{"id":1227,"date":"2022-04-07T22:43:36","date_gmt":"2022-04-07T22:43:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&p=1227"},"modified":"2022-05-20T21:43:55","modified_gmt":"2022-05-20T21:43:55","slug":"comparing-variability-of-data-sets-what-to-know-4","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data-sets-what-to-know-4\/","title":{"raw":"Comparing Variability of Data Sets: What to Know 4","rendered":"Comparing Variability of Data Sets: What to Know 4"},"content":{"raw":"
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Standard Deviation<\/h3>\r\n[perspective video -- a 3-instructor video showing how to think about standard deviation as a measure of variability. Cover the parts of the formula (go into\u00a0why squaring, why\u00a0df<\/em> if desired) but emphasize the concept of variability from std dev and variance more so than the technical use of the formula.]<\/span>\r\n\r\n<\/div>\r\n

Standard Deviation<\/h3>\r\nStandard deviation<\/strong> is a measure of how spread out observations are from the mean. The symbol we use to denote standard deviation differs depending on whether we are discussing a sample or a population. We use the Greek letter [latex]\\sigma[\/latex] (sigma) to denote the standard deviation of a population of observations.\u00a0We use the Latin letter\u00a0[latex]s[\/latex]\u00a0to denote the standard deviation of a sample of observations.\r\n
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Standard Deviation<\/h3>\r\nThe following formulas are used to calculate the standard deviation of a population and a sample:\r\n

Standard deviation of a population<\/strong>: [latex]\\sigma = \\sqrt{\\dfrac{\\sum \\left(x-\\mu\\right)^2}{n}}[\/latex], where [latex]\\mu[\/latex] represents the population mean.<\/p>\r\n

Standard deviation of a sample<\/strong>: [latex]s=\\sqrt{\\dfrac{\\sum \\left(x-\\bar{x}\\right)^2}{n-1}} [\/latex], where [latex]\\bar{x}[\/latex] represents the sample mean.<\/p>\r\n\r\n<\/div>\r\nThe following steps can be applied to calculate a standard deviation by hand.<\/span>\r\n

    \r\n \t
  1. Calculate the mean of the population or sample.<\/li>\r\n \t
  2. Take the difference between each data value and the mean. Then square each difference.<\/li>\r\n \t
  3. Add up all the squared differences<\/li>\r\n \t
  4. Divide by either the total number of observations in the case of a population or by 1 fewer than the total in the case of a sample.<\/li>\r\n \t
  5. Take the square root of the result of the division in step 4.<\/li>\r\n<\/ol>\r\n
    \r\n

    example<\/h3>\r\nA sample of observations is listed below. Find its standard deviation.\r\n

    [latex]8, 7, 13, 15, 23, 18[\/latex]<\/p>\r\n[reveal-answer q=\"911375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"911375\"]\r\n\r\nFirst, calculate the mean of the sample observations.\r\n\r\n[latex]\\bar{x}=\\frac{84}{6}=14[\/latex]\r\n\r\nIdentify all the squared differences.\r\n\r\n[latex]\\left(8-14\\right)^{2}=36\\text{, } \\left(7-14\\right)^{2}=49\\text{, } \\left(13-14\\right)^{2}=1\\text{, } \\left(15-14\\right)^{2}=1\\text{, } \\left(23-14\\right)^{2}=81\\text{, } \\left(18-14\\right)^{2}=16\\\\ [\/latex]\r\n\r\nThen, take the square root of the sum of the squared differences divided by 1 less than the sample size.\r\n\r\n[latex]\\sqrt{\\dfrac{26+49+1+1+81+16}{6-1}}=\\sqrt{\\dfrac{184}{5}}\\approx{6.07}[\/latex]\r\n\r\nThis process would be too tedious for large samples or populations! We'll use technology to calculate standard deviation from now on. Try it now using the\u00a0Describing and Exploring Quantitative Variables<\/em> tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.\u00a0 Choose \"Your Own\" under \"Enter Data\" and enter the observations\u00a0[latex]8, 7, 13, 15, 23, 18[\/latex] to read the Std. Dev. from Descriptive Statistics.\r\n\r\nDid you obtain the same result? That was much better than calculating it by hand!\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is a breakdown of the formula for standard deviation of a sample, [latex]s[\/latex].\r\n

    [latex]s=\\sqrt{\\dfrac{\\sum \\left(x-\\bar{x}\\right)^2}{n-1}} [\/latex]<\/p>\r\n\r\n