{"id":29,"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-corequisite-support-activity-3-2\/"},"modified":"2022-07-11T18:34:31","modified_gmt":"2022-07-11T18:34:31","slug":"comparing-variability-of-data-sets-corequisite-support-activity-3-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/comparing-variability-of-data-sets-corequisite-support-activity-3-2\/","title":{"raw":"Comparing Variability of Data Sets: Background You'll Need 3","rendered":"Comparing Variability of Data Sets: Background You’ll Need 3"},"content":{"raw":"

Representations of large numbers<\/h2>\r\nTake a moment to consider the ways in which large numbers can be represented. In the table above, we see hurricane damage in millions of dollars in the column on the left. Look at the the bottom number in the column:\u00a0[latex]11,227[\/latex]. Presumably, that means\u00a0[latex]11,227[\/latex] millions of dollars. But what does that mean in terms of a pure number?\u00a0The hurricanes contributing to this data were catastrophic, causing billions of dollars of damage. Use the recall box below to see how to write a number like [latex]11,227[\/latex]\u00a0million dollars\u00a0<\/em>as $[latex]11.227[\/latex]\u00a0billion<\/em>.<\/span>\u00a0You may also see the Student Resource: Number-Word Combinations<\/em><\/a>.<\/span>\r\n
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recall<\/h3>\r\nIt can be helpful to communicate large numbers using a combination of numbers and words.\r\n\r\nWhen reading text containing a large value, we generally comprehend a number written as a combination of numbers and words more quickly than we do the pure number form. For example, it may take a moment to make sense of $[latex]1,250,000[\/latex] but we understand immediately what $[latex]1.25[\/latex] million represents.\r\n\r\nTake a moment to refresh your understanding of combining numbers and words to express large numbers.\r\n\r\nCore Skill:\r\n[reveal-answer q=\"301175\"]Express and interpret large numbers[\/reveal-answer]\r\n[hidden-answer a=\"301175\"]\r\n\r\nWhen a number is so large that it would be unwieldy to list all of its digits on a page, we often use a power of ten to represent some of the digits.\r\n\r\nFor example, one million is written\u00a0[latex]1,000,000[\/latex]: a one followed by six zeros: [latex]10^{6}[\/latex].\r\n\r\nA billion is a thousand million. It's written\u00a0[latex]1,000,000,000[\/latex]: a one followed by nine zeros; [latex]10^{9}[\/latex].\r\n

Or,\u00a0[latex]1,000[\/latex] followed by six zeros, since a thousand million is\u00a0[latex]1,000[\/latex] times [latex]1,000,000[\/latex].<\/p>\r\n

[latex]10^{3}\\times10^{6}=10^{3+6}=10^{9}[\/latex]<\/p>\r\nRecall, when we multiply by a million, we move the decimal point six places to the right in the number we are multiplying. That is, we multiply by [latex]10^{6}[\/latex]\r\n\r\nWe can express multiples of millions or billions using a combination of digits and words.\r\n\r\nEx. Write\u00a0[latex]35[\/latex] million as a number.\r\n

[latex]35\\times1,000,000=35,000,000[\/latex]<\/p>\r\nEx. Write\u00a0[latex]350[\/latex] million as a number.\r\n

[latex]350\\times1,000,000=350,000,000[\/latex]<\/p>\r\nEx. Write\u00a0[latex]3500[\/latex] million as a number.\r\n

[latex]3500\\times1,000,000=3,500,000,000[\/latex], which is 3 billion, 500 million.<\/p>\r\nNote that the final row of the table above gives\u00a0[latex]11,227[\/latex] millions of dollars in hurricane damage. How much is that in billions?\r\n\r\nEx. Write\u00a0[latex]11,227[\/latex] million as a number.\r\n

[latex]11,227\\times1,000,000=11,227,000,000[\/latex], which is\u00a0[latex]11[\/latex] billion,\u00a0[latex]227[\/latex] million.<\/p>\r\nWe can also write this as\u00a0[latex]11.227[\/latex] billion.\r\n\r\nEx. [latex]11.227\\times1,000,000,000=11.227\\times10^{9}=11,227,000,000[\/latex] by moving the decimal\u00a0[latex]9[\/latex] places to the right.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow that you have refreshed your understanding of combining numbers and words to express large numbers,\r\n

\r\n

question 5<\/h3>\r\n[ohm_question hide_question_numbers=1]241050[\/ohm_question]\r\n\r\n[reveal-answer q=\"101343\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"101343\"]What do you<\/em> think? Use the recall box above as a guide.[\/hidden-answer]\u00a0<\/span>\r\n\r\n<\/div>\r\n

Signed numbers as proximities<\/h2>\r\nBefore answering Question 6 and 7, you may wish to refresh your understanding of distance as an absolute value.\r\n
\r\n

recall<\/h3>\r\nWhen discussing the difference between two numbers as a distance, use the concept of absolute value to help make sense of the result. For example, we would say the difference between\u00a0[latex]-1[\/latex] and\u00a0[latex]3[\/latex] is four units even though taking their difference may result in a negative or a positive depending upon which we subtract from which.\r\n

[latex]-1-3=-4\\qquad\\text{ and }\\qquad3 - \\left(-1\\right)=4[\/latex]<\/p>\r\n

[latex]|-1-3|=4\\qquad\\text{ and }\\qquad|3 - \\left(-1\\right)|=4[\/latex]<\/p>\r\nSee the skill below if needed for an example of how absolute value can be applied in Questions 6 and 7 and how to interpret positive and negative results when calculating deviation from the mean.\r\n\r\nCore skill:\r\n[reveal-answer q=\"761688\"]Express a distance as an absolute value.[\/reveal-answer]\r\n[hidden-answer a=\"761688\"]\r\n\r\nSay the mean of a sample is given as [latex]\\bar{x}=12[\/latex] and the observations\u00a0[latex]7[\/latex] and\u00a0[latex]15[\/latex] are contained in the sample. Which value is closer to the mean?\r\n

