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ACC Module #2 Unit 2.8

ACC Module #2 Unit 2.8

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ACC Module #2 Unit 2.8

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  1. DemingEarly College High SchoolUnit 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics

  2. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.1 What Are Statistics The field of statistics describes relationships between quantities that are related, but not necessarily in a deterministic manner. For example, a graduating student’s salary will often be higher when the student graduates with a higher GPA, but this is not always the case.Likewise, people who smoke tobacco are more likely to develop lung cancer, but, in fact, it is possible for non- smokers to develop the disease as well.Statistics describes these kinds of situations, where the likelihood of some outcome depends on the starting data.

  3. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.1 What Are Statistics Descriptive statistics involves analyzing a collection of data to describe its broad properties such as average (or mean), what percent of the data falls within a given range, and other such properties.An example of this would be taking all of the test scores from a given class and calculating the average test score.Differential statistics attempts to use date about a subset of some population to make inferences about the rest of the population.An example of this would be taking a collection of students who received tutoring and comparing their results to a collection of students who did not receive tutoring, then using that comparison to try and predict whether the tutoring program is beneficial.

  4. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.1 What Are Statistics To be sure that inferences have a high probability of being true for the whole population, the subset that is analyzed needs to resemble a miniature version of the population as closely as possible. For this reason, statisticians like to choose random samples from the population to study, rather than picking a specific group of people based on some similarity. For example, studying the incomes of people who live in Portland does not tell anything useful about the incomes of people who live in Tallahassee.

  5. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Suppose that X is a set of data points some description of the general properties of this data need to be found.The first property that can be defined for this data set is the mean. To find the mean, add up all the data points, then divide by the total number of data points. This can be expressed using summation notation as:

  6. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range For example, suppose that in a class of 10 students, the scores on a test were 50, 60,65, 65, 75, 80, 85, 85, 90, 100. Therefore, the average (or mean) test score will be:The mean is a useful number if the distribution of data is normal , which roughly means that the frequency of different outcomes has a single peak and is roughly equally distributed on both sides of that peak.

  7. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range A normal distribution of data (sometimes called a Gaussian Distribution or Bell Curve) looks like the curve below.. However, it is less useful in some cases where the data might be split or where there are some outliers.

  8. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Outliers are data points that are far from the rest of the data.For example, suppose there are 90 employees and 10 executives at a company. The executives make $1000 per hour, and the employees make $10 per hour. Therefore, the average pay rate will be or $109 per hour. In this case the average (or mean) is not very descriptive.

  9. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range What is the mean of the following data set (10, 7, 5, 8, 0, 6, 5, -1)?The mean is the average of the numbers. You find the mean by adding the numbers together and then dividing by the number of numbers in the group.First, count how many numbers are in the group. There are 8 numbers. Now add all the numbers together: 10 + 7 + 5 + 8 + 0 + 6 + 5 + -1 = 40Now divide the sum by the number of numbers: 40 ÷ 8 = 5The mean is 5.

  10. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Another useful measurement is the median, In a data set X consisting of data points , the median is the point in the middle. The middle refers to the point where half the data comes before it and half comes after, when the data is recorded in numerical order.If n is odd, then the median is . If n is even, it is defined as )

  11. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range In the previous example of test scores, the two middle points are 75 and 80. Since there is no single point, the average of these two points needs to be found. The average is The median is generally a good value to use if there are a few outliers in the data. It prevents those outliers from affecting the “middle” as much as when using the mean. Since the outlier is a data point that is far from most of the other data points in a data set, this means an outlier also is any point far from the median of the target set. The outliers can have a substantial effect on the mean of the data set, but do not change the median or mode, or do not change them by a large quantity.

  12. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Example: What is the median in the data set ( 1, 8, 0, 1, 0, 2)?When the numbers are arranged from least to greatest, the median is the number in the middle. So first, arrange the numbers from least to greatest: 0 0 1 1 2 8.There is an even number of numbers, so there are two numbers in the middle 0 0 1 1 2 8The median is the mean of the two middle numbers. Find the mean of 1 and 1.1 + 1 = 2 2 ÷ 2 = 1 The median is 1.

