std of 68,65,58,55,49

{ "query": { "display": "standard deviation $$68,\\:65,\\:58,\\:55,\\:49$$", "symbolab_question": "STATISTICS#std 68,65,58,55,49" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "std", "subTopic": "Other", "default": "7.64852…" }, "steps": { "type": "interim", "title": "Standard Deviation of $$68,\\:65,\\:58,\\:55,\\:49:{\\quad}7.64852…$$", "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:$${\\quad}58.5$$", "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, $$\\bar{x}:{\\quad}59$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=295$$", "steps": [ { "type": "step", "primary": "Take the sum of $$68,\\:65,\\:58,\\:55,\\:49$$", "result": "68+65+58+55+49" }, { "type": "step", "primary": "Simplify", "result": "295" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=5$$", "input": "68,\\:65,\\:58,\\:55,\\:49", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}68&65&58&55&49\\\\1&2&3&4&5\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "5" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAFZ9aeyZgsUaHG8WQcjMt1c6urOB/xTn7Xw/h3iKrWOU3kCh3oevUunZ7/b0qFKBQaX3Ff8s2oJCF512+gbP1WndBrgWO4fjYIqbXYybmyzvmJ1uLYQ2OEB6nvMp+GF4I=" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}59$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{295}{5}$$", "result": "\\frac{295}{5}" }, { "type": "step", "primary": "Simplify", "result": "59" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=59" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}234$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(68-59\\right)^{2},\\:\\left(65-59\\right)^{2},\\:\\left(58-59\\right)^{2},\\:\\left(55-59\\right)^{2},\\:\\left(49-59\\right)^{2}$$", "result": "\\left(68-59\\right)^{2}+\\left(65-59\\right)^{2}+\\left(58-59\\right)^{2}+\\left(55-59\\right)^{2}+\\left(49-59\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "234" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=5$$", "input": "68,\\:65,\\:58,\\:55,\\:49", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}68&65&58&55&49\\\\1&2&3&4&5\\end{Bmatrix}" }, { "type": "step", "primary": "The number of terms in the data set is", "result": "5" } ], "meta": { "interimType": "Compute Number Terms Specific 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAFZ9aeyZgsUaHG8WQcjMt1c6urOB/xTn7Xw/h3iKrWOU3kCh3oevUunZ7/b0qFKBQaX3Ff8s2oJCF512+gbP1WndBrgWO4fjYIqbXYybmyzvmJ1uLYQ2OEB6nvMp+GF4I=" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}58.5$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{234}{4}$$", "result": "\\frac{234}{4}" }, { "type": "step", "primary": "Simplify", "result": "58.5" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "58.5" } ], "meta": { "interimType": "Variance Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sigma\\left(X\\right)=\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}:{\\quad}7.64852…$$", "steps": [ { "type": "step", "primary": "The variance is $$58.5$$ , therefore $$\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}=\\sqrt{58.5}$$", "result": "\\sqrt{58.5}" }, { "type": "step", "primary": "Simplify", "result": "7.64852…" } ], "meta": { "interimType": "Compute The STDV Title 0Eq" } }, { "type": "step", "result": "7.64852…" } ] } }