std of 3,832,3,779,3,655,3,642,3,579

{ "query": { "display": "standard deviation $$3,\\:832,\\:3,\\:779,\\:3,\\:655,\\:3,\\:642,\\:3,\\:579$$", "symbolab_question": "STATISTICS#std 3,832,3,779,3,655,3,642,3,579" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "std", "subTopic": "Other", "default": "372.54971…" }, "steps": { "type": "interim", "title": "Standard Deviation of $$3,\\:832,\\:3,\\:779,\\:3,\\:655,\\:3,\\:642,\\:3,\\:579:{\\quad}372.54971…$$", "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:$${\\quad}138793.28888…$$", "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, $$\\bar{x}:{\\quad}350.2$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=3502$$", "steps": [ { "type": "step", "primary": "Take the sum of $$3,\\:832,\\:3,\\:779,\\:3,\\:655,\\:3,\\:642,\\:3,\\:579$$", "result": "3+832+3+779+3+655+3+642+3+579" }, { "type": "step", "primary": "Simplify", "result": "3502" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=10$$", "input": "3,\\:832,\\:3,\\:779,\\:3,\\:655,\\:3,\\:642,\\:3,\\:579", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}3&832&3&779&3&655&3&642&3&579\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAUfXNYlQQ2JHJt6Z+qf+vXtxS4dZirQRaESp1548O3HlDZ132F90gY8mrVGx3DO7ec0UAm97e3ZdFs70jM7AY+HtBxF4YoCq+yXnRYGozDl5B00w6UQ2s6IWYitDhNtoeji/kK9z21uZiIOjEhPZ+e" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}350.2$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{3502}{10}$$", "result": "\\frac{3502}{10}" }, { "type": "step", "primary": "Simplify", "result": "350.2" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=350.2" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}1249139.6$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(3-350.2\\right)^{2},\\:\\left(832-350.2\\right)^{2},\\:\\left(3-350.2\\right)^{2},\\:\\left(779-350.2\\right)^{2},\\:\\left(3-350.2\\right)^{2},\\:\\left(655-350.2\\right)^{2},\\:\\left(3-350.2\\right)^{2},\\:\\left(642-350.2\\right)^{2},\\:\\left(3-350.2\\right)^{2},\\:\\left(579-350.2\\right)^{2}$$", "result": "\\left(3-350.2\\right)^{2}+\\left(832-350.2\\right)^{2}+\\left(3-350.2\\right)^{2}+\\left(779-350.2\\right)^{2}+\\left(3-350.2\\right)^{2}+\\left(655-350.2\\right)^{2}+\\left(3-350.2\\right)^{2}+\\left(642-350.2\\right)^{2}+\\left(3-350.2\\right)^{2}+\\left(579-350.2\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "1249139.6" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=10$$", "input": "3,\\:832,\\:3,\\:779,\\:3,\\:655,\\:3,\\:642,\\:3,\\:579", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}3&832&3&779&3&655&3&642&3&579\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAUfXNYlQQ2JHJt6Z+qf+vXtxS4dZirQRaESp1548O3HlDZ132F90gY8mrVGx3DO7ec0UAm97e3ZdFs70jM7AY+HtBxF4YoCq+yXnRYGozDl5B00w6UQ2s6IWYitDhNtoeji/kK9z21uZiIOjEhPZ+e" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}138793.28888…$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{1249139.6}{9}$$", "result": "\\frac{1249139.6}{9}" }, { "type": "step", "primary": "Simplify", "result": "138793.28888…" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "138793.28888…" } ], "meta": { "interimType": "Variance Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sigma\\left(X\\right)=\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}:{\\quad}372.54971…$$", "steps": [ { "type": "step", "primary": "The variance is $$138793.28888…$$ , therefore $$\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}=\\sqrt{138793.28888…}$$", "result": "\\sqrt{138793.28888…}" }, { "type": "step", "primary": "Simplify", "result": "372.54971…" } ], "meta": { "interimType": "Compute The STDV Title 0Eq" } }, { "type": "step", "result": "372.54971…" } ] } }