For the value of\u00a0[latex]7[\/latex], [latex]x-\\bar{x} = 7-12=-5[\/latex]<\/p>\r\n

For the value of\u00a0[latex]15[\/latex], [latex]x-\\bar{x} = 15-12=3[\/latex]<\/p>\r\nWe might be tempted to conclude that\u00a0[latex]7[\/latex] is closer since\u00a0[latex]-5[\/latex] is a smaller number than\u00a0[latex]3[\/latex]. But distance is calculated using absolute value. The value of\u00a0[latex]7[\/latex] is\u00a0[latex]5[\/latex] units away from the mean (to the left) while the value of\u00a0[latex]15[\/latex] is only\u00a0[latex]3[\/latex] units away from the mean (to the right). To calculate which is closer, use absolute value.\r\n

For the value of\u00a0[latex]7[\/latex], [latex]|7-12|=|-5|=5[\/latex]<\/p>\r\n

For the value of\u00a0[latex]15[\/latex], [latex]|15-12|=|3|=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

\r\n

question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241051[\/ohm_question]\r\n\r\n[reveal-answer q=\"282551\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"282551\"]For each calculation, you subtracted the mean from the observed value. Why would some result in a negative deviation? See the interactive example above for an explanation.[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n

question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241052[\/ohm_question]\r\n\r\n[reveal-answer q=\"421294\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"421294\"]Think of closer<\/em> as being a distance (i.e., absolute value).[\/hidden-answer]\r\n\r\n<\/div>\r\nYou've learned how to calculate the deviation from the mean in this activity, which you'll be using in the upcoming section and following activity. You've also refreshed several mathematical skills and statistical definitions. Hopefully, you are feeling comfortable enough with these concepts to move on to the next section.","rendered":"

Representations of large numbers<\/h2>\n

Take a moment to consider the ways in which large numbers can be represented. In the table above, we see hurricane damage in millions of dollars in the column on the left. Look at the the bottom number in the column:\u00a0[latex]11,227[\/latex]. Presumably, that means\u00a0[latex]11,227[\/latex] millions of dollars. But what does that mean in terms of a pure number?\u00a0The hurricanes contributing to this data were catastrophic, causing billions of dollars of damage. Use the recall box below to see how to write a number like [latex]11,227[\/latex]\u00a0million dollars\u00a0<\/em>as $[latex]11.227[\/latex]\u00a0billion<\/em>.<\/span>\u00a0You may also see the Student Resource: Number-Word Combinations<\/em><\/a>.<\/span><\/p>\n

\n

recall<\/h3>\n

It can be helpful to communicate large numbers using a combination of numbers and words.<\/p>\n

When reading text containing a large value, we generally comprehend a number written as a combination of numbers and words more quickly than we do the pure number form. For example, it may take a moment to make sense of $[latex]1,250,000[\/latex] but we understand immediately what $[latex]1.25[\/latex] million represents.<\/p>\n

Take a moment to refresh your understanding of combining numbers and words to express large numbers.<\/p>\n

Core Skill:<\/p>\n

Express and interpret large numbers<\/span><\/p>\n
\n

When a number is so large that it would be unwieldy to list all of its digits on a page, we often use a power of ten to represent some of the digits.<\/p>\n

For example, one million is written\u00a0[latex]1,000,000[\/latex]: a one followed by six zeros: [latex]10^{6}[\/latex].<\/p>\n

A billion is a thousand million. It’s written\u00a0[latex]1,000,000,000[\/latex]: a one followed by nine zeros; [latex]10^{9}[\/latex].<\/p>\n

Or,\u00a0[latex]1,000[\/latex] followed by six zeros, since a thousand million is\u00a0[latex]1,000[\/latex] times [latex]1,000,000[\/latex].<\/p>\n

[latex]10^{3}\\times10^{6}=10^{3+6}=10^{9}[\/latex]<\/p>\n

Recall, when we multiply by a million, we move the decimal point six places to the right in the number we are multiplying. That is, we multiply by [latex]10^{6}[\/latex]<\/p>\n

We can express multiples of millions or billions using a combination of digits and words.<\/p>\n

Ex. Write\u00a0[latex]35[\/latex] million as a number.<\/p>\n

[latex]35\\times1,000,000=35,000,000[\/latex]<\/p>\n

Ex. Write\u00a0[latex]350[\/latex] million as a number.<\/p>\n

[latex]350\\times1,000,000=350,000,000[\/latex]<\/p>\n

Ex. Write\u00a0[latex]3500[\/latex] million as a number.<\/p>\n

[latex]3500\\times1,000,000=3,500,000,000[\/latex], which is 3 billion, 500 million.<\/p>\n

Note that the final row of the table above gives\u00a0[latex]11,227[\/latex] millions of dollars in hurricane damage. How much is that in billions?<\/p>\n

Ex. Write\u00a0[latex]11,227[\/latex] million as a number.<\/p>\n

[latex]11,227\\times1,000,000=11,227,000,000[\/latex], which is\u00a0[latex]11[\/latex] billion,\u00a0[latex]227[\/latex] million.<\/p>\n

We can also write this as\u00a0[latex]11.227[\/latex] billion.<\/p>\n

Ex. [latex]11.227\\times1,000,000,000=11.227\\times10^{9}=11,227,000,000[\/latex] by moving the decimal\u00a0[latex]9[\/latex] places to the right.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

Now that you have refreshed your understanding of combining numbers and words to express large numbers,<\/p>\n

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question 5<\/h3>\n