{"id":3572,"date":"2021-07-29T11:04:59","date_gmt":"2021-07-29T11:04:59","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3572"},"modified":"2021-11-29T18:43:29","modified_gmt":"2021-11-29T13:13:29","slug":"formula-for-variance-and-standard-deviation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-for-variance-and-standard-deviation\/","title":{"rendered":"Formula for Variance and Standard Deviation"},"content":{"rendered":"

Here, you will learn learn formula for variance and standard deviation and relationship between variance and standard deviation.<\/p>\n

Let’s begin –<\/p>\n

Variance and Standard Deviation :<\/h2>\n

The variance of a distribution is, the mean of squares of deviation of variate from their mean. It is denoted by \\(\\sigma^2\\) or var(x).<\/p>\n

The positive square root of the variance are called the standard deviation. It is denoted by \\(\\sigma\\) or S.D.<\/p>\n

\n

Hence standard deviation = + \\(\\sqrt{variance}\\)<\/p>\n<\/blockquote>\n

Formula for Variance :<\/h2>\n

(i) <\/strong>for ungrouped distribution :<\/strong><\/p>\n

\n

\\({\\sigma^2}_x\\) = \\(\\sum(x_i – \\bar{x})^2\\over n\\)<\/p>\n

\\({\\sigma^2}_x\\) = \\(\\sum{x_i}^2\\over n\\) – \\(\\bar{x}^2\\)<\/p>\n

= \\(\\sum{x_i}^2\\over n\\) – \\(({\\sum{x_i}\\over n})^2\\)<\/p>\n

\\({\\sigma^2}_d\\) = \\(\\sum{d_i}^2\\over n\\) – \\(({\\sum{d_i}\\over n})^2\\), where \\(d_i\\) = \\(x_i\\) – a<\/p>\n<\/blockquote>\n

(ii) for frequency distribution :<\/strong><\/p>\n

\n

\\({\\sigma^2}_x\\) = \\(\\sum f_i(x_i – \\bar{x})^2\\over N\\)<\/p>\n

\\({\\sigma^2}_x\\) = \\(\\sum f_i{x_i}^2\\over N\\) – \\(\\bar{x}^2\\)<\/p>\n

= \\(\\sum f_i{x_i}^2\\over N\\) – \\(({\\sum f_i{x_i}\\over N})^2\\)<\/p>\n

\\({\\sigma^2}_d\\) = \\(\\sum f_i{d_i}^2\\over n\\) – \\(({\\sum f_i{d_i}\\over n})^2\\), where \\(d_i\\) = \\(x_i\\) – a<\/p>\n

\\({\\sigma^2}_d\\) = \\(h^2\\)[\\(\\sum f_i{u_i}^2\\over n\\) – \\(({\\sum f_i{u_i}\\over n})^2\\)],  where \\(u_i\\) = \\(x_i\\over h\\)<\/p>\n<\/blockquote>\n

(iii) Coefficient of Standard Deviation = \\(\\sigma\\over \\bar{x}\\)<\/strong><\/p>\n

\n

Coefficient of variation = \\(\\sigma\\over \\bar{x}\\) \\(\\times\\) 100   (in percentage)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Find the variance and standard deviation of first n natural numbers.<\/p>\n

Solution : <\/span> We know that, \n\\({\\sigma^2}_x\\) = \\(\\sum{x_i}^2\\over n\\) – \\(({\\sum{x_i}\\over n})^2\\)

\n = \\(\\sum{n}^2\\over n\\) – \\(({\\sum{n}\\over n})^2\\) = \\(n(n + 1)(2n + 1)\\over {6n}\\) – \\([{n(n + 1)\\over {2n}}]^2\\) = \\(n^2 – 1\\over 12\\)

\nStandard Deviation = \\(\\sqrt{variance}\\) = \\(\\sqrt{n^2 – 1\\over 12}\\)

<\/br
<\/p>\n\n\n\n

Example : <\/span>Find the Coefficient of variation in percentage of first n natural numbers.<\/p>\n

Solution : <\/span> We know that \nMean \\(\\bar{x}\\) = \\(n + 1\\over 2\\),\nVariance = \\({\\sigma^2}_x\\) = \\(n^2 – 1\\over 12\\)

\nStandard Deviation \\(\\sigma\\) = \\(\\sqrt{variance}\\) = \\(\\sqrt{n^2 – 1\\over 12}\\)

\nCoefficient of variation = \\(\\sigma\\over \\bar{x}\\) \\(\\times\\) 100 

\n= \\(\\sqrt{n^2 – 1\\over 12}\\) \\(\\times\\) (\\(2\\over n + 1\\)) \\(\\times\\) 100

\n= \\(\\sqrt{(n – 1)\\over 3(n + 1)}\\) \\(\\times\\) 100

\n<\/br
<\/p>\n\n\n

Hope you learnt what is the formula for variance and standard deviation, learn more concepts of statistics and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – Mean Square Deviation Formula and Example<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – What are Measures of Dispersion Statistics ?<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn learn formula for variance and standard deviation and relationship between variance and standard deviation. Let’s begin – Variance and Standard Deviation : The variance of a distribution is, the mean of squares of deviation of variate from their mean. It is denoted by \\(\\sigma^2\\) or var(x). The positive square root of …<\/p>\n

Formula for Variance and Standard Deviation<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[22],"tags":[610,609,608,607],"yoast_head":"\nFormula for Variance and Standard Deviation - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn formula for variance and standard deviation, relationship between variance and standard deviation with example.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/formula-for-variance-and-standard-deviation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formula for Variance and Standard Deviation - 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