{"id":154386,"date":"2022-03-25T23:28:38","date_gmt":"2022-03-25T17:58:38","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/average-definition-symbol-formula-and-solved-examples\/"},"modified":"2024-08-29T12:58:46","modified_gmt":"2024-08-29T07:28:46","slug":"average-definition-symbol-formula-and-solved-examples","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/average\/","title":{"rendered":"Average \u2013 Definition, Symbol, Formula and Solved Examples"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 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\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><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><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Average_Symbol\" title=\"Average Symbol\">Average Symbol<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Average_Formula\" title=\"Average Formula\">Average Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#How_to_Calculate_Average\" title=\"How to Calculate Average?\">How to Calculate Average?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#What_is_Average_Used_For\" title=\"What is Average Used For?\">What is Average Used For?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#What_is_Mean\" title=\"What is Mean?\">What is Mean?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Types_of_Mean\" title=\"Types of Mean\">Types of Mean<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Arithmetic_Mean\" title=\"Arithmetic Mean\">Arithmetic Mean<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Geometric_Mean\" title=\"Geometric Mean\">Geometric Mean<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Harmonic_Mean\" title=\"Harmonic Mean\">Harmonic Mean<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Important_Formulas_for_Calculating_Averages\" title=\"Important Formulas for Calculating Averages\">Important Formulas for Calculating Averages<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Average_of_the_First_n_Natural_Numbers\" title=\"Average of the First n Natural Numbers \">Average of the First n Natural Numbers <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Average_of_the_Squares_of_the_First_n_Natural_Numbers\" title=\"Average of the Squares of the First n Natural Numbers\">Average of the Squares of the First n Natural Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Average_of_the_Cubes_of_the_First_n_Natural_Numbers\" title=\"Average of the Cubes of the First n Natural Numbers \">Average of the Cubes of the First n Natural Numbers <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Average_of_the_First_n_Natural_Odd_Numbers\" title=\"Average of the First n Natural Odd Numbers\">Average of the First n Natural Odd Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Average_of_the_First_n_Natural_Even_Numbers\" title=\"Average of the First n Natural Even Numbers\">Average of the First n Natural Even Numbers<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Solved_Examples_of_Average\" title=\"Solved Examples of Average \">Solved Examples of Average <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Practice_Questions_of_Average\" title=\"Practice Questions of Average\">Practice Questions of Average<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#FAQs_on_Average\" title=\"FAQs on Average\">FAQs on Average<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#Whats_the_difference_between_average_mean_and_arithmetic_mean\" title=\"What&#039;s the difference between average, mean, and arithmetic mean?\">What&#039;s the difference between average, mean, and arithmetic mean?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#How_do_you_calculate_the_average_of_a_set_of_numbers\" title=\"How do you calculate the average of a set of numbers?\">How do you calculate the average of a set of numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/#What_are_the_types_of_means_used_in_statistics\" title=\"What are the types of means used in statistics?\">What are the types of means used in statistics?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p><span style=\"font-weight: 400;\">The average is also known as the arithmetic mean. It is calculated by adding up all the numbers and then dividing the sum by the total number of values in the set. This Average formula helps us find the average for any group of data. This article will discuss the use of the average formula, its examples and more. <\/span><\/p>\n<p style=\"text-align: center;\"><strong>Also Check:<a href=\"https:\/\/infinitylearn.com\/surge\/maths\/faces-edges-and-vertices\/\"> Vertices, Faces and Edges<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Average_Symbol\"><\/span><span style=\"font-weight: 400;\">Average Symbol<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">The average is also known as the mean. Mean is a commonly used mathematical term. It is often represented by the symbol x\u0304, pronounced as &#8220;x bar&#8221;. Additionally, the Greek letter \u201c\u03bc\u201d is sometimes used to denote the average, especially in statistics.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Average_Formula\"><\/span><span style=\"font-weight: 400;\">Average Formula<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">To calculate the average in mathematics, you use the following formula:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average Formula = <\/span><span style=\"font-weight: 400;\">sum of observations <\/span><span style=\"font-weight: 400;\">total number of observations <\/span><\/p>\n<p><span style=\"font-weight: 400;\">If you have a set of numbers, say n numbers like x\u2081, x\u2082, x\u2083, \u2026, x\u2099, the average can be calculated using the formula:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average Formula = <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">4<\/span><span style=\"font-weight: 400;\"> + . . . <\/span><span style=\"font-weight: 400;\"> + x<\/span><span style=\"font-weight: 400;\">n<\/span> <span style=\"font-weight: 400;\">n<\/span><\/p>\n<p style=\"text-align: center;\"><strong>Also Check: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/area-of-a-circle\/\">Area of a Circle<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_to_Calculate_Average\"><\/span><span style=\"font-weight: 400;\">How to Calculate Average?