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AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? Chapter 2: The Normal Distributions 2.1 Density Curves and the Normal Distribution A density curve is a smooth curve drawn from a histogram that describes the proportions of the observations that fall into a given range of values, not the counts. The areas under the curve represent specific proportions of the observations The total area under the curve is ALWAYS exactly 1 AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution β’ Skewed Right Density Curve β’π₯>π Skewed Left Density Curve π>π₯ AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? β’ Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution β’ π₯βπ AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? β’ Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution β’ The Normal Distribution β’ Symmetric, mounded, single-peaked, bell-shaped distributions β’ The mean and standard deviation of a normal distribution are π & π respectively. AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution β’ All Normal distributions have the same overall shape. The exact density curve for a particular normal distribution is described by giving the mean and standard deviation. The standard deviation, π, can be approximated by inspection as the point of inflection of the normal curve. The point is also known as the point where the graph changes from concave down to concave up. AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution β’ The Empirical Rule β’ AKA The 68-95-99.7 Rule AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution β’ Since the Normal Distribution is described by the men & standard deviation, a short-hand notation for normal distribution is π π, π AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What are the characteristics of a distribution that is normal, or somewhat normal? Chapter 2: The Normal Distributions β’ 2.1 Density Curves and the Normal Distribution AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns β’ Homework: AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What does the scatterplot of a normal probability plot look like? β’ Chapter 2: The Normal Distributions β’ 2.2 Standard Normal Calculations β’ π π, π describes all normal distributions β’ All distributions that are truly normal will also observe the Empirical Rule, or the 68-9599.7 rule. If two different populations are also normal, then we can compare both populations by βStandardizingβ the data. This new statistic is known as the standardized score, or the π§ score β’ π§= π₯βπ π β’ The value of π§ refers to the number of standard deviations from the mean an observation lies in a normal distribution. β’ Since π§ describes the number of standard deviations from the mean, then π§ can range in value from about β3.4 β€ π§ β€ 3.4 AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What does the scatterplot of a normal probability plot look like? β’ Chapter 2: The Normal Distributions β’ 2.2 Standard Normal Calculations β’ The π§ scores can also be plotted along the normal curve, more specifically known as the standard normal curve. AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What does the scatterplot of a normal probability plot look like? β’ Chapter 2: The Normal Distributions β’ 2.2 Standard Normal Calculations β’ Table A & TI Methods β’ π 100, 15 βΉFind the proportion of the population whose IQ lies above 108 β’ π 500, 100 βΉFind the proportion of the population whose SATM score lies below 540 β’ π 500, 100 βΉFind the proportion of the population whose SATV score lies between 490 & 560 AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What does the scatterplot of a normal probability plot look like? β’ Chapter 2: The Normal Distributions β’ 2.2 Standard Normal Calculations β’ Un-Standardizing a score β’ Table A & TI Methods β’ On a given standardized assessment, a standardized score of β.86 was received by a person. If the mean and standard deviation of the assessment were 78 & 8.4 respectively, what raw score was received by the person? β’ the height of an individual was taken and it was determined to be in the 88th percentile. If the mean height is 72 inches with a standard deviation of 2.5 inches, how tall the individual? AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns Essential Question: What does the scatterplot of a normal probability plot look like? β’ Chapter 2: The Normal Distributions β’ 2.2 Standard Normal Calculations β’ Assessing Normality β’ How can we tell for sure if a data is normal, or approximately normal? οConstruct a histogram or stem-plot ο§ Is the shape approximately bell-shaped? οCheck the Empirical Rule β’ How closely does the data observe the Empirical Rule? ο Construct the Normal Probability Plot β’ Normal data will reveal a linear scatter plot in which the independent variables are the raw data & the dependent variables are the π§ β scores of the percentiles of the raw data AP Statistics Part 1: Organizing Data: Looking for patterns and departures from patterns β’ Homework:
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