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MDM4U Chapter 3 Review Normal Distribution Mr. Lieff 3.1 Graphical Displays name and be able to interpret the various types of distributions ex: when would we use a histogram vs. a bar graph? Histogram necessary for continuous data Bar graph for qualitative data Discrete data depends on the spread ex: how do you calculate bin width? (range) ÷ (# of bars) 3.2 Central Tendency Be able to calculate mean, median, mode and weighted mean ex: determine which measure is appropriate Be aware of the effect of outliers Affect the mean more than the median Recognize the location of the measures with respect to skewed distributions if mode < median < mean…right skew If mean < median < mode…left skew 3.3 Measures of Spread Be able to calculate and interpret range, IQR and (population) standard deviation Up to 6 data points for std.dev. A larger value for any measure of spread (range, IQR, std.dev.) means the data has more spread range gives the size of the interval containing all of the data IQR gives the size of the interval containing the middle 50% of the data Std dev measures the variation from the mean 3.3 Measures of Spread cont’d How to calculate IQR Order the data!!! 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 – mean) Square the deviations Average the deviations variance σ2 Take square root std. dev. σ 3.4 Normal Distribution Be familiar with the characteristics of a Normal Distribution (68–95–99.7% rule) Calculate the ranges of expected data based on 1, 2 or 3 std.dev. above and/or below the mean 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 diagram) Normal Distribution 99.7% 95% 68% X ~ N ( x, 2 ) 34% 34% 0.15% 13.5% 13.5% 2.35% 2.35% x - 3σ x - 2σ 0.15% x - 1σ x x + 1σ x + 2σ x + 3σ 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% 3.5 Z-Scores Standard normal distribution X ~ N (0,12 ) mean 0, std dev 1 x 1) Be able to calculate a z-score z x 2) Be able to calculate the % of data below / above a value (z-table) 3) Given the standard deviation and the mean, be able to calculate the percentile for a piece of data (round z-table pct to whole number) 4) Be able to calculate the percent of data between 2 population values 3.5 Z-Scores Ex: Given that X~N(10,22), what percent of the population is between 7 and 11? Ans: calculate z-scores for the two data values, look up their respective percents in the z-table and subtract 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 3.6 Mathematical Indices These are arbitrary numbers that provide a measure of something e.g. BMI, Slugging Percentage, Moving Average You should be able to work with a given formula and interpret the meaning of calculated results Review p. 199 #1a, 3a, 4-6 20 Marks MC 30 marks Short Answer / Problem You will be provided with: Formulas in Back Of Book z-score table on p. 398 Tomorrow: bring an object of chance (coin, cards, dice, spinner, etc.)