Chapter 6 The Normal Distribution

2 Chapter 6 The Normal Distribution Major Points Distributions and area Distributions and area The normal distribution The normal distribution The standard normal distribution The standard normal distribution Setting probable limits on an observation Setting probable limits on an observation Measures related to z Measures related to z

3 Chapter 6 The Normal Distribution Distributions and Area We can apply the concepts of We can apply the concepts of Area Area Percentages Percentages Probability Probability Addition of areas Addition of areas Addition of probabilities Addition of probabilities When using the normal distribution

4 Chapter 6 The Normal Distribution 12 Convert Histogram to Frequency Polygon of Distribution: connect the dots…….. N=50

5 Chapter 6 The Normal Distribution The Normal Distribution “Normal” refers to the general shape of the distribution “Normal” refers to the general shape of the distribution See next slide See next slide Remember our descriptors for distributions Remember our descriptors for distributions The formula for the normal distribution allows us to The formula for the normal distribution allows us to Predict a proportion of cases that fall under the normal curve between 2 given x values Predict a proportion of cases that fall under the normal curve between 2 given x values Cont.

6 Chapter 6 The Normal Distribution Normal Distribution --cont. X is the value on the abscissa X is the value on the abscissa Y is the resulting height of the curve at X Y is the resulting height of the curve at X There are constants also in the formula There are constants also in the formula Do you need to memorize or use the formula? No! It’s just there to show you how they arrive at the percentages discussed in a minute. Do you need to memorize or use the formula? No! It’s just there to show you how they arrive at the percentages discussed in a minute.

7 Chapter 6 The Normal Distribution The Standard Normal Distribution We simply transform all X values to have a mean = 0 and a standard deviation = 1 We simply transform all X values to have a mean = 0 and a standard deviation = 1 Call these new values z Call these new values z Define the area under the curve to be 1.0, can look at the proportion of cases falling above mean and below mean Define the area under the curve to be 1.0, can look at the proportion of cases falling above mean and below mean

8 Chapter 6 The Normal Distribution Ex. of a transformed Distribution

9 Chapter 6 The Normal Distribution z Scores Calculation of z Calculation of z where  is the mean of the population and  is its standard deviation where  is the mean of the population and  is its standard deviation This is a simple linear transformation of X. This is a simple linear transformation of X.

10 Chapter 6 The Normal Distribution Tables of z We use tables to find areas under the distribution We use tables to find areas under the distribution A sample table is on the next slide A sample table is on the next slide The following slide illustrates areas under the distribution The following slide illustrates areas under the distribution

11 Chapter 6 The Normal Distribution z Table

12 Chapter 6 The Normal Distribution Using the Tables Define “larger” versus “smaller” portion Define “larger” versus “smaller” portion Distribution is symmetrical, so we don’t need negative values of z Distribution is symmetrical, so we don’t need negative values of z Areas between z = +1.5 and z = -1.0 Areas between z = +1.5 and z = -1.0 See next slide See next slide

13 Chapter 6 The Normal Distribution Calculating areas Area between mean and +1.5 = Area between mean and +1.5 = Area between mean and -1.0 = Area between mean and -1.0 = Sum equals Sum equals Therefore about 77% of the observations would be expected to fall between z = -1.0 and z = +1.5 Therefore about 77% of the observations would be expected to fall between z = -1.0 and z = +1.5

14 Chapter 6 The Normal Distribution Converting Back to X Assume  = 30 and  = 5 Assume  = 30 and  = 5 So 77% of the distribution is expected to lie between 25 and 37.5

15 Chapter 6 The Normal Distribution Probable Limits X =  + z   X =  + z   Our last example has  = 30 and  = 5 Our last example has  = 30 and  = 5 We want to cut off 2.5% in each tail, so We want to cut off 2.5% in each tail, so z = (found from a z table, mean to z of 1.96 =.475, but we usually round to z = 2). z = (found from a z table, mean to z of 1.96 =.475, but we usually round to z = 2). Cont.

16 Chapter 6 The Normal Distribution Probable Limits --cont. We have just shown that 95% of the normal distribution lies between 20.2 and 39.8, or +/ SD’s We have just shown that 95% of the normal distribution lies between 20.2 and 39.8, or +/ SD’s Therefore the probability is.95 that an observation drawn at random will lie between those two values Therefore the probability is.95 that an observation drawn at random will lie between those two values

17 Chapter 6 The Normal Distribution 20 Statistical Methods: The Normal Distributions Walsh & Betz (1995), p29

18 Chapter 6 The Normal Distribution Measures Related to z Standard score Standard score Another name for a z score, there are other standard scores, such as ….. Another name for a z score, there are other standard scores, such as ….. T scores: Scores with a mean of 50 and and standard deviation of 10, no negative values T scores: Scores with a mean of 50 and and standard deviation of 10, no negative values Others include IQ, SAT’s, etc Others include IQ, SAT’s, etc Percentile score/rank Percentile score/rank Percent of cases which fall at or below a given score in the norming sample- THIS is how we typically find a percentile rank, not calculating using that somewhat confusing formula Percent of cases which fall at or below a given score in the norming sample- THIS is how we typically find a percentile rank, not calculating using that somewhat confusing formula Cont.

19 Chapter 6 The Normal Distribution Related Measures --cont. Stanines Stanines Scores which are integers between 1 and 9, with mean = 5 and standard deviation = 2 Scores which are integers between 1 and 9, with mean = 5 and standard deviation = 2 Mainly used in education Mainly used in education

20 Chapter 6 The Normal Distribution Review Questions Why do you suppose we call it the “normal” distribution? Why do you suppose we call it the “normal” distribution? What do we gain by knowing that something is normally distributed? What do we gain by knowing that something is normally distributed? How is a “standard” normal distribution different? How is a “standard” normal distribution different? Cont.

21 Chapter 6 The Normal Distribution Review Questions --cont. How do we convert X to z? How do we convert X to z? How do we use the tables of z? How do we use the tables of z? If we know your test score, how do we calculate your percentile? If we know your test score, how do we calculate your percentile? What is a T score and why do we care? What is a T score and why do we care?