{"id":1204,"date":"2022-04-07T19:21:59","date_gmt":"2022-04-07T19:21:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&p=1204"},"modified":"2022-05-20T16:44:17","modified_gmt":"2022-05-20T16:44:17","slug":"comparing-variability-of-data-sets-corequisite-support-activity-continued","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/comparing-variability-of-data-sets-corequisite-support-activity-continued\/","title":{"raw":"Comparing Variability of Data Sets: Corequisite Support Activity 2","rendered":"Comparing Variability of Data Sets: Corequisite Support Activity 2"},"content":{"raw":"

Deviation from the mean<\/h3>\r\nA deviation from the mean<\/strong> is the distance between an observation, [latex]x[\/latex], in a data set and the mean, [latex]\\bar{x}[\/latex], of the data set. To calculate the deviation from the mean, subtract the sample mean from each observation in the data set [latex]\\left(x-\\bar{x}\\right)[\/latex].\r\n\r\nPractice calculating the deviation from the mean in the following interactive example. Then, for the data table that follows, calculate the deviation from the mean to answer Question 4.\r\n
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Example<\/h3>\r\nSeven of\u00a0[latex]50[\/latex] observations a student made about her commute time by bicycle from her apartment to school are listed in the following table. Her mean commute time was\u00a0[latex]10[\/latex] minutes,\u00a0[latex]\\bar{x}=10[\/latex]. Use this information to calculate the deviation from the mean [latex]\\left(x-\\bar{x}\\right)[\/latex] for each of the seven observations given.\r\n\r\nExample: See the first entry in the table: an observed 8 minute commute time.\u00a0 Given a mean commute time of 10 minutes, the deviation from the mean for the observation of 8 minutes is [latex]8-10=-2[\/latex].\r\n\r\nComplete the table then check your answers below.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

Bike Ride to School (in minutes)<\/strong><\/p>\r\n

[latex]x[\/latex]<\/strong><\/p>\r\n<\/td>\r\n

\r\n

Deviation from the Mean (in minutes)<\/strong><\/p>\r\n

[latex]x-\\bar{x}[\/latex]<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n

8<\/strong><\/td>\r\n[latex]8 - 10 = -2[\/latex]<\/td>\r\n<\/tr>\r\n
10<\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n
12<\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n
14<\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n
5<\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n
15<\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n
9<\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"129517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"129517\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

Bike Ride to School (in minutes)<\/strong><\/p>\r\n

[latex]x[\/latex]<\/strong><\/p>\r\n<\/td>\r\n

\r\n

Deviation from the Mean (in minutes)<\/strong><\/p>\r\n

[latex]x-\\bar{x}[\/latex]<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n

8<\/strong><\/td>\r\n[latex]8 - 10 = -2[\/latex]<\/td>\r\n<\/tr>\r\n
10<\/strong><\/td>\r\n[latex]10 - 10 = 0[\/latex]<\/td>\r\n<\/tr>\r\n
12<\/strong><\/td>\r\n[latex]12 - 10 = 2[\/latex]<\/td>\r\n<\/tr>\r\n
14<\/strong><\/td>\r\n[latex]14 - 10 = 4[\/latex]<\/td>\r\n<\/tr>\r\n
5<\/strong><\/td>\r\n[latex]5 - 10 = -5[\/latex]<\/td>\r\n<\/tr>\r\n
15<\/strong><\/td>\r\n[latex]15 - 10 = 5[\/latex]<\/td>\r\n<\/tr>\r\n
9<\/strong><\/td>\r\n[latex]9 - 10 = -1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEach value listed in the first column is identified as an observation. Subtract the given mean, [latex]\\bar{x}=10[\/latex] from each observation to obtain the deviation from the mean. For example, the observation of\u00a0[latex]12[\/latex] minutes results in that ride to school being\u00a0[latex]2[\/latex] minutes above the mean (or a longer ride), while the observation of\u00a0[latex]5[\/latex] results in that ride to school being\u00a0[latex]5[\/latex] minutes below the mean (or a shorter ride).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow it's your turn to calculate the deviation from the mean for ten of the 30 hurricane damage observations from the data set \"Hurricane Damage.\" Follow the instructions above to reopen the data set if you have closed it.\r\n
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question 4<\/h3>\r\n[ohm_question hide_question_numbers=1]241046[\/ohm_question]\r\n
\r\n\r\n[reveal-answer q=\"719277\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"719277\"]Locate the value for [latex]\\bar{x}[\/latex] in Descriptive Statistics.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Representations of large numbers<\/h3>\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, try Question 5 on the next page.","rendered":"

