11. Can I calculate probabilities for non-symmetric normal distributions?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#12_What_if_the_mean_or_standard_deviation_of_the_normal_distribution_is_not_known\" title=\"12. What if the mean or standard deviation of the normal distribution is not known?\">12. What if the mean or standard deviation of the normal distribution is not known?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Calculating_the_probability_using_the_cumulative_distribution_function_CDF\"><\/span>Calculating the probability using the cumulative distribution function (CDF)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The cumulative distribution function (CDF) represents the probability that a random variable X is less than or equal to a given value x. In MATLAB, the normcdf function can be used to calculate the CDF of a normal distribution. The syntax to calculate the probability below a certain value using the CDF is as follows:<\/p>\n<p>&#8220;`matlab<br \/>\nprob = normcdf(x, mu, sigma)<br \/>\n&#8220;`<\/p>\n<p>Where `x` is the threshold value, `mu` is the mean of the normal distribution, and `sigma` is the standard deviation. The variable `prob` represents the probability of observing a value below `x` in the normal distribution.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"What_do_the_mean_and_standard_deviation_represent_in_a_normal_distribution\"><\/span>What do the mean and standard deviation represent in a normal distribution?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nThe mean (mu) represents the central tendency or average of the distribution, while the standard deviation (sigma) indicates the spread or variability of the distribution.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Can_the_value_x_be_any_real_number\"><\/span>Can the value x be any real number?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, the value `x` can be any real number. It represents the threshold below which we want to calculate the probability in the normal distribution.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"What_is_the_range_of_the_probability_calculated\"><\/span>What is the range of the probability calculated?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nThe probability calculated using the CDF ranges from 0 to 1, where 0 represents no probability and 1 represents a certainty.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Example_Usage\"><\/span>Example Usage<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let&#8217;s consider an example to further understand how to calculate the probability below a certain value in a normal distribution using MATLAB.<\/p>\n<p>Suppose we have a normal distribution with a mean of 10 and a standard deviation of 2. We want to find the probability of observing a value below 8. To calculate this, we can use the normcdf function in MATLAB:<\/p>\n<p>&#8220;`matlab<br \/>\nmu = 10;<br \/>\nsigma = 2;<br \/>\nx = 8;<br \/>\nprob = normcdf(x, mu, sigma);<br \/>\n&#8220;`<\/p>\n<p>The calculated probability will be stored in the variable `prob`. In this case, the probability of observing a value below 8 in the given normal distribution is approximately 0.1587.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"What_is_the_difference_between_using_normcdf_and_using_CDF_directly\"><\/span>What is the difference between using normcdf and using CDF directly?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nThe normcdf function is a convenient way to calculate the CDF of a normal distribution in MATLAB. It automatically standardizes the threshold value (`x`) using the provided mean and standard deviation. Using the CDF directly would require manual calculation of the z-score and looking up the probability in a standard normal distribution table.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Can_the_normcdf_function_handle_vector_input\"><\/span>Can the normcdf function handle vector input?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, the normcdf function can handle vector input for the threshold value (`x`), mean (`mu`), and standard deviation (`sigma`). In such cases, it will return a vector of probabilities corresponding to each value in the input vectors.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Using_the_inverse_cumulative_distribution_function_invCDF\"><\/span>Using the inverse cumulative distribution function (invCDF)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Alternatively, instead of finding the probability below a certain threshold, we can also find the value that corresponds to a given probability using the inverse cumulative distribution function (invCDF) or the norminv function in MATLAB.<\/p>\n<p>The syntax to calculate the value below which a certain probability lies in a normal distribution is as follows:<\/p>\n<p>&#8220;`matlab<br \/>\nthreshold = norminv(prob, mu, sigma)<br \/>\n&#8220;`<\/p>\n<p>Where `prob` is the probability and `threshold` is the value corresponding to that probability.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Can_the_norminv_function_handle_vector_input\"><\/span>Can the norminv function handle vector input?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nSimilar to the normcdf function, the norminv function can also handle vector input for the probability (`prob`), mean (`mu`), and standard deviation (`sigma`). It will return a vector of threshold values corresponding to each probability.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"How_can_I_verify_the_calculated_probabilities\"><\/span>How can I verify the calculated probabilities?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nOne way to verify the calculated probabilities is to generate a large number of random samples from the same normal distribution using the normrnd function. Then, count the proportion of samples that fall below the threshold value `x`. This proportion should be close to the calculated probability.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"1_Can_MATLAB_be_used_for_other_probability_distributions\"><\/span>1. Can MATLAB be used for other probability distributions?