MDM4U Chapter 3/5 Review Normal Distribution 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 Order the data!!! Find the median, Q2 Find the 1 st half median, Q1 Find the 2 nd 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 σ 2 Take square root  std. dev. σ OMLUD* * = credit to Chris/Jasmine/Holly

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 34% 13.5% 2.35% 68% 95% 99.7% %

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% % = 81.5%

3.5 Z-Scores Standard Normal Distribution mean 0, std dev 1 z-scores map any data to this distribution 1) Calculate a z-score 2) Calculate the % of data below / above a value (z- table on p ) 3) Calculate the percentile for a piece of data (round z-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(10,2 2 ), 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% – 6.68 = 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 Moving averages – use for data that fluctuates over time

5.3 Binomial Distributions recognize a binomial experiment situation n identical trials two possible outcomes independent events (constant probability) calculate probabilities for these situations

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 getting 3 males? ans: using binomial probability distribution formula:

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 Verify that a binomial distribution can be approximated by a Normal distribution np > 5 n(1 – p) > 5 Calculate and σ, given the number of trials, n, and the probability, p Use z-scores to calculate the probability of a range of data (below or above a value, or between two values)

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: 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:

Review Read through the class slides p. 199 #1a, 3a, 4-6 pp. 324 – 325 #2, 3, 4ac, 7cd, Multiple Choice? You will be provided with: Formulas in Back of Book z-score table on p. 398