Deviation from the mean<\/h3>\n

A deviation from the mean<\/strong> is the distance between an observation, [latex]x[\/latex], in a data set and the mean, [latex]\\bar{x}[\/latex], of the data set. To calculate the deviation from the mean, subtract the sample mean from each observation in the data set [latex]\\left(x-\\bar{x}\\right)[\/latex].<\/p>\n

Practice calculating the deviation from the mean in the following interactive example. Then, for the data table that follows, calculate the deviation from the mean to answer Question 4.<\/p>\n

\n

Example<\/h3>\n

Seven of\u00a0[latex]50[\/latex] observations a student made about her commute time by bicycle from her apartment to school are listed in the following table. Her mean commute time was\u00a0[latex]10[\/latex] minutes,\u00a0[latex]\\bar{x}=10[\/latex]. Use this information to calculate the deviation from the mean [latex]\\left(x-\\bar{x}\\right)[\/latex] for each of the seven observations given.<\/p>\n

Example: See the first entry in the table: an observed 8 minute commute time.\u00a0 Given a mean commute time of 10 minutes, the deviation from the mean for the observation of 8 minutes is [latex]8-10=-2[\/latex].<\/p>\n

Complete the table then check your answers below.<\/p>\n\n\n\n\n\n\n\n\n\n\n
\n

Bike Ride to School (in minutes)<\/strong><\/p>\n

[latex]x[\/latex]<\/strong><\/p>\n<\/td>\n

\n

Deviation from the Mean (in minutes)<\/strong><\/p>\n

[latex]x-\\bar{x}[\/latex]<\/strong><\/p>\n<\/td>\n<\/tr>\n

8<\/strong><\/td>\n[latex]8 - 10 = -2[\/latex]<\/td>\n<\/tr>\n
10<\/strong><\/td>\n<\/td>\n<\/tr>\n
12<\/strong><\/td>\n<\/td>\n<\/tr>\n
14<\/strong><\/td>\n<\/td>\n<\/tr>\n
5<\/strong><\/td>\n<\/td>\n<\/tr>\n
15<\/strong><\/td>\n<\/td>\n<\/tr>\n
9<\/strong><\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Show Solution<\/span><\/p>\n
\n\n\n\n\n\n\n\n\n\n\n
\n

Bike Ride to School (in minutes)<\/strong><\/p>\n

[latex]x[\/latex]<\/strong><\/p>\n<\/td>\n

\n

Deviation from the Mean (in minutes)<\/strong><\/p>\n

[latex]x-\\bar{x}[\/latex]<\/strong><\/p>\n<\/td>\n<\/tr>\n

8<\/strong><\/td>\n[latex]8 - 10 = -2[\/latex]<\/td>\n<\/tr>\n
10<\/strong><\/td>\n[latex]10 - 10 = 0[\/latex]<\/td>\n<\/tr>\n
12<\/strong><\/td>\n[latex]12 - 10 = 2[\/latex]<\/td>\n<\/tr>\n
14<\/strong><\/td>\n[latex]14 - 10 = 4[\/latex]<\/td>\n<\/tr>\n
5<\/strong><\/td>\n[latex]5 - 10 = -5[\/latex]<\/td>\n<\/tr>\n
15<\/strong><\/td>\n[latex]15 - 10 = 5[\/latex]<\/td>\n<\/tr>\n
9<\/strong><\/td>\n[latex]9 - 10 = -1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Each value listed in the first column is identified as an observation. Subtract the given mean, [latex]\\bar{x}=10[\/latex] from each observation to obtain the deviation from the mean. For example, the observation of\u00a0[latex]12[\/latex] minutes results in that ride to school being\u00a0[latex]2[\/latex] minutes above the mean (or a longer ride), while the observation of\u00a0[latex]5[\/latex] results in that ride to school being\u00a0[latex]5[\/latex] minutes below the mean (or a shorter ride).<\/p>\n<\/div>\n<\/div>\n<\/div>\n

Now it’s your turn to calculate the deviation from the mean for ten of the 30 hurricane damage observations from the data set “Hurricane Damage.” Follow the instructions above to reopen the data set if you have closed it.<\/p>\n

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question 4<\/h3>\n