std of 3,4,2,1,2,3,5,4,4,3,2

{ "query": { "display": "standard deviation $$3,\\:4,\\:2,\\:1,\\:2,\\:3,\\:5,\\:4,\\:4,\\:3,\\:2$$", "symbolab_question": "STATISTICS#std 3,4,2,1,2,3,5,4,4,3,2" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "std", "subTopic": "Other", "default": "1.18321…" }, "steps": { "type": "interim", "title": "Standard Deviation of $$3,\\:4,\\:2,\\:1,\\:2,\\:3,\\:5,\\:4,\\:4,\\:3,\\:2:{\\quad}1.18321…$$", "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:$${\\quad}1.4$$", "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, $$\\bar{x}:{\\quad}3$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=33$$", "steps": [ { "type": "step", "primary": "Take the sum of $$3,\\:4,\\:2,\\:1,\\:2,\\:3,\\:5,\\:4,\\:4,\\:3,\\:2$$", "result": "3+4+2+1+2+3+5+4+4+3+2" }, { "type": "step", "primary": "Simplify", "result": "33" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=11$$", "input": "3,\\:4,\\:2,\\:1,\\:2,\\:3,\\:5,\\:4,\\:4,\\:3,\\:2", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}3&4&2&1&2&3&5&4&4&3&2\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCCRGA3gWHAE0SFYC9YUOuV8m+k9gDdMT3BgfQvJFnMsIKN6Hv6MoTMtvtU0IQwXdn+dP39/b4v4puy4FddUb1Xuv84jJF16/iOOzDkZH7gikaKKt7jFqJG6jXkS2BhPkmI99UODWR5Ekeo4yWJoO3k1" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}3$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{33}{11}$$", "result": "\\frac{33}{11}" }, { "type": "step", "primary": "Simplify", "result": "3" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=3" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}14$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(3-3\\right)^{2},\\:\\left(4-3\\right)^{2},\\:\\left(2-3\\right)^{2},\\:\\left(1-3\\right)^{2},\\:\\left(2-3\\right)^{2},\\:\\left(3-3\\right)^{2},\\:\\left(5-3\\right)^{2},\\:\\left(4-3\\right)^{2},\\:\\left(4-3\\right)^{2},\\:\\left(3-3\\right)^{2},\\:\\left(2-3\\right)^{2}$$", "result": "\\left(3-3\\right)^{2}+\\left(4-3\\right)^{2}+\\left(2-3\\right)^{2}+\\left(1-3\\right)^{2}+\\left(2-3\\right)^{2}+\\left(3-3\\right)^{2}+\\left(5-3\\right)^{2}+\\left(4-3\\right)^{2}+\\left(4-3\\right)^{2}+\\left(3-3\\right)^{2}+\\left(2-3\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "14" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=11$$", "input": "3,\\:4,\\:2,\\:1,\\:2,\\:3,\\:5,\\:4,\\:4,\\:3,\\:2", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}3&4&2&1&2&3&5&4&4&3&2\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCCRGA3gWHAE0SFYC9YUOuV8m+k9gDdMT3BgfQvJFnMsIKN6Hv6MoTMtvtU0IQwXdn+dP39/b4v4puy4FddUb1Xuv84jJF16/iOOzDkZH7gikaKKt7jFqJG6jXkS2BhPkmI99UODWR5Ekeo4yWJoO3k1" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}1.4$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{14}{10}$$", "result": "\\frac{14}{10}" }, { "type": "step", "primary": "Simplify", "result": "1.4" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "1.4" } ], "meta": { "interimType": "Variance Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sigma\\left(X\\right)=\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}:{\\quad}1.18321…$$", "steps": [ { "type": "step", "primary": "The variance is $$1.4$$ , therefore $$\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}=\\sqrt{1.4}$$", "result": "\\sqrt{1.4}" }, { "type": "step", "primary": "Simplify", "result": "1.18321…" } ], "meta": { "interimType": "Compute The STDV Title 0Eq" } }, { "type": "step", "result": "1.18321…" } ] } }