std of 8,4,9,2,8,2,6,2,2,4

{ "query": { "display": "standard deviation $$8,\\:4,\\:9,\\:2,\\:8,\\:2,\\:6,\\:2,\\:2,\\:4$$", "symbolab_question": "STATISTICS#std 8,4,9,2,8,2,6,2,2,4" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "std", "subTopic": "Other", "default": "2.83039…" }, "steps": { "type": "interim", "title": "Standard Deviation of $$8,\\:4,\\:9,\\:2,\\:8,\\:2,\\:6,\\:2,\\:2,\\:4:{\\quad}2.83039…$$", "steps": [ { "type": "definition", "title": "Standard Deviation", "text": "The standard deviation, $$\\sigma\\left(X\\right)$$, is the square root of the variance:$${\\quad}\\sigma\\left(X\\right)=\\sqrt{\\frac{\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}$$" }, { "type": "interim", "title": "Compute the variance:$${\\quad}8.01111…$$", "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, $$\\bar{x}:{\\quad}4.7$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=47$$", "steps": [ { "type": "step", "primary": "Take the sum of $$8,\\:4,\\:9,\\:2,\\:8,\\:2,\\:6,\\:2,\\:2,\\:4$$", "result": "8+4+9+2+8+2+6+2+2+4" }, { "type": "step", "primary": "Simplify", "result": "47" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=10$$", "input": "8,\\:4,\\:9,\\:2,\\:8,\\:2,\\:6,\\:2,\\:2,\\:4", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}8&4&9&2&8&2&6&2&2&4\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCB6FnKgOM7KRyrRuO4TO0Sm9HCEuPP4G5Xb5b8405QTnu9sGZu5A1MXROmEpnxG69p8Kc4dHKiaZXtUIPOGIntk8DHUbcdO0jo6qYvJ8S6ga+eZum20Vj8ULi7vwcxTxmeJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}4.7$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{47}{10}$$", "result": "\\frac{47}{10}" }, { "type": "step", "primary": "Simplify", "result": "4.7" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=4.7" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}72.1$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(8-4.7\\right)^{2},\\:\\left(4-4.7\\right)^{2},\\:\\left(9-4.7\\right)^{2},\\:\\left(2-4.7\\right)^{2},\\:\\left(8-4.7\\right)^{2},\\:\\left(2-4.7\\right)^{2},\\:\\left(6-4.7\\right)^{2},\\:\\left(2-4.7\\right)^{2},\\:\\left(2-4.7\\right)^{2},\\:\\left(4-4.7\\right)^{2}$$", "result": "\\left(8-4.7\\right)^{2}+\\left(4-4.7\\right)^{2}+\\left(9-4.7\\right)^{2}+\\left(2-4.7\\right)^{2}+\\left(8-4.7\\right)^{2}+\\left(2-4.7\\right)^{2}+\\left(6-4.7\\right)^{2}+\\left(2-4.7\\right)^{2}+\\left(2-4.7\\right)^{2}+\\left(4-4.7\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "72.1" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=10$$", "input": "8,\\:4,\\:9,\\:2,\\:8,\\:2,\\:6,\\:2,\\:2,\\:4", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}8&4&9&2&8&2&6&2&2&4\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCB6FnKgOM7KRyrRuO4TO0Sm9HCEuPP4G5Xb5b8405QTnu9sGZu5A1MXROmEpnxG69p8Kc4dHKiaZXtUIPOGIntk8DHUbcdO0jo6qYvJ8S6ga+eZum20Vj8ULi7vwcxTxmeJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}8.01111…$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{72.1}{9}$$", "result": "\\frac{72.1}{9}" }, { "type": "step", "primary": "Simplify", "result": "8.01111…" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "8.01111…" } ], "meta": { "interimType": "Variance Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sigma\\left(X\\right)=\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}:{\\quad}2.83039…$$", "steps": [ { "type": "step", "primary": "The variance is $$8.01111…$$ , therefore $$\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}=\\sqrt{8.01111…}$$", "result": "\\sqrt{8.01111…}" }, { "type": "step", "primary": "Simplify", "result": "2.83039…" } ], "meta": { "interimType": "Compute The STDV Title 0Eq" } }, { "type": "step", "result": "2.83039…" } ] } }