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Statistics and Probability 13.2 Measures of Center and Spread

Essential Question: What are the different graphical displays of data?. Statistics and Probability 13.2 Measures of Center and Spread. 13.2 Measures of Center and Spread. Mean → Average Example 1: Mean Number of Accidents

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Statistics and Probability 13.2 Measures of Center and Spread

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  1. Essential Question: What are the different graphical displays of data? Statistics and Probability13.2 Measures of Center and Spread

  2. 13.2 Measures of Center and Spread • Mean → Average • Example 1: Mean Number of Accidents • A six-month study of a busy intersection reports the number of accidents per month as 3, 8, 5, 6, 6, 10. Find the mean number of accidents per month at the site. Solution: Add all the values, divide by the number of values

  3. 13.2 Measures of Center and Spread • Example 2, Mean Home Prices • In the real-estate section of the Sunday paper, the following houses were listed: • 2-bedroom fixer-upper: $98,000 • 2-bedroom ranch: $136,700 • 3-bedroom colonial: $210,000 • 3-bedroom contemporary: $289,900 • 4-bedroom contemporary: $315,500 • 8-bedroom mansion: $2,456,500 • Find the mean price, and discuss how well it represents the center of the data. $584,433.33

  4. 13.2 Measures of Center and Spread • Median → middle value of a data set • If the number of values is odd, the median is the number in the middle • If the number of values is even, the median is the average of the two middle numbers • Example 3: Median Home Prices • Find the median of the data set in example 2, and discuss how well it represents the center of data.

  5. 13.2 Measures of Center and Spread • Example 3: Median Home Prices • Find the median of the data set in example 2, and discuss how well it represents the center of data. • 2-bedroom fixer-upper: $98,000 • 2-bedroom ranch: $136,700 • 3-bedroom colonial: $210,000 • 3-bedroom contemporary: $289,900 • 4-bedroom contemporary: $315,500 • 8-bedroom mansion: $2,456,500 $249,950

  6. 13.2 Measures of Center and Spread • Mode → data value with the highest frequency • Most often used for qualitative data • Why? • If every value appears the same number of times, there is no mode • If two or more scores have equal frequency, the data is called bimodal (2 modes), trimodal (3 modes), or multimodal.

  7. 13.2 Measures of Center and Spread • Example 4: Mode of a Data Set • Find the mode of the data represented by the bar graph below

  8. 13.2 Measures of Center and Spread • Mean, Median, and Mode of a Distribution • Symmetric Distribution: mean = median • Skewed Left: mean is to the left of the median • Skewed Right: mean is to the right of the median

  9. 13.2 Measures of Center and Spread • Measures of Spread • Variability → spread of the data most least

  10. 13.2 Measures of Center and Spread • Standard Deviation → most common measure of variability • Best used with symmetric distribution (bell curve) • Measures the average distance of an element from the mean • Deviation→ individual distance of an element from the mean

  11. 13.2 Measures of Center and Spread • Standard Deviation • Find the mean • Determine each individual deviation • Square each individual deviation • Find the average of those squared values • This gives you the variance (σ2) • Take the square root of the variance • Denoted using the Greek letter sigma (σ) • Population versus Sample • When dealing with a sample of a population, divide by n-1 instead of n. The result is called the sample standard deviation, and is denoted by s. • As samples become larger, the deviation approaches the population standard deviation

  12. 13.2 Measures of Center and Spread • Find the standard deviation for the data set: • 2, 5, 7, 8, 10 • Find the mean: • Find each individual deviation: • Square each individual deviation: • Find the variance: • Population? Average n: • Sample? Use n – 1: • Take square root of each: • Population standard deviation: • Sample standard deviation: 32/5 = 6.4 4.4, 1.4, 0.6, 1.6, 3.6 19.36, 1.96, 0.36, 2.56, 12.96 37.2/5 = 7.44 37.2/4 = 9.3 σ ≈ 2.73 s = 3.05

  13. 13.2 Measures of Center and Spread • What is cool (but not necessary) to know: • 68% - 96% - 99% of population within 1-2-3 standard distributions • What I want you to know • What a standard deviation is • How to calculate it based on a population • How to calculate it based on a sample

  14. 13.2 Measures of Center and Spread • Assignment (Wed) • Page 862 – 863 • Problems 1 – 13 (odd) • Assignment (Thurs) • Page 862 – 863 • Problems 19 – 25, 35 & 37 (odds)

  15. 13.2 Measures of Center and Spread • Interquartile Range • Measure of variability that is resistant to extreme values • A median divides a data set into an upper & lower half • The first quartile, Q1, is the median of the lower half • The third quartile, Q3, is the median of the upper half • The interquartile range is the difference between the two quartiles, which represents the spread of the middle 50% of data

  16. 13.2 Measures of Center and Spread • Box & Whisker Plot • Need five pieces of data: minimum, Q1, median, Q3, maximum • Box is drawn, with the Q1 and Q3 representing the left and right sides of the box, respectively • Vertical line is drawn at the median • “Whiskers” are horizontal lines drawn from the left side of the box to the minimum, and right side to the maximum

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