  13. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Another measure to define for X is the mode. This is the data point that appears more frequently. If two or more data points all tie for the most frequent appearance, then each of them is considered a mode.In the case of the test scores, where the numbers were 50, 60, 65, 65, 75, 80, 85, 85, 90, 100, there are two modes 65 and 85 (both appear two times).

  14. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Consider the data set (3, 5, 6, 6, 6, 8). This has a median of 6 and a mode of 6, with a mean of . Now suppose a new data point of 1000 is added so that the data set is now (3, 5, 6, 6, 6, 8, 1000). This does NOT change the median or mode, which are both still 6. However, the mean (or average) is now which is approximately 147.7. In this case, the median and mode will be better descriptions for most of the data points.

  15. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range What is the mode of the data set (-3, 5, 5, -3, -6, 5)?The mode is the number that appears most often.First, arrange the numbers from least to greatest. -6 -3 -3 5 5 5 Now count how many times each number appears.-6 appears 1 time, 3 appears 2 times, 5 appears 3 times.The number that appears most often is 5. The mode is 5.

  16. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range The final measurement to define is range. The range is the difference between the greatest number and the least number (sometimes called the “spread” of the data set). Subtracting a negative number is the same as adding a positive number.For example, given the data set (-4, 4, -1, 6, -1, 1, 1), the range is 6 - (-4) = 10. Or given that the data set ( 24, 24, 24, x, 24) and the range is 0, which number could x be 16, 24, or 32? Since the range is 0, x must be 24.

  17. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range What is the range of the data set (-1, -2, -2, -4, -3, 0, 0, 0, -1)?The range is the difference between the greatest number and the least number. First, find the greatest number. The greatest number is 0.Next, find the least number. The least number is -4.Subtract the least number from the greatest number:0 − -4 = 0 + 4 = 4 The range is 4.

  18. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Mean - same as average, add up all the data points and divide by the number of data points.Median - is the point in the middle of the data set. If the data set is odd, the median is the middle point. If the data set is even, the median is the mean of the two middle points.Mode - is the data point that appears most frequently. We can have multiple modes but they must be equal.Range - is the difference between the greatest number and the least number.

  19. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range What is the range of the data set ( -4, 4, -1, 6, -1, 1, 1)? What is the mean the data set ( 75, 87, 68, 62, 88, 89, 95, 69)?

  20. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range What is the range of the data set ( -4, 4, -1, 6, -1, 1, 1)? 7 What is the mean the data set ( 75, 87, 68, 62, 88, 89, 95, 69)? 79.1

  21. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Amanda bought 6 items at a school sale. The items cost: $4.00, $4.00, $7.00, $8.00, $6.00, $7.00 What was the range of the prices of the items? What is the median of the data set? -51, -46, -51, -45, -49, -35, -56, -41

  22. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Amanda bought 6 items at a school sale. The items cost: $4.00, $4.00, $7.00, $8.00, $6.00, $7.00 What was the range of the prices of the items? $4.00 What is the median of the data set? -51, -46, -51, -45, -49, -35, -56, -41 -47.5

  23. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Linda's Diner had 5 entrees on the menu. The prices were: $27.00, $28.00, $26.00 $30.00 and $27.00 What was the mode of the entree prices? What is the median of the data set? -51, -46, -51, -45, -49, -35, -56, -41

  24. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range Linda's Diner had 5 entrees on the menu. The prices were: $27.00, $28.00, $26.00 $30.00 and $27.00 What was the mode of the entree prices? $27.00 What is the median of the data set? -51, -46, -51, -45, -49, -35, -56, -41 -45.5

  25. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range What is the mean of the data set? 5, 9, 9, 2, 2, 7, 10, 4 What is the mode of the data set? 2, 4, 2, 4, 6, 6, 6, 4, 6, 2

  26. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.8 Descriptive Statistics 2.8.2 Mean, Median, Mode, and Range What is the mean of the data set? 5, 9, 9, 2, 2, 7, 10, 4 6 6 What is the mode of the data set? 2, 4, 2, 4, 6, 6, 6, 4, 6, 2

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