Normal Distributions and z Scores PDF

Chapter 6 Normal Distributions and z Scores  Outline Normal Distributions The Standard Normal Distribution Standardizing Scores for Comparisons Calculating Probabilities with z Scores 2 ...

Chapter 6 Normal Distributions and z Scores  Outline Normal Distributions The Standard Normal Distribution Standardizing Scores for Comparisons Calculating Probabilities with z Scores 2  Normal Distributions Normal distribution: a theoretical distribution with data that are symmetrically distributed above and below the mean, median, and mode at the center of the distribution 3 Characteristics of the normal distribution: It’s mathematically defined, i.e., it has an equation It’s theoretical, i.e., rarely are behaviors exactly normally distributed The mean, median, and mode are located at the 50th percentile It’s symmetrical (which follows from the previous characteristic) The mean can be any value 4 Characteristics of the normal distribution (continued): The standard deviation can be any positive value because the smallest amount that scores vary from the mean is none, i.e., 0 The area under the curve is equal to 1.0 because it represents probabilities 5 Examples of normal distributions: 6 Because many, if not most, behaviors can be represented by a normal distribution, the empirical rule can be applied to predict behaviors, and calculate probabilities of behaviors occurring Empirical rule: the rule that for data that are normally distributed, approximately: 68% of data lie within 1 standard deviation of the mean 95% of data lie within 2 standard deviations of the mean 99.7% of data lie within 3 standard deviations of the mean 7 8  The Standard Normal Distribution Standard normal distribution: a normal distribution with a mean equal to 0 and a standard deviation equal to 1 It’s also called a z distribution It’s distributed in z-score units along the x-axis 9 z score: the number of standard deviations that a data point is from the mean on a standard normal distribution The standard normal distribution is a normal distribution, but not all normal distributions are standard normal distributions 10 The standard normal distribution: 11 z scores correspond to percentiles Percentile: a score that represents the percent of a distribution that is equal to or below that score E.g., the 50th percentile means 50% of scores are lower (i.e., to the left), and 50% are higher (i.e., to the right). 50% + 50% = 100% E.g., the 84th percentile means 84% of scores are lower (i.e., to the left), and 16% are higher (i.e., to the right). 84% + 16% = 100% 12 Percents equal area under curve. Nearly 100% of scores are within 3 SD of the mean. (z scores are on the x axis.) 13 z score: –3 = 0th percentile –2 = 2nd percentile –1 = 16th percentile 0 = 50th percentile 1 = 84th percentile 2 = 98th percentile 3 = 100th percentile 14 z table for z scores 0.00 to 3.49 List the percentile to 2 decimals: z score 1.00 = percentile z score –1.00 = percentile z score 2.22 = percentile z score.43 = percentile z score –.04 = percentile 15 List the z score to 2 decimals: 99.87th percentile = z score 85.99th percentile = z score 7.93rd percentile = z score 33rd percentile = z score 16  Standardizing Scores for Comparisons Transforming a normal distribution of scores to a standard normal distribution allows for comparisons, e.g., on which of the two exams below did a student perform relatively better? Exam A: Grade = 79%, µ = 70%, σ = 4% Exam B: Grade = 90%, µ = 80%, σ = 5% 17 Exam A: Grade = 79%, µ = 70%, σ = 4% Exam B: Grade = 90%, µ = 80%, σ = 5% Transform scores to z scores with the formula: z = (x − µ)/σ 18  Calculating Probabilities with z Scores Transforming a normal distribution of scores to a standard normal distribution allows for the calculation of probabilities of outcomes (and therefore behaviors) 19 E.g., the Achenbach Youth Self-Report form measures behavior problems and has the following parameters: u = 50 and σ = 10 What is the probability that a randomly selected youth will have a score below 60? This is another way of asking for the percentile that represents a score below 60 20 What is the probability that a randomly selected youth will have a score below 60, if u = 50 and σ = 10? 10 20 30 40 50 60 70 80 90 Youth Scores 21 What is the probability that a randomly selected youth will have a score of less than 60, if u = 50 and σ = 10? z = (x – u)/σ 22 What is the probability that a randomly selected a youth will have a score > 55, if u = 50; σ = 10? 10 20 30 40 50 60 70 80 90 Youth Scores 23 What is the probability that a randomly selected a youth will have a score > 55, if u = 50; σ = 10? z = (x – u)/σ 24 What percent of youths have a score of < 37, if u = 50; σ = 10? 10 20 30 40 50 60 70 80 90 Youth Scores 25 What percent of youths have a score of < 37, if u = 50; σ = 10? z = (x – u)/σ 26 What percent is between scores 42 and 56, if u = 50; σ = 10? 10 20 30 40 50 60 70 80 90 Youth Scores 27 What percent is between scores 42 and 56, if u = 50; σ = 10? z = (x – u)/σ 28 What score would 2.5% of youths be above, if u = 50; σ = 10? 10 20 30 40 50 60 70 80 90 Youth Scores 29 What score would 2.5% of youths be above, if u = 50; σ = 10? Notice you’re solving for x but you need z first: z = (x – u)/σ 30 How many youths’ scores are < 61, if N = 110; u = 50; σ = 10? 10 20 30 40 50 60 70 80 90 Youth Scores 31 Approximately how many youths’ scores are < 61, if N = 110; u = 50; σ = 10? z = (x – u)/σ 32 The previous problems calculated probabilities of ranges; not single scores because the standard normal distribution is continuous The probability of a single score occurring is theoretically infinitely small 33 The standard normal table can't be used, i.e., probabilities cannot be calculated, if scores to be transformed to z scores aren’t normally distributed E.g., if a skewed distribution is transformed, a z score of 0 is indeed the mean, but it's not the 50th percentile The transformation doesn’t make the data normally distributed 34 The z table is also known as the unit normal table or standard normal table 35