<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">One must follow the steps given below to calculate the average of a set of numbers:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">List all the numbers (observations) and count how many there are (let\u2019s call this number n).<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Add up all the numbers to find their total sum.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Divide the sum you found in Step 2 by the total number of observations (n).<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Simplify the result to get the average.<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><strong>Also Check: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/cube\/\">CUBE<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Average_Used_For\"><\/span><span style=\"font-weight: 400;\">What is Average Used For?<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">The average is a useful way to summarise a large set of data with just one number, representing the central value of the data. Here are some practical applications of averages:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-checked=\"false\" aria-level=\"1\"><span style=\"font-weight: 400;\">By calculating the average time it takes to commute to work or school, you can better plan your daily schedule.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-checked=\"false\" aria-level=\"1\"><span style=\"font-weight: 400;\">In cricket, a player&#8217;s batting average helps assess their performance over time.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-checked=\"false\" aria-level=\"1\"><span style=\"font-weight: 400;\">Before purchasing a product, the average customer review score can guide your decision.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-checked=\"false\" aria-level=\"1\"><span style=\"font-weight: 400;\">Averages like household income, unemployment rate, or inflation rate are used to understand economic trends.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-checked=\"false\" aria-level=\"1\"><span style=\"font-weight: 400;\">Businesses calculate the average daily sales of a product to determine how much stock they need.<\/span><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Mean\"><\/span><span style=\"font-weight: 400;\">What is Mean?<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">In mathematics, the mean is a measure of central tendency, commonly known as the average or arithmetic mean. Mean is calculated by dividing the sum of all values by the total number of values.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Mean = <\/span><span style=\"font-weight: 400;\">sum of observations <\/span><span style=\"font-weight: 400;\">total number of observations<\/span><\/p>\n<p style=\"text-align: center;\"><strong>Also Check:<a href=\"https:\/\/infinitylearn.com\/surge\/maths\/circumference-of-a-circle\/\"> Circumstance of Circle<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Types_of_Mean\"><\/span><span style=\"font-weight: 400;\">Types of Mean<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">There are three types of means in mathematics:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Arithmetic Mean<\/b><span style=\"font-weight: 400;\">: Arithmetic Mean is the most common type, calculated as described above.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Geometric Mean<\/b><span style=\"font-weight: 400;\">: Geometric Mean is the nth root of the product of n values, often used for growth rates.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Harmonic Mean<\/b><span style=\"font-weight: 400;\">: Harmonic Mean is the reciprocal of the average of the reciprocals, useful in certain types of averages like rates.<\/span><\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Arithmetic_Mean\"><\/span><span style=\"font-weight: 400;\">Arithmetic Mean<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The arithmetic mean, often simply called the average, is calculated by adding up all the values and then dividing by the number of values. The formula to calculate the arithmetic mean for a set of values <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\">, . . . <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> is <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Arithmetic Mean (A. M.) = <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">4<\/span><span style=\"font-weight: 400;\"> + . . . <\/span><span style=\"font-weight: 400;\"> + x<\/span><span style=\"font-weight: 400;\">n<\/span> <span style=\"font-weight: 400;\">n<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Geometric_Mean\"><\/span><span style=\"font-weight: 400;\">Geometric Mean<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The geometric mean is a method to measure central tendency. It is calculated by multiplying all the values together and then taking the nth root of the product, where n is the total number of values. The formula to calculate the geometric mean for values <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\">, . . . <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> is<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Geometric Mean (G. M.) = <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\"> . <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\">. <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\"> . . . <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">n<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Harmonic_Mean\"><\/span><span