std of 0.154,0.15,0.146,0.162,0.162,0.152,0.156,0.152,0.158,0.152,0.178

{ "query": { "display": "standard deviation $$0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178$$", "symbolab_question": "STATISTICS#std 0.154,0.15,0.146,0.162,0.162,0.152,0.156,0.152,0.158,0.152,0.178" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "std", "subTopic": "Other", "default": "0.00862…" }, "steps": { "type": "interim", "title": "Standard Deviation of $$0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178:{\\quad}0.00862…$$", "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:$${\\quad}0.00007…$$", "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, $$\\bar{x}:{\\quad}0.15654…$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=1.722$$", "steps": [ { "type": "step", "primary": "Take the sum of $$0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178$$", "result": "0.154+0.15+0.146+0.162+0.162+0.152+0.156+0.152+0.158+0.152+0.178" }, { "type": "step", "primary": "Simplify", "result": "1.722" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=11$$", "input": "0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}0.154&0.15&0.146&0.162&0.162&0.152&0.156&0.152&0.158&0.152&0.178\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCD7Qw6ykTeoDaFv1B9imsi8iLOVs7thsurDEKJBF4k9Qo3kqf235XlFxYuGx8Gy2hsjEFqG8NOzmVbJlpWdfroRj7VOYytIpJPPs4EzRLbvJdbA+zX4bD3u3gx65o2NJhPs1H69IJAo34JOa7+RHmOq3oYuMlAvpO2dke81PW5P2+73IFe9TCuC6TXh6hddSsUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}0.15654…$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{1.722}{11}$$", "result": "\\frac{1.722}{11}" }, { "type": "step", "primary": "Simplify", "result": "0.15654…" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=0.15654…" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}0.00074…$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(0.154-0.15654…\\right)^{2},\\:\\left(0.15-0.15654…\\right)^{2},\\:\\left(0.146-0.15654…\\right)^{2},\\:\\left(0.162-0.15654…\\right)^{2},\\:\\left(0.162-0.15654…\\right)^{2},\\:\\left(0.152-0.15654…\\right)^{2},\\:\\left(0.156-0.15654…\\right)^{2},\\:\\left(0.152-0.15654…\\right)^{2},\\:\\left(0.158-0.15654…\\right)^{2},\\:\\left(0.152-0.15654…\\right)^{2},\\:\\left(0.178-0.15654…\\right)^{2}$$", "result": "\\left(0.154-0.15654…\\right)^{2}+\\left(0.15-0.15654…\\right)^{2}+\\left(0.146-0.15654…\\right)^{2}+\\left(0.162-0.15654…\\right)^{2}+\\left(0.162-0.15654…\\right)^{2}+\\left(0.152-0.15654…\\right)^{2}+\\left(0.156-0.15654…\\right)^{2}+\\left(0.152-0.15654…\\right)^{2}+\\left(0.158-0.15654…\\right)^{2}+\\left(0.152-0.15654…\\right)^{2}+\\left(0.178-0.15654…\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "0.00074…" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=11$$", "input": "0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}0.154&0.15&0.146&0.162&0.162&0.152&0.156&0.152&0.158&0.152&0.178\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCD7Qw6ykTeoDaFv1B9imsi8iLOVs7thsurDEKJBF4k9Qo3kqf235XlFxYuGx8Gy2hsjEFqG8NOzmVbJlpWdfroRj7VOYytIpJPPs4EzRLbvJdbA+zX4bD3u3gx65o2NJhPs1H69IJAo34JOa7+RHmOq3oYuMlAvpO2dke81PW5P2+73IFe9TCuC6TXh6hddSsUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}0.00007…$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{0.00074…}{10}$$", "result": "\\frac{0.00074…}{10}" }, { "type": "step", "primary": "Simplify", "result": "0.00007…" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "0.00007…" } ], "meta": { "interimType": "Variance Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sigma\\left(X\\right)=\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}:{\\quad}0.00862…$$", "steps": [ { "type": "step", "primary": "The variance is $$0.00007…$$ , therefore $$\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}=\\sqrt{0.00007…}$$", "result": "\\sqrt{0.00007…}" }, { "type": "step", "primary": "Simplify", "result": "0.00862…" } ], "meta": { "interimType": "Compute The STDV Title 0Eq" } }, { "type": "step", "result": "0.00862…" } ] } }