# Ch. 3 - Edinboro University of Pennsylvania Ch. 3 -- part 2 Review of Ch. 2: Data - stem and leaf, dotplots, Summary of Ch. 3: Averages, standard deviation 5 number summaries Empirical Rule Z scores ma260notes_ch3.pptx Notation

Mean Sample Population Variance Standard deviation s Chebychevs Theorem Chebychevs Theorem: for k>1, at least (1- 1/k 2 )*100% of data falls within k stand dev of the mean.

For k=2: at least 1- 1/2 2 = 75% of data falls within 2 standard deviations of the mean (that is, between -2 and +2) For k =3: at least 1- 1/3 2 = 89% of data falls within 3 standard deviations of the mean (that is, between -3 and +3 ) Empirical Rule (discussed fully in later chapters) Empirical Rule: For normally distributed data, 68% of data falls with 1 standard deviation of the mean (between - and +) 95% of data falls with 2 standard deviation of the mean (between -2 and +2) 99.7% of data falls with 3 standard deviation of

the mean (between -3 and +3) Example #1 Example #1 test scores 30 32 51 42 41 51 25 26 47 88 61 52 42 22 35 53 44 60 51 48 90 52 40 Organize with a stem and leaf Organize with a dotplot _______________________________________ Mean, standard deviation, and outliers

= _______ Find limits: s=_________ How much of our data falls here? Chebychev says: +s= -s =

+2s= -2s= +3s= -3s= Note: One theory is that outliers fall outside of 2.5 st dev from mean Empirical rule says: 68% in within 1 st dev of mean

At least 75% falls within 2 st dev 95% within 2 st dev At least 89% 99.7% falls within within 3 st 3 st dev dev 5 number summary Low

Q1= P25 (median of the bottom half) Q2=M=P50 (median) Q3=P75 (median of the top half) High 5 number summary, IQR, and outliers Low_________ Position of Q1, the 25th percentile=(.25)(n+1)_________ Q1_______________ Position of M=Q2, the 50th percentile=(.5)(n+1)________ M=Q2__________ Position of Q3, the 75th percentile=(.75)(n+1)_________ Q3_______________ High_________ Interquartile Range= IQR=Q3-Q1= _______________ lower fence=LF=Q1-1.5(IQR)= __________________

upper fence=UF= Q3+1.5(IQR)=__________________ Another theory for Outliers:- Look outside of fence______ Box plot ___________________________________________________________ Empirical Rule Example #2 Example temperatures 48 53 72 61 51 34 41 43 63 39

88 47 47 Organize with a stem and leaf Organize with a dotplot _______________________________________ 35 32 Ex 2 = _______ Find limits: s=_________ How much

of our data falls here? Chebychev says: +s= -s = +2s= -2s= +3s= -3s= Note: One theory is that outliers fall

outside of 2.5 st dev from mean Empirical rule says: 68% in within 1 st dev of mean At least 75% falls within 2 st dev 95% within

2 st dev At least 89% 99.7% falls within within 3 st 3 st dev dev Ex 2 Low_________ Position of Q1, the 25th percentile=(.25)(n+1)_________ Position of M=Q2, the 50th percentile=(.5)(n+1)________ Position of Q3, the 75th percentile=(.75)(n+1)_________ Q1_______________ M=Q2__________

Q3_______________ High_________ Interquartile Range= IQR=Q3-Q1= _______________ lower fence=LF=Q1-1.5(IQR)= __________________ upper fence=UF= Q3+1.5(IQR)=__________________ Another theory for Outliers:- Look outside of fence______ Box plot ___________________________________________________________ Example #3 Example Organize with a stem and leaf

Organize with a dotplot _______________________________________ Continue Z scores Calculate z scores A z-score represents the distance from the mean in terms of standard deviation. Z= Ex: On a test with mean=100 and stdev=10, find the z scores for raw scores of: X=110 z= X=90z= X=80z=

X=105 z= Use the Empirical Rule to find percents Same Ex: A test has mean=100 and stdev=10. Using the Empirical Rule, find the percent of students with raw scores (x): Between 90 and 110 Between 80 and 120 Between 100 and 110 Less than 100 Greater than 100 Using z scores to compare Example: You got a score of 85 on a math test, where

the mean was 80 and the stdev was 10. You got a score of 92 on a physics test, where the mean was 90 and the stdev was 20. Use z scores to see which score was relatively better than the other.