style=\"font-weight: 400;\">Harmonic Mean<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The harmonic mean is a type of Pythagorean mean, alongside the arithmetic and geometric means. It is calculated by dividing the number of values by the sum of the reciprocals of the values. The harmonic mean tends to be lower than both the arithmetic and geometric means. The formula for the harmonic mean for values <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\">, . . . <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> is <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Harmonic Mean (H. M.) =<\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\">(<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">) + (<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">) + . . . + (<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\">)<\/span><\/p>\n<p style=\"text-align: center;\"><strong>Also Check: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/fibonacci-sequence\/\">Fibonacci Sequence<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Important_Formulas_for_Calculating_Averages\"><\/span><span style=\"font-weight: 400;\">Important Formulas for Calculating Averages<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">Below are some key formulas related to averages. These formulas are useful for board exams and competitive tests. <\/span><\/p>\n<h3><span style=\"font-weight: 400;\">Average of the First n Natural Numbers <\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The sum of the first n natural numbers <\/span><\/p>\n<p><span style=\"font-weight: 400;\">sum = <\/span><span style=\"font-weight: 400;\">n(n + 1)<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average of the first n natural numbers<\/span><\/p>\n<p><span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">n + 1<\/span><span style=\"font-weight: 400;\">2<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Average_of_the_Squares_of_the_First_n_Natural_Numbers\"><\/span><span style=\"font-weight: 400;\">Average of the Squares of the First n Natural Numbers<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The sum of the squares of the first n natural numbers <\/span><\/p>\n<p><span style=\"font-weight: 400;\">sum = <\/span><span style=\"font-weight: 400;\">n(n + 1)(2n + 1)<\/span><span style=\"font-weight: 400;\">6<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average of the squares of the first n natural numbers<\/span><\/p>\n<p><span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">(n + 1)(2n + 1)<\/span><span style=\"font-weight: 400;\">6<\/span><span style=\"font-weight: 400;\">  <\/span><\/p>\n<h3><span style=\"font-weight: 400;\">Average of the Cubes of the First n Natural Numbers <\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The sum of the cubes of the first n natural numbers:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Sum = <\/span><span style=\"font-weight: 400;\">(<\/span><span style=\"font-weight: 400;\">n(n + 1)<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">)<\/span><span style=\"font-weight: 400;\">2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average of the cubes of the first n natural numbers:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average = <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\">(n + 1)<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">4<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Average_of_the_First_n_Natural_Odd_Numbers\"><\/span><span style=\"font-weight: 400;\">Average of the First <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> Natural Odd Numbers<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The sum of the first n natural odd numbers:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Sum = <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\">2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average of the first n natural odd numbers:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average=n<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Average_of_the_First_n_Natural_Even_Numbers\"><\/span><span style=\"font-weight: 400;\">Average of the First <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> Natural Even Numbers<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The sum of the first n natural even numbers:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Sum=n(n+1)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average of the first n natural even numbers:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average=n+1<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">Solved Examples of Average <\/span><\/h2>\n<p><b>Example 1: Find the average of the first 10 natural numbers.<\/b><\/p>\n<p><b>Solution: <\/b><span style=\"font-weight: 400;\">To find the average of the first 10 natural numbers we can use the formula for the average of the first n natural numbers:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">n + 1<\/span><span style=\"font-weight: 400;\">2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Here, n = 10<\/span><\/p>\n<p><span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">10 + 1<\/span><span style=\"font-weight: 400;\">2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">11<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> = 5.5 <\/span><\/p>\n<p><span style=\"font-weight: 400;\"> Answer: The average of the first 10 natural numbers is 5.5.<\/span><\/p>\n<p><b>Example 2: Calculate the average of the squares of the first 5 natural numbers.