Decimals (Concepts & Computation)
Normal Distributions &
z score
Dec. 9
Normal Distribution
• Tend to cluster around the mean
• Symmetrical
• bell-shaped frequency polygon
• Mode, median, mean all occur at the highest point, in the center of distribution
• 3 units of SD to the left of the mean and
3 units of SD to the right of the mean
• 99.74% of polygon area lies btw –3SD and
+3SD
Skewed Distribution
• A large percentage the values is on either left or right side
• Positively skewed
– Has a tail to the right
• Negatively skewed
– Has a tail to the left
z scores
• Used to compare values of interval or ratio
–level variables
– Rita’s score of 21 out of 24 vs. Mary’s score 85 out of 100
• Variables should be normally distributed
• z score equals raw score minus mean divided by SD
• Can be converted to percentiles
• Known as standard scores
Areas of Normal Curve
Table
Examples of z Scores & corresponding Areas and percentiles
z score row column area percentile
.12
1.78
-2.90
1.15
-1.15
0.1
1.7
2.9
1.1
1.1
.02
.08
.00
.05
.05
04.78
46.25
49.81
37.49
37.49
54.78
96.25
.19
87.49
12.51
Exercise
1. On an IQ test with a mean of 100, SD of 16, what is the percentile of IQ 104?
• Debora leads a treatment group of college students diagnosed as chronic anxiety.
Group members have been selected for treatment on the basis their scores on
Anxiety Scale A with a mean of 70 and SD of 10 or Scale B with mean of 50 and SD of 12. Gina had score of 78 on the Scale A and Tom received 66 on the Scale B.
There is only one vacancy in the group.
Who should be in the treatment group?
Deriving Raw Scores from Percentiles
• Lauren wishes to form a treatment group of college students with high anxiety levels using the results of testing with Scale B (mean=50,
SD=12). She wants to include the top
10% of all students measured by
Scale B. What is the cutoff point?
• Subtract the percentile of mean (50) from
90 to determine the area of a normal curve that would fall between the 90 th percentile and the mean
• Find the z score corresponding to the area
• Use the z score formula, add 50 if the z score is positive
Exercise
One hundred students took mid-term exam designed to measure knowledge of statistics. Their scores on it formed a normal distribution with a mean of 80 and a standard deviation of 6.
Instructor decided to assign A to the top 10% of the scores. What score does
John have to get to earn?
Exercise
• 89,56,45,78,98,45,55,77,88,99,98,97,54,34,94
• 77,88,87,67,98,87,55,77,45,44,88,99,69,67,98
• Calculate the mode, median, mean, range, variance, and standard deviation for both sections. Which section did better overall on the exam?
