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Unit 1.2 – Descriptive Statistics Standard Deviation Degrees of Freedom Variance 68-95-99.7 Rule Data Types Individuals Categorical Quantitative Bar Graphs Pie Charts Stem Plots Histograms Graphing Categorical Variables Graphing Quantitative Variables Dot Plots Unit 1.2 – Descriptive Statistics Part 1 Standard Deviation Degrees of Freedom Variance 68-95-99.7 Rule Unit 1 – Descriptive Statistics Our data set on temperature readings has been modified using a transformation and is shown below: 70.2 77.4 74.7 90.9 104.4 81.9 85.5 86.4 86.4 75.6 74.7 68.4 94.5 84.6 81.9 The most commonly used measure of spread in AP Statistics is standard deviation. Find both the variance and standard deviation for the data set above. Make sure you understand the relationship between variance and standard deviation. The degrees of freedom is simply n – 1 where n is the sample size. We will use n and n – 1 very often throughout the year. Unit 1 – Descriptive Statistics 70.2 77.4 74.7 90.9 104.4 81.9 85.5 86.4 86.4 75.6 74.7 68.4 94.5 84.6 81.9 Understanding the 68-95-99.7 Rule Often times we will talk about a data point or observation with respect to the mean and standard deviation. Ex 1. Mean = 82.5 Standard Deviation = 9.577 Unit 1 – Descriptive Statistics 70.2 77.4 74.7 90.9 104.4 81.9 85.5 86.4 86.4 75.6 74.7 68.4 94.5 84.6 81.9 Ex 1. Mean = 82.5 Standard Deviation = 9.577 There is a major assumption being made when using the 68-95 Rule and that is that the data is normally distributed. We will talk more about this idea in Unit 1.3 but for now, know that this means the distribution is spread about the mean proportionally to it’s standard deviation. In otherwords N(x,s) Unit 1 – Descriptive Statistics 70.2 77.4 74.7 90.9 104.4 81.9 85.5 86.4 86.4 75.6 74.7 68.4 94.5 84.6 81.9 Ex 1. Mean = 82.5 Standard Deviation = 9.577 Checking for Understanding… 1. What percent of days can we expect to have a temperature lower than 53.769˚ F? 2. What percent of days can we expect to have a temperature lower than 72.923˚ F? 3. What percent of days can we expect to have a temperature lower than 82.5˚ F? 4. What percent of days can we expect to have a temperature lower than 101.654˚ F? Unit 1 – Descriptive Statistics 70.2 77.4 74.7 90.9 104.4 81.9 85.5 86.4 86.4 75.6 74.7 68.4 94.5 84.6 81.9 Ex 1. Mean = 82.5 Standard Deviation = 9.577 Checking for Understanding… continued… 1. What percent of days can we expect to have a temperature higher than 63.346˚ F? 2. What percent of days can we expect to have a temperature higher than 72.923˚ F? 3. What percent of days can we expect to have a temperature higher than 82.5˚ F? 4. What percent of days can we expect to have a temperature higher than 92.077˚ F? Unit 1 – Descriptive Statistics 70.2 77.4 74.7 90.9 104.4 81.9 85.5 86.4 86.4 75.6 74.7 68.4 94.5 84.6 81.9 Ex 1. Mean = 82.5 Standard Deviation = 9.577 Checking for Understanding… continued… 1. What percent of days can we expect to have a temperature between 72.923˚ F and 92.077˚ F ? 2. What percent of days can we expect to have a temperature between 53.769˚ F and 111.231˚ F ? 3. What percent of days can we expect to have a temperature between 63.346˚ F and 92.077˚ F ? 4. What percent of days can we expect to have a temperature between 92.077˚ F and 111.231˚ F ? Unit 1.2 – Descriptive Statistics Part 2 Data Types Individuals Categorical Quantitative Unit 1.2 – Descriptive Statistics Our next task was to gather more detailed data over a one week period. Our data is shown below: Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Low 63 66 58 59 60 63 64 High 78 88 90 92 97 96 98 Humidity 45% 13% 10% 8% 18% 15% 8% Precipitation Air Quality Light Good No Fair No Poor No Poor No Fair No Fair No Poor Unit 1.2 – Descriptive Statistics Day Monday Low 63 High 78 Humidity 45% Precipitation Air Quality Light Good Tuesday Wednesday Thursday 66 58 59 88 90 92 13% 10% 8% No No No Fair Poor Poor Friday Saturday Sunday 60 63 64 97 96 98 18% 15% 8% No No No Fair Fair Poor 1. Identify the individuals and categories Unit 1.2 – Descriptive Statistics Day Monday Low 63 High 78 Humidity 45% Precipitation Air Quality Light Good Tuesday Wednesday Thursday 66 58 59 88 90 92 13% 10% 8% No No No Fair Poor Poor Friday Saturday Sunday 60 63 64 97 96 98 18% 15% 8% No No No Fair Fair Poor 2. Classify each category as categorical or quantitative Unit 1.2 – Descriptive Statistics Day Monday Low 63 High 78 Humidity 45% Precipitation Air Quality Light Good Tuesday Wednesday Thursday 66 58 59 88 90 92 13% 10% 8% No No No Fair Poor Poor Friday Saturday Sunday 60 63 64 97 96 98 18% 15% 8% No No No Fair Fair Poor 4. For each column, identify the most appropriate graphing type Unit 1.2 – Descriptive Statistics Day Monday Low 63 High 78 Humidity 45% Tuesday Wednesday Thursday 66 58 59 88 90 92 13% 10% 8% No No No Fair Poor Poor Friday Saturday Sunday 60 63 64 97 96 98 18% 15% 8% No No No Fair Fair Poor Ticket out the Door Precipitation Air Quality Light Good 5. Come up with your own example of a data set that includes all 4 vocab words and check with Mr. Newton Unit 1.2 – Descriptive Statistics Part 3 Graphing Categorical Variables Bar Graphs Pie Charts Graphing Quantitative Variables Dot Plots Stem Plots Histograms Unit 1.2 – Descriptive Statistics Categorical Data - Bar Graph Unit 1.2 – Descriptive Statistics Categorical Data – Pie Chart Unit 1.2 – Descriptive Statistics Quantitative Data – Dot Plot Unit 1.2 – Descriptive Statistics Quantitative Data – Stem and Leaf Plot Unit 1.2 – Descriptive Statistics Quantitative Data – Histogram Unit 1.2 – Descriptive Statistics Quantitative Data – Histogram