std of 20,0.08

{ "query": { "display": "standard deviation $$20,\\:0.08$$", "symbolab_question": "STATISTICS#std 20,0.08" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "std", "subTopic": "Other", "default": "14.08556…" }, "steps": { "type": "interim", "title": "Standard Deviation of $$20,\\:0.08:{\\quad}14.08556…$$", "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:$${\\quad}198.4032$$", "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, $$\\bar{x}:{\\quad}10.04$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=20.08$$", "steps": [ { "type": "step", "primary": "Take the sum of $$20,\\:0.08$$", "result": "20+0.08" }, { "type": "step", "primary": "Simplify", "result": "20.08" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=2$$", "input": "20,\\:0.08", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}20&0.08\\\\1&2\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "2" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAEPZGpZCGnIViqJyUZ2cuRP8vQyhiD4JSfqjIvcQ7tis3yYC8dn4PQQou1afG6tXmNKNjY9+ZcbbnQJ40VZJCma87YR9Kait5JqLb32Q/sBIde4aJ3P6c20Oh4zI4iJlI=" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}10.04$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{20.08}{2}$$", "result": "\\frac{20.08}{2}" }, { "type": "step", "primary": "Simplify", "result": "10.04" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=10.04" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}198.4032$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(20-10.04\\right)^{2},\\:\\left(0.08-10.04\\right)^{2}$$", "result": "\\left(20-10.04\\right)^{2}+\\left(0.08-10.04\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "198.4032" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=2$$", "input": "20,\\:0.08", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}20&0.08\\\\1&2\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "2" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAEPZGpZCGnIViqJyUZ2cuRP8vQyhiD4JSfqjIvcQ7tis3yYC8dn4PQQou1afG6tXmNKNjY9+ZcbbnQJ40VZJCma87YR9Kait5JqLb32Q/sBIde4aJ3P6c20Oh4zI4iJlI=" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}198.4032$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{198.4032}{1}$$", "result": "\\frac{198.4032}{1}" }, { "type": "step", "primary": "Simplify", "result": "198.4032" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "198.4032" } ], "meta": { "interimType": "Variance Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sigma\\left(X\\right)=\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}:{\\quad}14.08556…$$", "steps": [ { "type": "step", "primary": "The variance is $$198.4032$$ , therefore $$\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}=\\sqrt{198.4032}$$", "result": "\\sqrt{198.4032}" }, { "type": "step", "primary": "Simplify", "result": "14.08556…" } ], "meta": { "interimType": "Compute The STDV Title 0Eq" } }, { "type": "step", "result": "14.08556…" } ] } }