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, MATLAB offers functions to work with various other probability distributions, including uniform, exponential, gamma, beta, and more.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2_What_if_I_want_to_calculate_the_probability_between_two_threshold_values\"><\/span>2. What if I want to calculate the probability between two threshold values?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nTo calculate the probability between two threshold values, you can subtract the probability below the lower threshold from the probability below the higher threshold.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3_Is_MATLAB_the_only_programming_language_that_can_calculate_probabilities_in_a_normal_distribution\"><\/span>3. Is MATLAB the only programming language that can calculate probabilities in a normal distribution?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nNo, other programming languages like Python, R, and Julia also offer libraries and functions to calculate probabilities in normal distributions.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4_What_if_the_threshold_value_is_outside_the_range_of_observed_data\"><\/span>4. What if the threshold value is outside the range of observed data?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nIf the threshold value is outside the range of observed data, the calculated probability will be either 0 or 1, depending on whether the value is below the minimum or above the maximum observed value.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5_Can_MATLAB_handle_non-normal_distributions\"><\/span>5. Can MATLAB handle non-normal distributions?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, MATLAB provides functions for working with various probability distributions beyond the normal distribution.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6_How_are_probabilities_in_a_normal_distribution_related_to_the_percentile_ranks\"><\/span>6. How are probabilities in a normal distribution related to the percentile ranks?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nThe probability of observing a value below a certain threshold is equivalent to the cumulative percentile rank of that threshold value.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7_Is_it_possible_to_calculate_probabilities_for_multi-dimensional_normal_distributions_in_MATLAB\"><\/span>7. Is it possible to calculate probabilities for multi-dimensional normal distributions in MATLAB?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, MATLAB offers functions like mvncdf and mvnpdf to work with multi-dimensional normal distributions.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"8_Can_I_use_normcdf_to_calculate_the_probability_of_a_value_above_a_certain_threshold\"><\/span>8. Can I use normcdf to calculate the probability of a value above a certain threshold?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nNo, normcdf specifically calculates the probability of values **below or equal to** a certain threshold. To calculate the probability of values above a threshold, you can subtract the normcdf result from 1.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"9_How_can_I_plot_the_normal_distribution_with_the_calculated_probability\"><\/span>9. How can I plot the normal distribution with the calculated probability?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYou can use the normpdf function in MATLAB to plot the normal distribution, and the probability calculated using normcdf can be represented by shading the area below the threshold value.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"10_Is_MATLAB_suitable_for_large-scale_data_analysis\"><\/span>10. Is MATLAB suitable for large-scale data analysis?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, MATLAB is widely used for large-scale data analysis due to its efficient handling of arrays and built-in functions for statistical analysis.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"11_Can_I_calculate_probabilities_for_non-symmetric_normal_distributions\"><\/span>11. Can I calculate probabilities for non-symmetric normal distributions?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, MATLAB allows calculation of probabilities for both symmetric and non-symmetric normal distributions.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"12_What_if_the_mean_or_standard_deviation_of_the_normal_distribution_is_not_known\"><\/span>12. What if the mean or standard deviation of the normal distribution is not known?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nIf the mean or standard deviation of the normal distribution is not known, you can estimate them from your data using appropriate statistical methods before calculating the probabilities with MATLAB.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The probability below a certain value in a normal distribution can be determined using MATLAB, a powerful programming language widely used for scientific computing and data analysis. In MATLAB, the probability of observing a value below a certain threshold in a normal distribution can be calculated using the cumulative distribution function (CDF) or the normcdf &#8230; <\/p>\n<p class=\"read-more-container\"><a title=\"What is probability normal distribution below a certain value in MATLAB?\" class=\"read-more button\" href=\"https:\/\/namso-gen.co\/blog\/what-is-probability-normal-distribution-below-a-certain-value-in-matlab\/#more-226256\">Read more<span class=\"screen-reader-text\">What is probability normal distribution below a certain value in MATLAB?<\/span><\/a><\/p>\n","protected":false},"author":57,"featured_media":107420,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[86279],"tags":[],"class_list":["post-226256","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-learn","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v22.1 - 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