# Statistics and Probability (Day 1) 13.2 Measures of Center and Spread

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Statistics and Probability (Day 1) 13.2 Measures of Center and Spread
Essential Question: What are the different graphical displays of data? Statistics and Probability (Day 1) 13.2 Measures of Center and Spread

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

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

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.

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

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.

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

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

13.2 Measures of Center and Spread
Assignment Page 862 – 863 Problems 1 – 17 (odd)

Statistics and Probability (Day 2) 13.2 Measures of Center and Spread
Essential Question: What are the different graphical displays of data? Statistics and Probability (Day 2) 13.2 Measures of Center and Spread

13.2 Measures of Center and Spread
Measures of Spread Variability → spread of the data 6 5 7 8 1 9 9 1 5 10 1 5 9 11 5 9 12 13 14 most least

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

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

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

13.2 Measures of Center and Spread
Using the calculator TI Calculators Make a list (2nd, minus sign, edit) Go into statistical functions (2nd, plus sign, Calc) Choose “OneVar” Go into list (2nd, minus sign, names) Choose the appropriate list Casio Calculators Menu (Stat – Menu item #2) Make a list Calc (F2) 1Var (F1)

13.2 Measures of Center and Spread
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 What is cool (but not necessary) to know: 68% - 96% - 99% of population within standard distributions

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

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 (Q3 – Q1), which represents the spread of the middle 50% of data

13.2 Measures of Center and Spread
Assignment Page 862 – 863 Problems 19 – 37 (odd)