variance of 82,44,67,52,120

{ "query": { "display": "variance $$82,\\:44,\\:67,\\:52,\\:120$$", "symbolab_question": "STATISTICS#variance 82,44,67,52,120" }, "solution": { "level": "PERFORMED", "subject": "Statistics", "topic": "variance", "subTopic": "Other", "default": "902" }, "steps": { "type": "interim", "title": "Sample Variance of $$82,\\:44,\\:67,\\:52,\\:120:{\\quad}902$$", "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, $$\\bar{x}:{\\quad}73$$", "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:$${\\quad}\\sum_{i=1}^{n}a_{i}=365$$", "steps": [ { "type": "step", "primary": "Take the sum of $$82,\\:44,\\:67,\\:52,\\:120$$", "result": "82+44+67+52+120" }, { "type": "step", "primary": "Simplify", "result": "365" } ], "meta": { "interimType": "Take Sum Of Set Title 0Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=5$$", "input": "82,\\:44,\\:67,\\:52,\\:120", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}82&44&67&52&120\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAPH6cV8dmqzpETS6G76c46bLMWDFrHTOo2D5n/oMNXp3ql8XXPq6bNQlMm+36iNhmdgIcqpKNB+/Ii+Z22dHUaDM5YpEOnFPH0Xui8+WLUCmGOmXsgz455KIYC+RBszao=" } }, { "type": "interim", "title": "Divide the sum by the number of terms and simplify:$${\\quad}73$$", "steps": [ { "type": "step", "primary": "Divide the sum by the number of terms:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{365}{5}$$", "result": "\\frac{365}{5}" }, { "type": "step", "primary": "Simplify", "result": "73" } ], "meta": { "interimType": "Compute The Average Title 0Eq" } }, { "type": "step", "result": "=73" } ], "meta": { "interimType": "Arithmetic Mean Top 1Eq" } }, { "type": "interim", "title": "Compute $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}3608$$", "steps": [ { "type": "step", "primary": "Take the sum of $$\\left(82-73\\right)^{2},\\:\\left(44-73\\right)^{2},\\:\\left(67-73\\right)^{2},\\:\\left(52-73\\right)^{2},\\:\\left(120-73\\right)^{2}$$", "result": "\\left(82-73\\right)^{2}+\\left(44-73\\right)^{2}+\\left(67-73\\right)^{2}+\\left(52-73\\right)^{2}+\\left(120-73\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "3608" } ], "meta": { "interimType": "Generic Compute Title 1Eq" } }, { "type": "interim", "title": "Compute the number of terms in the data set:$${\\quad}n=5$$", "input": "82,\\:44,\\:67,\\:52,\\:120", "steps": [ { "type": "step", "primary": "Count the number of terms in the data set", "result": "\\begin{Bmatrix}82&44&67&52&120\\\\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+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAPH6cV8dmqzpETS6G76c46bLMWDFrHTOo2D5n/oMNXp3ql8XXPq6bNQlMm+36iNhmdgIcqpKNB+/Ii+Z22dHUaDM5YpEOnFPH0Xui8+WLUCmGOmXsgz455KIYC+RBszao=" } }, { "type": "interim", "title": "Compute $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}902$$", "steps": [ { "type": "step", "primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{3608}{4}$$", "result": "\\frac{3608}{4}" }, { "type": "step", "primary": "Simplify", "result": "902" } ], "meta": { "interimType": "Compute The Variance Title 0Eq" } }, { "type": "step", "result": "902" } ] } }