This document provides an overview of key concepts related to the normal distribution including:
- Properties and shapes of the normal distribution
- Using z-scores to transform between raw scores, proportions, and percentiles
- Finding areas under the normal curve using z-scores and the standard normal table
- Calculating percentiles and raw scores corresponding to given percentiles or proportions
The document uses examples based on a dataset of final grades to demonstrate techniques for working with the normal distribution.
Copyright:
Attribution Non-Commercial (BY-NC)
Available Formats
Download as PPT, PDF, TXT or read online from Scribd
This document provides an overview of key concepts related to the normal distribution including:
- Properties and shapes of the normal distribution
- Using z-scores to transform between raw scores, proportions, and percentiles
- Finding areas under the normal curve using z-scores and the standard normal table
- Calculating percentiles and raw scores corresponding to given percentiles or proportions
The document uses examples based on a dataset of final grades to demonstrate techniques for working with the normal distribution.
This document provides an overview of key concepts related to the normal distribution including:
- Properties and shapes of the normal distribution
- Using z-scores to transform between raw scores, proportions, and percentiles
- Finding areas under the normal curve using z-scores and the standard normal table
- Calculating percentiles and raw scores corresponding to given percentiles or proportions
The document uses examples based on a dataset of final grades to demonstrate techniques for working with the normal distribution.
Copyright:
Attribution Non-Commercial (BY-NC)
Available Formats
Download as PPT, PDF, TXT or read online from Scribd
This document provides an overview of key concepts related to the normal distribution including:
- Properties and shapes of the normal distribution
- Using z-scores to transform between raw scores, proportions, and percentiles
- Finding areas under the normal curve using z-scores and the standard normal table
- Calculating percentiles and raw scores corresponding to given percentiles or proportions
The document uses examples based on a dataset of final grades to demonstrate techniques for working with the normal distribution.
Copyright:
Attribution Non-Commercial (BY-NC)
Available Formats
Download as PPT, PDF, TXT or read online from Scribd
Download as ppt, pdf, or txt
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Chapter 10 The Normal Distribution
Properties of the Normal Distribution Shapes of Normal Distributions Standard (Z) Scores The Standard Normal Distribution Transforming Z Scores into Proportions Transforming Proportions into Z Scores Finding the Percentile Rank of a Raw Score Finding the Raw Score for a Percentile Chapter 10 1
Normal Distributions Normal Distribution A bell-shaped and symmetrical theoretical distribution, with the mean, the median, and the mode all coinciding at its peak and with frequencies gradually decreasing at both ends of the curve. The normal distribution is a theoretical ideal distribution. Real-life empirical distributions never match this model perfectly. However, many things in life do approximate the normal distribution, and are said to be normally distributed. Chapter 10 2
Scores Normally Distributed?
Table 10.1 Final Grades in Social Statistics of 1,200 Students (1983-1993) Midpoint Cum. Freq. Cum % Score Frequency Bar Chart Freq. (below) % (below) 40 * 4 4 0.33 0.33 50 ******* 78 82 6.5 6.83 60 *************** 275 357 22.92 29.75 70 *********************** 483 840 40.25 70 80 *************** 274 1114 22.83 92.83 90 ******* 81 1195 6.75 99.58 100 * 5 1200 0.42 100
Is this distribution normal? There are two things to initially examine: (1) look at the shape illustrated by the bar chart, and (2) calculate the mean, median, and mode.
Chapter 10 3
Scores Normally Distributed!
The Mean = 70.07 The Median = 70 The Mode = 70 Since all three are essentially equal, and this supports the bar graph in expressing that these data are normally distributed. Also, since the median is approximately equal to the mean, we know that the distribution is symmetrical. Chapter 10 4
The Shape of a Normal Distribution: The Normal Curve
Chapter 10 5
The Shape of a Normal Distribution
Notice the shape of the normal curve in this graph. Some normal distributions are tall and thin, while others are short and wide. All normal distributions, though, are wider in the middle and symmetrical. Chapter 10 6
Different Shapes of the Normal Distribution
Notice that the standard deviation changes the relative width of the distribution; the larger the standard deviation, the wider the curve. Chapter 10 7
Areas Under the Normal Curve by Measuring Standard Deviations
Chapter 10 8
Standard (Z) Scores
A standard score (also called Z score) is the number of standard deviations that a given raw score is above or below the mean.
