Ch 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 MeanVarianceStandard deviation Sample s Population 

Chebychev’s Theorem Chebychev’s 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 Organize with a stem and leaf Organize with a dotplot _______________________________________

Mean, standard deviation, and outliers = _______ s=_________ Find limits: + s = -s = +2s= -2s= +3s= -3s= Note: One theory is that outliers fall outside of 2.5 st dev from mean How much of our data falls here? Chebychev says: 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% falls within 3 st dev 99.7% within 3 st dev

5 number summary Low Q 1 = P 25 (median of the bottom half) Q 2 =M=P 50 (median) Q 3 =P 75 (median of the top half) High

5 number summary, IQR, and outliers Low_________ Position of Q 1, the 25 th percentile=(.25)(n+1)_________ Q 1_______________ Position of M=Q 2, the 50 th percentile=(.5)(n+1)________ M=Q 2__________ Position of Q 3, the 75 th percentile=(.75)(n+1)_________ Q 3_______________ High_________ Interquartile Range= IQR=Q 3 -Q 1= _______________ lower fence=LF=Q (IQR)= __________________ upper fence=UF= Q (IQR)=__________________ Another theory for Outliers:- Look outside of fence______ Box plot ___________________________________________________________

Empirical Rule

Example #2 Example temperatures Organize with a stem and leaf Organize with a dotplot _______________________________________

Ex 2 = _______ s=_________ Find limits: + s = -s = +2s= -2s= +3s= -3s= Note: One theory is that outliers fall outside of 2.5 st dev from mean How much of our data falls here? Chebychev says: 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% falls within 3 st dev 99.7% within 3 st dev

…Ex 2 Low_________ Position of Q 1, the 25 th percentile=(.25)(n+1)_________ Q 1_______________ Position of M=Q 2, the 50 th percentile=(.5)(n+1)________ M=Q 2__________ Position of Q 3, the 75 th percentile=(.75)(n+1)_________ Q 3_______________ High_________ Interquartile Range= IQR=Q 3 -Q 1= _______________ lower fence=LF=Q (IQR)= __________________ upper fence=UF= Q (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=110z= X=90z= X=80z= X=105z=

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.