<\/b><\/p>\n<p><b>Solution: <\/b><span style=\"font-weight: 400;\">To calculate the average of the squares of the first n natural numbers, we use the below-given formula: <\/span><\/p>\n<p><span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">(n + 1)(2n + 1)<\/span><span style=\"font-weight: 400;\">6<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Here, n=5<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Therefore, <\/span> <span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">(5 + 1)(10 + 1)<\/span><span style=\"font-weight: 400;\">6<\/span><span style=\"font-weight: 400;\"> <\/span><\/p>\n<p><span style=\"font-weight: 400;\">average = <\/span><span style=\"font-weight: 400;\">(6)(11)<\/span><span style=\"font-weight: 400;\">6<\/span><span style=\"font-weight: 400;\"> = 11<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Answer: The average of the squares of the first 5 natural numbers is 11.<\/span><\/p>\n<p><b>Example 3: Find the average of the cubes of the first 4 natural numbers.<\/b><\/p>\n<p><b>Solution: <\/b><span style=\"font-weight: 400;\">To calculate the average of the cubes of the first n natural numbers, we can use the below-given formula:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average = <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\">(n + 1)<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">4<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Here, n = 4<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average = <\/span><span style=\"font-weight: 400;\">4<\/span><span style=\"font-weight: 400;\"> (4 + 1)<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">4<\/span><span style=\"font-weight: 400;\"> <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average = 25<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Answer: The average of the cubes of the first 4 natural numbers is 25.<\/span><\/p>\n<p><b>Example 4: Determine the average of the first 7 natural odd numbers.<\/b><\/p>\n<p><b>Solution: <\/b><span style=\"font-weight: 400;\">To calculate the average of first n natural odd numbers, the average is simply n.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average=n<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Here, n = 7 <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Average=7<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Answer: The average of the first 7 natural odd numbers is 7. <\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Practice_Questions_of_Average\"><\/span><span style=\"font-weight: 400;\">Practice Questions of Average<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">Question 1: Find the average of the first 15 natural numbers.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Question 2: Calculate the average of the squares of the first 6 natural numbers.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Question 3: Determine the average of the cubes of the first 5 natural numbers.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Question 4: What is the average of the first 10 natural odd numbers?<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Question 5: Find the average of the first 8 natural even numbers.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Question 6: If the average of the cubes of the first 7 natural numbers is calculated, what will be the result?<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"FAQs_on_Average\"><\/span><span style=\"font-weight: 400;\">FAQs on Average<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Whats_the_difference_between_average_mean_and_arithmetic_mean\"><\/span>What&#039;s the difference between average, mean, and arithmetic mean?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\t&quot;Average,&quot; &quot;mean,&quot; and &quot;arithmetic mean&quot; are often used interchangeably. &quot;Mean&quot; and &quot;arithmetic mean&quot; specifically refer to the sum of values divided by the number of values. &quot;Average&quot; is a broader term that can refer to different types of central tendencies.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_do_you_calculate_the_average_of_a_set_of_numbers\"><\/span>How do you calculate the average of a set of numbers?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tTo calculate the average of a set of numbers, we need to add all the numbers together, and then divide by the total count of numbers.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_the_types_of_means_used_in_statistics\"><\/span>What are the types of means used in statistics?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tArithmetic Mean: Arithmetic Mean is the sum of values divided by the count. Geometric Mean: Geometric Mean refers to the nth root of the product of values. Harmonic Mean: Harmonic Mean is the Reciprocal of the average of reciprocals.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What's the difference between average, mean, and arithmetic mean?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"&amp;quot;Average,&amp;quot; &amp;quot;mean,&amp;quot; and &amp;quot;arithmetic mean&amp;quot; are often used interchangeably. &amp;quot;Mean&amp;quot; and &amp;quot;arithmetic mean&amp;quot; specifically refer to the sum of values divided by the number of values. &amp;quot;Average&amp;quot; is a broader term that can refer to different types of central tendencies.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How do you calculate the average of a set of numbers?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"To calculate the average of a set of numbers, we need to add all the numbers together, and then divide by the total count of numbers.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are the types of means used in statistics?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Arithmetic Mean: Arithmetic Mean is the sum of values divided by the count. Geometric Mean: Geometric Mean refers to the nth root of the product of values. Harmonic Mean: Harmonic Mean is the Reciprocal of the average of reciprocals.