variance of 0,4,2,8,2,3,1,0,5,7

{ "query": { "display": "variance $$0,\\:4,\\:2,\\:8,\\:2,\\:3,\\:1,\\:0,\\:5,\\:7$$", "symbolab_question": "STATISTICS#variance 0,4,2,8,2,3,1,0,5,7" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "variance", "subTopic": "Other", "default": "7.73333…" }, "steps": { "type": "interim", "title": "Sample Variance of $$0,\\:4,\\:2,\\:8,\\:2,\\:3,\\:1,\\:0,\\:5,\\:7:{\\quad}7.73333…$$", "steps": [ { "type": "definition", "title": "Sample Variance", "text": "The sample variance measures how much the data is spread out in the sample.<br/>For a data set $$x_{1},\\:\\ldots\\:,\\:x_{n}$$ (n elements) with an average $$\\bar{x}$$, $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}$$" }, { "type": "interim", "title": "Compute the average, $$\\bar{x}:{\\quad}3.2$$", "steps": [ { "type": "definition", "title": "Arithmetic Mean", "text": "The arithemtic mean (average) is the sum of the values in the set divided by the number of elements in that set.<br/>If our data set contains the values $$a_{1},\\:\\ldots\\:,\\:a_{n}$$ (n elements) then the average$$=\\frac{1}{n}\\sum_{i=1}^{n}a_{i}\\:$$" }, { "type": "interim", "title": "Compute the sum of the data set:$${\\quad}\\sum_{i=1}^{n}a_{i}=32$$", "steps": [ { "type": "step", "primary": "Take the sum of $$0,\\:4,\\:2,\\:8,\\:2,\\:3,\\:1,\\:0,\\:5,\\:7$$", "result": "0+4+2+8+2+3+1+0+5+7" }, { "type": "step", "primary": "Simplify", "result": "32" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=10$$", "input": "0,\\:4,\\:2,\\:8,\\:2,\\:3,\\:1,\\:0,\\:5,\\:7", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}0&4&2&8&2&3&1&0&5&7\\\\1&2&3&4&5&6&7&8&9&10\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "10" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAqhyZSij6Y+Y15pb74n8MY6+9pYHSoyiRUFMD+vwcWZO9sGZu5A1MXROmEpnxG69p8Kc4dHKiaZXtUIPOGIntk8DHUbcdO0jo6qYvJ8S6ga+eZum20Vj8ULi7vwcxTxmeJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}3.2$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{32}{10}$$", "result": "\\frac{32}{10}" }, { "type": "step", "primary": "Simplify", "result": "3.2" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=3.2" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}69.6$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(0-3.2\\right)^{2},\\:\\left(4-3.2\\right)^{2},\\:\\left(2-3.2\\right)^{2},\\:\\left(8-3.2\\right)^{2},\\:\\left(2-3.2\\right)^{2},\\:\\left(3-3.2\\right)^{2},\\:\\left(1-3.2\\right)^{2},\\:\\left(0-3.2\\right)^{2},\\:\\left(5-3.2\\right)^{2},\\:\\left(7-3.2\\right)^{2}$$", "result": "\\left(0-3.2\\right)^{2}+\\left(4-3.2\\right)^{2}+\\left(2-3.2\\right)^{2}+\\left(8-3.2\\right)^{2}+\\left(2-3.2\\right)^{2}+\\left(3-3.2\\right)^{2}+\\left(1-3.2\\right)^{2}+\\left(0-3.2\\right)^{2}+\\left(5-3.2\\right)^{2}+\\left(7-3.2\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "69.6" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=10$$", "input": "0,\\:4,\\:2,\\:8,\\:2,\\:3,\\:1,\\:0,\\:5,\\:7", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}0&4&2&8&2&3&1&0&5&7\\\\1&2&3&4&5&6&7&8&9&10\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "10" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAqhyZSij6Y+Y15pb74n8MY6+9pYHSoyiRUFMD+vwcWZO9sGZu5A1MXROmEpnxG69p8Kc4dHKiaZXtUIPOGIntk8DHUbcdO0jo6qYvJ8S6ga+eZum20Vj8ULi7vwcxTxmeJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}7.73333…$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{69.6}{9}$$", "result": "\\frac{69.6}{9}" }, { "type": "step", "primary": "Simplify", "result": "7.73333…" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "7.73333…" } ] } }