{"id":33,"date":"2022-05-20T16:59:03","date_gmt":"2022-05-20T16:59:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/comparing-variability-of-data-sets-what-to-know-4\/"},"modified":"2022-06-16T15:21:17","modified_gmt":"2022-06-16T15:21:17","slug":"comparing-variability-of-data-sets-what-to-know-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/comparing-variability-of-data-sets-what-to-know-3\/","title":{"raw":"Comparing Variability of Data Sets: Learn It 3","rendered":"Comparing Variability of Data Sets: Learn It 3"},"content":{"raw":"

Standard Deviation<\/h2>\r\nIn statistics, we are particularly interested in understanding how data are distributed and where each observation is in reference to the mean. This measurement of variability<\/strong> is called\u00a0standard deviation, <\/strong>which tells us 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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Recall<\/h3>\r\nWe'll be using statistical formulas, symbols, and language to discuss measures of variability. Take a moment to recall the formula you learned to calculate the mean of a sample. What symbols do we use to represent sample mean, summation, and sample size?\r\n\r\nCore skill: [reveal-answer q=\"450894\"]Express the formula for calculating the mean of a sample[\/reveal-answer]\r\n[hidden-answer a=\"450894\"]\r\n

[latex]\\bar{x}=\\dfrac{\\sum{x}}{n}[\/latex]<\/p>\r\nwhere,\r\n

[latex]\\bar{x}[\/latex] = the sample mean<\/p>\r\n

[latex]\\sum{x}[\/latex] = the sum of all the sample values [latex]x[\/latex]<\/p>\r\n

[latex]n[\/latex] = the sample size<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

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Standard Deviation VIDEO ANIMATED CONCEPT<\/h3>\r\nvideo about standard deviation and why we find it and it's purpose<\/span>\r\n\r\n<\/div>\r\n
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Standard Deviation<\/h3>\r\n\r\n
\r\n\r\n \r\n\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\nThe following steps can be applied to calculate a standard deviation by hand:\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 step 4<\/li>\r\n<\/ol>\r\n<\/div>\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

    [latex]\\bar{x}=\\frac{84}{6}=14[\/latex]<\/p>\r\nThen, identify all the squared differences.\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]<\/p>\r\nNext, take the square root of the sum of the squared differences divided by 1 less than the sample size.\r\n

    [latex]\\sqrt{\\dfrac{26+49+1+1+81+16}{6-1}}=\\sqrt{\\dfrac{184}{5}}\\approx{6.07}[\/latex]<\/p>\r\nThis process would be too tedious for large samples or populations, so we'll use that statistical technology tool to calculate standard deviation. Try it now by\u00a0choosing \"Your Own\" under \"Enter Data\" and enter the observations\u00a0[latex]8, 7, 13, 15, 23, 18[\/latex] to read the Std. Dev. from the Descriptive Statistics section.\r\n\r\nDid you obtain the same result?\r\n\r\n[\/hidden-answer]\r\n\r\nDATASET CHOICE OPTION\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