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>The average is also known as the arithmetic mean. It is calculated by adding up all the numbers and then [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Average","_yoast_wpseo_title":"Average \u2013 Definition, Symbol, Formula and Solved Examples","_yoast_wpseo_metadesc":"Learn about Average topic of Maths in details explained by subject experts. To know more at infinitylearn.com.","custom_permalink":"maths\/average\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Average \u2013 Definition, Symbol, Formula and Solved Examples<\/title>\n<meta name=\"description\" content=\"Learn about Average topic of Maths in details explained by subject experts. To know more at infinitylearn.com.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Average \u2013 Definition, Symbol, Formula and Solved Examples\" \/>\n<meta property=\"og:description\" content=\"Learn about Average topic of Maths in details explained by subject experts. To know more at infinitylearn.com.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/\" \/>\n<meta property=\"og:site_name\" content=\"Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-25T17:58:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-29T07:28:46+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2025\/04\/infinitylearn.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1920\" \/>\n\t<meta property=\"og:image:height\" content=\"1008\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:site\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"vipin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Average \u2013 Definition, Symbol, Formula and Solved Examples","description":"Learn about Average topic of Maths in details explained by subject experts. To know more at infinitylearn.com.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/infinitylearn.com\/surge\/maths\/average\/","og_locale":"en_US","og_type":"article","og_title":"Average \u2013 Definition, Symbol, Formula and Solved Examples","og_description":"Learn about Average topic of Maths in details explained by subject experts. To know more at infinitylearn.com.","og_url":"https:\/\/infinitylearn.com\/surge\/maths\/average\/","og_site_name":"Infinity Learn by Sri Chaitanya","article_publisher":"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","article_published_time":"2022-03-25T17:58:38+00:00","article_modified_time":"2024-08-29T07:28:46+00:00","og_image":[{"width":1920,"height":1008,"url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2025\/04\/infinitylearn.jpg","type":"image\/jpeg"}],"twitter_card":"summary_large_image","twitter_creator":"@InfinityLearn_","twitter_site":"@InfinityLearn_","twitter_misc":{"Written by":"vipin","Est. reading time":"6 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/infinitylearn.com\/surge\/#organization","name":"Infinity Learn","url":"https:\/\/infinitylearn.com\/surge\/","sameAs":["https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","https:\/\/www.instagram.com\/infinitylearn_by_srichaitanya\/","https:\/\/www.linkedin.com\/company\/infinity-learn-by-sri-chaitanya\/","https:\/\/www.youtube.com\/c\/InfinityLearnEdu","https:\/\/twitter.com\/InfinityLearn_"],"logo":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#logo","inLanguage":"en-US","url":"","contentUrl":"","caption":"Infinity Learn"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/#logo"}},{"@type":"WebSite","@id":"https:\/\/infinitylearn.com\/surge\/#website","url":"https:\/\/infinitylearn.com\/surge\/","name":"Infinity Learn by Sri Chaitanya","description":"Surge","publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/infinitylearn.com\/surge\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/infinitylearn.com\/surge\/maths\/average\/#webpage","url":"https:\/\/infinitylearn.com\/surge\/maths\/average\/","name":"Average \u2013 Definition, Symbol, Formula and Solved Examples","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/#website"},"datePublished":"2022-03-25T17:58:38+00:00","dateModified":"2024-08-29T07:28:46+00:00","description":"Learn about Average topic of Maths in details explained by subject experts. To know more at infinitylearn.com.","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/average\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/maths\/average\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/maths\/average\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"Average \u2013 Definition, Symbol, Formula and Solved Examples"}]},{"@type":"Article","@id":"https:\/\/infinitylearn.com\/surge\/maths\/average\/#article","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/average\/#webpage"},"author":{"@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2"},"headline":"Average \u2013 Definition, Symbol, Formula and Solved Examples","datePublished":"2022-03-25T17:58:38+00:00","dateModified":"2024-08-29T07:28:46+00:00","mainEntityOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/average\/#webpage"},"wordCount":1299,"publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"articleSection":["Maths"],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2","name":"vipin","image":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","caption":"vipin"},"sameAs":["http:\/\/surge.infinitylearn.com"],"url":"https:\/\/infinitylearn.com\/surge\/author\/vipin\/"}]}},"_links":{"self":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/154386"}],"collection":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/comments?post=154386"}],"version-history":[{"count":0,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/154386\/revisions"}],"wp:attachment":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/media?parent=154386"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/categories?post=154386"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/tags?post=154386"},{"taxonomy":"table_tags","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/table_tags?post=154386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}