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3.1-3.4 Review Will be assessed on the Ch3 Summative Mr. Lieff 3.1 Graphical Displays Name and be able to analyze the various types of distributions Symmetric: Uniform, U-shaped, Mound-shaped Asymmetric: Left/Right-skewed How do you calculate bin width? (range) ÷ (# of bars) 3.2 Central Tendency Calculate mean, median, mode and weighted mean Determine which measure is appropriate Symmetric Mean/Median No outliers Mean Outliers Median Qualitative data; frequency important Mode Skewed distributions Right skewed: mode < median < mean Left-skewed: mean < median < mode 3.3 Measures of Spread Calculate and interpret Range, IQR and Standard Deviation (4-6 data points) A larger value for ANY measure of spread means the data has more spread (less consistent) Range size of the interval containing all of the data IQR size of the interval containing the middle 50% Std. dev. average deviation from the mean 3.3 Measures of Spread cont’d How to calculate IQR OMLUD* Order the data!!! * = credit to Chris/Jasmine/Holly Find the median, Q2 Find the 1st half median, Q1 Find the 2nd half median, Q3 IQR = Q3 – Q1 How to calculate Std.dev. Find the mean Find the deviations (data point – mean) Square the deviations Average the deviations variance σ2 Take square root std. dev. σ 3.4 Normal Distribution Know the characteristics of a Normal Distribution (68–95–99.7% Rule) Calculate the % of data in an interval based on std.dev. Ex: If a set of data has mean 10 and standard deviation 2, what percent of the data lie between 6 and 14? ans: 6 is 2 std dev below the mean and 14 is 2 std dev above. So 95% of the data falls in the range (see next slide) Normal Distribution 99.7% 95% 68% X ~ N (10,2 2 ) 34% 34% 0.15% 13.5% 13.5% 2.35% 2.35% 4 0.15% 6 8 10 12 14 16 Normal Distribution Ex: If a set of data has mean 10 and standard deviation 2, what percent of the data lie between 8 and 14? Ans: 34% + 34% + 13.5% = 81.5% Chapter 3/5 Review Normal Distribution and Binomial Probabilities Mr. Lieff 3.5 Z-Scores 2 Standard Normal Distribution X ~ N (0, 1 ) mean 0, std dev 1 z-scores map any data to this distribution 1) Calculate a z-score # of std. dev. above/below the mean z xx 2) Calculate the % of data below / above a value (z-score table on p. 398-399) 3) Calculate the percentile for a piece of data (round z-score table percentage to a whole number) 4) Calculate the percentage of data between 2 values (find z-scores, look up %s below both, subtract smaller from larger) 3.5 Z-Scores Ex: Given that X~N(10,22), what percent of the data is between 7 and 11? Ans: for 7: z = (7 – 10) ÷ 2 = -1.5 6.68% for 11: z = (11 - 10) ÷ 2 = 0.5 69.15% 69.15 – 6.68 = 62.47 So 62.47% of the data lies between these two values 5.3 Binomial Distributions recognize a binomial experiment situation n identical trials two possible outcomes independent events (constant probability) calculate probabilities for these situations n k nk P( X k ) p 1 p k 5.3 Binomial Distributions ex: A family decides to buy 5 dogs. If the chances of picking a male and female are equal, what is the probability of picking exactly 3 males? ans: using binomial probability distribution formula: 5 1 P( X 3) 3 2 3 1 2 5 3 0.3125 5.3 Binomial Distributions Calculate the expected value for # of passes on 4 tests if you have a 60% chance of passing each time Ans: E(X) = np = 4(0.60) = 2.4 So you are expected to pass 2.4 tests Expected value is an average continuous Don’t round in these situations 5.4 Normal Approximation of Binomial Distribution What about finding a range of probabilities e.g., What is the probability of picking between 1 and 4 males? e.g., P(1 ≤ X ≤ 4) = P(X=1) + P(X=2) + P(X=3) + P(X=4) If the number of trials is large enough, the binomial distribution approximates a Normal Distribution 5.4 Normal Approximation of Binomial Distribution Verify that a binomial distribution can be approximated by a Normal distribution np > 5 n(1 – p) > 5 Calculate 𝑥 and σ, given the number of trials, n, and the probability of success, p Use z-scores to calculate the probability of a range of data (below or above a value, or between two values) Use boundary values ending in .5! 5.4 Normal Approximation of Binomial Distribution ex: A die is rolled 100 times. What is the probability of getting fewer than 15 sixes? Ans: 1 5 Check : np 100 16.66 n(1 p ) 100 83.33 6 6 1 1 1 x 100 16.66 100 1 3.72 6 6 6 14.5 16.66 z 0.58 28.1% 3.72 From the z-score table, the probability is 28.1% 5.4 Normal Approximation of Binomial Distribution ex: what is the probability of getting between 15 and 20 sixes? ans: 1 1 1 x 100 16.66 100 1 3.72 6 6 6 14.5 16.66 for 15 : z 0.58 28.1% 3.72 20.5 16.66 for 20 : z 1.03 84.9% 3.72 p 84.9 28.1% 56.8% 3.6 Mathematical Indices Arbitrary numbers that provide a measure of something Given a set of data: Work with a given Index formula or create your own Interpret the meaning of calculated results Moving averages – use for data that fluctuates over time e.g., 3-day moving average is the average of every consecutive 3-day period (no wraparounds) Ex: BMI, Slugging Percentage, Moving Average Review Read through the class slides p. 199 #6; p. 200 #3-7 (use a table for #6) pp. 324 – 325 #7, 9-12 16 Multiple Choice 8 Problems You will be provided with: Formulas in Back of Book z-score table on p. 398