{"id":4033,"date":"2021-08-14T20:32:21","date_gmt":"2021-08-14T20:32:21","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4033"},"modified":"2021-11-29T18:38:43","modified_gmt":"2021-11-29T13:08:43","slug":"mean-square-deviation-formula","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/mean-square-deviation-formula\/","title":{"rendered":"Mean Square Deviation Formula and Example"},"content":{"rendered":"

Here you will learn mean square deviation formula and relation between mean square deviation and variance with example.<\/p>\n

Let’s begin –<\/p>\n

Mean Square Deviation Formula<\/h2>\n

The mean square deviation of a distribution is the mean of the square of deviations of variate from assumed mean. It is denoted by \\(S^2\\).<\/p>\n

\n

Hence \\(S^2\\) = \\(\\sum{x_i – a}^2\\over n\\) = \\(\\sum{d_i}^2\\over n\\)\u00a0 \u00a0 \u00a0 (for ungrouped dist.)<\/p>\n

\\(S^2\\) = \\(\\sum{x_i – a}^2\\over N\\) = \\(\\sum{f_id_i}^2\\over N\\)\u00a0 \u00a0 (for frequency dist.),\u00a0 \u00a0 where \\(d_i\\) = \\(x_i – a\\)<\/p>\n<\/blockquote>\n

Relation between variance and mean square deviation<\/h2>\n

\\(\\because\\)\u00a0 \u00a0 \\({\\sigma}^2\\) = \\(\\sum{f_id_i}^2\\over N\\) – \\(({\\sum f_i{d_i}\\over N})^2\\)<\/p>\n

\n

\\(\\implies\\)\u00a0 \u00a0 \\({\\sigma}^2\\) = \\(s^2\\) – \\(d^2\\),\u00a0 \u00a0 where d = \\(\\bar{x} – a\\) = \\({\\sum f_i{d_i}\\over N}\\)<\/p>\n

\\(\\implies\\)\u00a0 \u00a0 \\(s^2\\) = \\({\\sigma}^2\\) + \\(d^2\\),\u00a0 \u00a0 \\(\\implies\\) \\(s^2\\) \\(\\geq\\) \\({\\sigma}^2\\)<\/p>\n<\/blockquote>\n

Hence the variance is the minimum value of mean square deviation of a distribution.<\/p>\n\n\n

Example : <\/span>Find the variance of the following freq. dist.\n <\/p>\n \n \n
class<\/td>\n 0 – 2<\/td>\n 2 – 4<\/td>\n 4 – 6<\/td>\n 6 – 8<\/td>\n 8 – 10<\/td>\n 10 – 12<\/td>\n <\/tr>\n
\\(f_i\\)<\/td>\n 2<\/td>\n 7<\/td>\n 12<\/td>\n 19<\/td>\n 9<\/td>\n 1<\/td>\n <\/tr>\n <\/tbody><\/table>

<\/p>\n

Solution : <\/span>Let a = 7 and h = 2\n <\/p>\n \n \n \n \n \n \n \n \n
class<\/td>\n \\(x_i\\)<\/td>\n \\(f_i\\)<\/td>\n \\(u_i\\) = \\(x_i – a\\over h\\)<\/td>\n \\(f_iu_i\\)<\/td>\n \\(f_iu_i^2\\)<\/td>\n <\/tr>\n
0 – 2<\/td>\n 1<\/td>\n 2<\/td>\n -3<\/td>\n -6<\/td>\n 18<\/td>\n <\/tr>\n
2 – 4<\/td>\n 3<\/td>\n 7<\/td>\n -2<\/td>\n -14<\/td>\n 28<\/td>\n <\/tr>\n
4 – 6<\/td>\n 5<\/td>\n 12<\/td>\n -1<\/td>\n -12<\/td>\n 12<\/td>\n <\/tr>\n
6 – 8<\/td>\n 7<\/td>\n 19<\/td>\n 0<\/td>\n 0<\/td>\n 0<\/td>\n <\/tr>\n
8 – 10<\/td>\n 9<\/td>\n 9<\/td>\n 1<\/td>\n 9<\/td>\n 9<\/td>\n <\/tr>\n
10 – 12<\/td>\n 11<\/td>\n 1<\/td>\n 2<\/td>\n 2<\/td>\n 4<\/td>\n <\/tr>\n
<\/td>\n <\/td>\n N = 50<\/td>\n <\/td>\n \\(\\sum{f_iu_i}\\) = -21<\/td>\n \\(\\sum{f_iu_i^2}\\) = 71<\/td>\n <\/tr>\n <\/tbody><\/table>

<\/p>\n

\\(\\because\\)     \\({\\sigma^2}\\) = \\(h^2\\)[\\(\\sum \n f_i{u_i}^2\\over n\\) – \\(({\\sum f_i{u_i}\\over n})^2\\)]

\n = 4[\\(71\\over 50\\) – (\\({-21\\over 50}^2\\))]

\n = 4[1.42 – 0.1764] = 4.97

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

Mathematical Properties of Variance<\/h2>\n

(i) Var.\\((x_i + p\\)) = Var.(\\(x_i\\))<\/p>\n

(ii) Var.\\((px_i\\)) = \\(p^2\\)Var.(\\(x_i\\))<\/p>\n

(iii) Var\\((ax_i + b\\)) = \\(a^2\\).Var(\\(x_i\\))<\/p>\n

where p, a, b are constants.<\/p>\n\n\n

\n
Previous – Formula for Variance and Standard Deviation<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn mean square deviation formula and relation between mean square deviation and variance with example. Let’s begin – Mean Square Deviation Formula The mean square deviation of a distribution is the mean of the square of deviations of variate from assumed mean. It is denoted by \\(S^2\\). Hence \\(S^2\\) = \\(\\sum{x_i – …<\/p>\n

Mean Square Deviation Formula and Example<\/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":[602,603],"yoast_head":"\nMean Square Deviation Formula and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn mean square deviation formula and relation between mean square deviation and variance 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\/mean-square-deviation-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Mean Square Deviation Formula and Example - 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