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Do Now

Do Now. Compare the distributions using SOCS This data represents number of tardies for 40 BHS students during their high school career. CHAPTER 1 Exploring Data. 1.3 Describing Quantitative Data with Numbers. What About Spread? The Standard Deviation.

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Do Now

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  1. Do Now • Compare the distributions using SOCS This data represents number of tardies for 40 BHS students during their high school career.

  2. CHAPTER 1Exploring Data 1.3Describing Quantitative Data with Numbers

  3. What About Spread? The Standard Deviation • A more powerful measure of spread than the IQR is the standard deviation, which takes into account how far each data value is from the mean. • A deviation is the distance that a data value is from the mean. • Since adding all deviations together would total zero, we square each deviation and find an average of sorts for the deviations.

  4. What About Spread? The Standard Deviation (cont.) • The variance, notated by s2, is found by summing the squared deviations and (almost) averaging them: • The variance will play a role later in our study, but it is problematic as a measure of spread—it is measured in squared units!

  5. What About Spread? The Standard Deviation (cont.) • The standard deviation, s,is just the square root of the variance and is measured in the same units as the original data.

  6. Thinking About Variation • Since Statistics is about variation, spread is an important fundamental concept of Statistics. • Measures of spread help us talk about what we don’t know. • When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small. • When the data values are scattered far from the center, the IQR and standard deviation will be large.

  7. Larson/Farber 4th ed. Interpreting Standard Deviation • Standard deviation is a measure of the typical amount an entry deviates from the mean. • The more the entries are spread out, the greater the standard deviation.

  8. Measuring Spread: The Standard Deviation The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. Consider the following data on the number of pets owned by a group of 9 children. • Calculate the mean. • Calculate each deviation. • deviation = observation – mean deviation: 1 - 5 = - 4 deviation: 8 - 5 = 3 = 5

  9. Measuring Spread: The Standard Deviation 3) Square each deviation. 4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance. 5) Calculate the square root of the variance…this is the standard deviation. “average” squared deviation = 52/(9-1) = 6.5 This is the variance. Standard deviation = square root of variance =

  10. Measuring Spread: The Standard Deviation The standard deviationsxmeasures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. The average squared distance is called the variance.

  11. Choosing Measures of Center and Spread We now have a choice between two descriptions for center and spread • Mean and Standard Deviation • Median and Interquartile Range • Choosing Measures of Center and Spread • The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. • Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. • NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA!

  12. You are the maintenance engineer for a local high school. You must purchase fluorescent light bulbs for the classrooms. Should you choose Type A with μ = 3000 hours and σ = 200 hours, or Type B with μ = 3000 hours and σ = 250 hours? Explain.

  13. There are two bus routes with the following statistics:Which bus route should you take if you want to have the option of maybe arriving the earliest? Explain.

  14. Organizing a Statistical Problem As you learn more about statistics, you will be asked to solve more complex problems. Here is a four-step process you can follow. • How to Organize a Statistical Problem: A Four-Step Process • State: What’s the question that you’re trying to answer? • Plan: How will you go about answering the question? What statistical techniques does this problem call for? • Do: Make graphs and carry out needed calculations. • Conclude: Give your conclusion in the setting of the real-world problem.

  15. Data Analysis: Making Sense of Data • CALCULATE measures of center (mean, median). • CALCULATE and INTERPRET measures of spread (range, IQR, standard deviation). • CHOOSE the most appropriate measure of center and spread in a given setting. • IDENTIFY outliers using the 1.5 × IQR rule. • MAKE and INTERPRET boxplots of quantitative data. • USE appropriate graphs and numerical summaries to compare distributions of quantitative variables.

  16. Homework • Finish AP Problem Sheets – due THURSDAY, AUGUST 29th • Quiz 1.3 TOMORROW • Review tomorrow, Wednesday • End of Chapter 1 test: THURSDAY (FRQ), FRIDAY (MCQ) • Study suggestions: • The AP problem sheets for HW • Looking over review guide (you’ll get it in class tomorrow) • Reading Chapter 1 sections • Practice problems in textbook: AP Statistics Practice Test - pg. 78-81

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