Y Y Z Sy Chapter 10 9
Using the Standard Normal Table (Appendix B)
The curve s above show the area that is given in columns B and C of the standard normal table. Chapter 10 10
Finding the Area Between the Mean and a Positive Z Score
Using the data presented in Table 10.1, find the percentage of students whose scores range from the mean (70.07) to 85. (1) Convert 85 to a Z score: Z = (85-70.07)/10.27 = 1.45 (2) Look up the Z score (1.45) in column A, finding the proportion (.4265)
Chapter 10 11
Finding the Area Between the Mean and a Positive Z Score
(3) Convert the proportion (.4265) to a percentage (42.65%); this is the percentage of students scoring between the mean and 85 in the course. Chapter 10 12
Finding the Area Between the Mean and a Negative Z Score
Using the data presented in Table 10.1, find the percentage of students scoring between 65 and the mean (70.07) (1) Convert 65 to a Z score: Z = (65-70.07)/10.27 = -.49 (2) Since the curve is symmetrical and negative area does not exist, use .49 to find the area in the standard normal table: .1879
Chapter 10 13
Finding the Area Between the Mean and a Negative Z Score
(3) Convert the proportion (.1879) to a percentage (18.79%); this is the percentage of students scoring between 65 and the mean (70.07) Chapter 10 14
Finding the Area Between 2 Z Scores on the Same Side of the Mean Using the same data presented in Table 10.1, find the percentage of students scoring between 74 and 84. (1) Find the Z scores for 74 and 84: Z = .38 and Z = 1.36 (2) Look up the corresponding areas for those Z scores: .1480 and .4131
Chapter 10 15
Finding the Area Between 2 Z Scores on the Same Side of the Mean
(3) To find the highlighted area above, subtract the smaller area from the larger area (.4131-.1480=.2651) Now, we have the percentage of students scoring between 74 and 84. Chapter 10 16
Finding the Area Between 2 Z Scores on Opposite Sides of the Mean
Using the same data, find the percentage of students scoring between 62 and 72. (1) Find the Z scores for 62 and 72: Z = (72-70.07)/10.27 = .19 Z = (62-70.07)/10.27 = -.79 (2) Look up the areas between these Z scores and the mean, like in the previous 2 examples: Z = .19 is .0753 and Z = -.79 is .2852 (3) Add the two areas together: .0753 + .2852 = .3605
Chapter 10 17
Finding the Area Between 2 Z Scores on Opposite Sides of the Mean
(4) Convert the proportion (.3605) to a percentage (36.05%); this is the percentage of students scoring between 62 and 72. Chapter 10 18
Finding Area Above a Positive Z Score or Below a Negative Z Score
Find the percentage of students who did (a) very well, scoring above 85, and (b) those students who did poorly, scoring below 50. (a) Convert 85 to a Z score, then look up the value in column C of the standard normal table: Z = (85-70.07)/10.27 = 1.45 7.35% (b) Convert 50 to a Z score, then look up the value (look for a positive Z score!) in column C: Z = (50-70.07)/10.27 = -1.95 2.56% Chapter 10 19
Finding Area Above a Positive Z Score or Below a Negative Z Score
Chapter 10 20
Finding a Z Score Bounding an Area Above It
Find the raw score that bounds the top 10 percent of the distribution (Table 10.1) (1) 10% = a proportion of .10 (2) Using the standard normal table, look in column C for .1000, then take the value in column A; this is the Z score (1.28) (3) Finally convert the Z score to a raw score: Y=70.07 + 1.28 (10.27) = 83.22 Chapter 10 21
Finding a Z Score Bounding an Area Above It
(4) 83.22 is the raw score that bounds the upper 10% of the distribution. The Z score associated with 83.22 in this distribution is 1.28 Chapter 10 22
Finding a Z Score Bounding an Area Below It
Find the raw score that bounds the lowest 5 percent of the distribution (Table 10.1) (1) 5% = a proportion of .05 (2) Using the standard normal table, look in column C for .05, then take the value in column A; this is the Z score (-1.65); negative, since on the left side of the distribution (3) Finally convert the Z score to a raw score: Y=70.07 + -1.65 (10.27) = 53.12 Chapter 10 23
Finding a Z Score Bounding an Area Below It
(4) 53.12 is the raw score that bounds the lower 5% of the distribution. The Z score associated with 53.12 in this distribution is -1.65 Chapter 10 24
Finding the Percentile Rank of a Score Higher than the Mean
Suppose your raw score was 85. You want to calculate the percentile (to see where in the class you rank.) (1) Convert the raw score to a Z score: Z = (85-70.07)/10.27 = 1.45 (2) Find the area beyond Z in the Standard Normal Table (Column C): .0735 (3) Subtract the area from 1.00 for the percentile, since .0735 is the only area not below the score: 1.00 - .0735 = .9265 (proportion of scores below 85)
Chapter 10 25
Finding the Percentile Rank of a Score Higher than the Mean
(4) .9265 represents the proportion of scores less than 85 corresponding to a percentile rank of 92.65% Chapter 10 26
Finding the Percentile Rank of a Score Lower than the Mean
Now, suppose your raw score was 65. (1) Convert the raw score to a Z score Z = (65-70.07)/10.27 = -.49 (2) Find the are beyond Z in the Standard Normal Table, Column C: .3121 (3) Multiply by 100 to obtain the percentile rank: .3121 x 100 = 31.21% Chapter 10 27
Finding the Percentile Rank of a Score Lower than the Mean
Chapter 10 28
Finding the Raw Score of a Percentile Higher than 50
Say you need to score in the 95th% to be accepted to a particular grad school program. Whats the cutoff for the 95th%? (1) Find the area associated with the percentile: 95/100 = .9500 (2) Subtract the area from 1.00 to find the area above & beyond the percentile rank: 1.00 - .9500 = .0500 (3) Find the Z Score by looking in Column C of the Standard Normal Table for .0500: Z = 1.65 Chapter 10 29
Finding the Raw Score of a Percentile Higher than 50
(4) Convert the Z score to a raw score. Y= 70.07 + 1.65(10.27) = 87.02
Chapter 10 30
Finding the Raw Score of a Percentile Lower than 50
What score is associated with the 40th%? (1) Find the area below the percentile: 40/100 = .4000 (2) Find the Z score associated with this area. Use column C, but remember that this is a negative Z score since it is less than the mean: -.25 (3) Convert the Z score to a raw score: Y = 70.07 + -.25(10.27) = 67.5 Chapter 10 31
Finding the Raw Score of a Percentile Lower than 50
