{"id":482,"date":"2022-07-11T19:37:40","date_gmt":"2022-07-11T19:37:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/?post_type=chapter&p=482"},"modified":"2022-07-11T19:37:40","modified_gmt":"2022-07-11T19:37:40","slug":"z-score-and-the-empirical-rule-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/z-score-and-the-empirical-rule-background-youll-need-3\/","title":{"raw":"Z-Score and the Empirical Rule: Background You'll Need 3","rendered":"Z-Score and the Empirical Rule: Background You’ll Need 3"},"content":{"raw":"

Standardizing a Value<\/h3>\r\nA natural question to consider might be, given any value any distance from the mean in any direction, if we find the difference between the value and mean, then divide by the standard deviation, would we be able to discover the number of standard deviations any value is from the mean and whether it lies to the right or to the left? Answer Questions 7 and 8 to explore this idea.\r\n
\r\n

Interactive example<\/h3>\r\nRecall what you've been doing in the second part of all the questions above. You purposefully found values one, two, and three standard deviations from the mean. Then you tested that the formula work to confirm the distances of those values.\r\n\r\nThis time, we'll start with a value that is an unknown distance from a mean.\r\n\r\nAnswer the following given a dataset with a mean of 14.2 and a standard deviation of 1.9.\r\n\r\nRecall the formula\u00a0[latex]\\dfrac{\\text{value}-\\text{mean}}{\\text{standard deviation}}[\/latex]<\/span>\r\n\r\nRound answers to two decimal places as needed.\r\n
    \r\n \t
  1. How many standard deviations from the mean is the value of 16.5? Is it above or below? Make sure your calculation justifies this.<\/li>\r\n \t
  2. How many standard deviations from the mean is the value of 11.5? Is it above or below? Make sure your calculation justifies this.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"778056\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"778056\"]\r\n
      \r\n \t
    1. [latex]\\dfrac{16.5-14.2}{1.9}=1.21[\/latex]. The value 16.5 is 1.21 standard deviations above the mean.<\/span><\/li>\r\n \t
    2. [latex]\\dfrac{11.5-14.2}{1.9}=-1.42[\/latex]. The value 11.5 is 1.42 standard deviations below the mean.<\/span><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
      \r\n

      question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241196[\/ohm_question]\r\n\r\n[reveal-answer q=\"914574\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"914574\"]Find the difference (value - mean) divided by (standard deviation)[\/hidden-answer]\r\n\r\n<\/div>\r\n
      \r\n

      question 8<\/h3>\r\n[ohm_question hide_question_numbers=1]241197[\/ohm_question]\r\n\r\n[reveal-answer q=\"55941\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"55941\"]Find the difference (value - mean) divided by (standard deviation)[\/hidden-answer]\r\n\r\n<\/div>","rendered":"

      Standardizing a Value<\/h3>\n

      A natural question to consider might be, given any value any distance from the mean in any direction, if we find the difference between the value and mean, then divide by the standard deviation, would we be able to discover the number of standard deviations any value is from the mean and whether it lies to the right or to the left? Answer Questions 7 and 8 to explore this idea.<\/p>\n

      \n

      Interactive example<\/h3>\n

      Recall what you’ve been doing in the second part of all the questions above. You purposefully found values one, two, and three standard deviations from the mean. Then you tested that the formula work to confirm the distances of those values.<\/p>\n

      This time, we’ll start with a value that is an unknown distance from a mean.<\/p>\n

      Answer the following given a dataset with a mean of 14.2 and a standard deviation of 1.9.<\/p>\n

      Recall the formula\u00a0[latex]\\dfrac{\\text{value}-\\text{mean}}{\\text{standard deviation}}[\/latex]<\/span><\/p>\n

      Round answers to two decimal places as needed.<\/p>\n

        \n
      1. How many standard deviations from the mean is the value of 16.5? Is it above or below? Make sure your calculation justifies this.<\/li>\n
      2. How many standard deviations from the mean is the value of 11.5? Is it above or below? Make sure your calculation justifies this.<\/li>\n<\/ol>\n
        Show Answer<\/span><\/p>\n
        \n
          \n
        1. [latex]\\dfrac{16.5-14.2}{1.9}=1.21[\/latex]. The value 16.5 is 1.21 standard deviations above the mean.<\/span><\/li>\n
        2. [latex]\\dfrac{11.5-14.2}{1.9}=-1.42[\/latex]. The value 11.5 is 1.42 standard deviations below the mean.<\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
          \n

          question 7<\/h3>\n