variance of 12,-25,2,3,21,8,10,14,3,18,21

{ "query": { "display": "variance $$12,\\:-25,\\:2,\\:3,\\:21,\\:8,\\:10,\\:14,\\:3,\\:18,\\:21$$", "symbolab_question": "STATISTICS#variance 12,-25,2,3,21,8,10,14,3,18,21" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "variance", "subTopic": "Other", "default": "166.89090…" }, "steps": { "type": "interim", "title": "Sample Variance of $$12,\\:-25,\\:2,\\:3,\\:21,\\:8,\\:10,\\:14,\\:3,\\:18,\\:21:{\\quad}166.89090…$$", "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, $$\\bar{x}:{\\quad}7.90909…$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=87$$", "steps": [ { "type": "step", "primary": "Take the sum of $$12,\\:-25,\\:2,\\:3,\\:21,\\:8,\\:10,\\:14,\\:3,\\:18,\\:21$$", "result": "12-25+2+3+21+8+10+14+3+18+21" }, { "type": "step", "primary": "Simplify", "result": "87" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=11$$", "input": "12,\\:-25,\\:2,\\:3,\\:21,\\:8,\\:10,\\:14,\\:3,\\:18,\\:21", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}12&-25&2&3&21&8&10&14&3&18&21\\\\1&2&3&4&5&6&7&8&9&10&11\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "11" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCDXTrsGfo3n+PWlOyV3yR9CaJ5r2bdW/RDx35a1G379dYSL8kGJgW8gGY++IASg5/Cc0UAm97e3ZdFs70jM7AY+HtBxF4YoCq+yXnRYGozDl5B00w6UQ2s6IWYitDhNtoeji/kK9z21uZiIOjEhPZ+e" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}7.90909…$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{87}{11}$$", "result": "\\frac{87}{11}" }, { "type": "step", "primary": "Simplify", "result": "7.90909…" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=7.90909…" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}1668.90909…$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(12-7.90909…\\right)^{2},\\:\\left(-25-7.90909…\\right)^{2},\\:\\left(2-7.90909…\\right)^{2},\\:\\left(3-7.90909…\\right)^{2},\\:\\left(21-7.90909…\\right)^{2},\\:\\left(8-7.90909…\\right)^{2},\\:\\left(10-7.90909…\\right)^{2},\\:\\left(14-7.90909…\\right)^{2},\\:\\left(3-7.90909…\\right)^{2},\\:\\left(18-7.90909…\\right)^{2},\\:\\left(21-7.90909…\\right)^{2}$$", "result": "\\left(12-7.90909…\\right)^{2}+\\left(-25-7.90909…\\right)^{2}+\\left(2-7.90909…\\right)^{2}+\\left(3-7.90909…\\right)^{2}+\\left(21-7.90909…\\right)^{2}+\\left(8-7.90909…\\right)^{2}+\\left(10-7.90909…\\right)^{2}+\\left(14-7.90909…\\right)^{2}+\\left(3-7.90909…\\right)^{2}+\\left(18-7.90909…\\right)^{2}+\\left(21-7.90909…\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "1668.90909…" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=11$$", "input": "12,\\:-25,\\:2,\\:3,\\:21,\\:8,\\:10,\\:14,\\:3,\\:18,\\:21", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}12&-25&2&3&21&8&10&14&3&18&21\\\\1&2&3&4&5&6&7&8&9&10&11\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "11" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCDXTrsGfo3n+PWlOyV3yR9CaJ5r2bdW/RDx35a1G379dYSL8kGJgW8gGY++IASg5/Cc0UAm97e3ZdFs70jM7AY+HtBxF4YoCq+yXnRYGozDl5B00w6UQ2s6IWYitDhNtoeji/kK9z21uZiIOjEhPZ+e" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}166.89090…$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{1668.90909…}{10}$$", "result": "\\frac{1668.90909…}{10}" }, { "type": "step", "primary": "Simplify", "result": "166.89090…" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "166.89090…" } ] } }