The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 2 2 Displaying Distributions with Graphs 8 31 06 Lecture 2 2 1 Recall Graphical tools for categorical data Data Individuals Variables Bar graph Pie chart Categorical variables Quantitative variables Graphical tools for quantitative data Stemplot Distribution of variables Any questions about homework 8 31 06 Lecture 2 2 2 Example A study on litter size Data 170 observations 4 8 6 8 7 5 4 4 8 8 31 06 6 2 6 8 7 8 7 8 8 5 7 7 5 5 9 2 3 6 6 7 6 6 7 7 7 9 9 7 7 6 8 3 5 8 8 5 3 9 7 5 7 7 5 3 5 6 3 5 5 6 5 8 6 6 4 7 4 4 5 5 6 5 6 4 5 5 9 3 5 6 4 7 6 4 4 9 5 10 6 6 5 6 7 7 4 5 5 6 7 6 7 8 6 6 1 3 4 7 5 4 7 5 6 7 3 7 7 5 4 6 9 6 7 10 5 6 8 7 5 5 7 5 6 3 7 8 7 7 6 3 4 4 5 6 4 7 5 5 6 9 3 5 7 8 4 8 6 8 5 6 4 3 6 8 6 11 6 5 6 6 3 Lecture 2 2 3 Stem and leaf plot for pups 0 122333333333333344 35 0 555555555555555555555555 132 1 001 8 31 06 Lecture 2 2 4 Histogram breaks the range of the values of a quantitative variable into intervals and displays only the count or percent of the observations that fall into each interval You can choose any convenient number of intervals Intervals must be of equal width 8 31 06 Lecture 2 2 5 Example A study on litter size 8 31 06 Lecture 2 2 6 Data analysis in action don t hang up on me 8 31 06 Lecture 2 2 7 Data analysis in action don t hang up on me 8 31 06 Lecture 2 2 8 Example Call Center Data Financial firm call center Calls handled by Avi within 60 seconds October 666 December 523 Avi Service Time Data 8 31 06 Lecture 2 2 9 October Frequency Histogram 120 100 80 60 40 20 0 Frequency 6 12 18 24 30 36 42 48 54 60 calling time 8 31 06 Lecture 2 2 10 December Histogram 120 Frequency 100 80 Frequency 60 40 20 0 6 12 18 24 30 36 42 48 54 60 calling time 8 31 06 Lecture 2 2 11 Notes for Making Histogram Choose the number of classes sensibly Fig 1 2 1 6 Intervals must be of equal width Areas of the bars are proportional to the frequency 8 31 06 Lecture 2 2 12 Examining Distributions Overall Pattern Shape Center numerical Lecture 3 midpoint Spread numerical Lecture 3 range Deviations Outliers some values that fall outside the overall pattern 8 31 06 Lecture 2 2 13 Shapes of Distributions Graphs can help to determine shapes Modes local peaks of a distribution Unimodal one peak Bimodal two peaks Symmetric or skewed 8 31 06 Lecture 2 2 14 Shakespeare s Words Uni modal 8 31 06 Lecture 2 2 15 Tuition and fees bimodal or trimodal 8 31 06 Lecture 2 2 16 A bimodal histogram A modal class 8 31 06 Lecture 2 2 A modal class 17 Shakespeare s Words 8 31 06 Lecture 2 2 18 Left skewed Right skewed 8 31 06 Lecture 2 2 19 Iowa Test of Basic Skills vocabulary scores 8 31 06 Lecture 2 2 20 A study on litter size 8 31 06 Lecture 2 2 21 Bell shaped Histograms 8 31 06 Lecture 2 2 22 Summary Shapes of Distributions Symmetric histogram in which the right half is a mirror image of the left half Skewed to the right histogram in which the right tail is more stretched out than the left long tail to the right Skewed to the left histogram the left tail is more stretched out than the right long tail to the left Number of modal classes the number of distinct peaks in a histogram Bell shaped A histogram looks like a bell 8 31 06 Lecture 2 2 23 Time plots A time plot of a variable plots each obs against the time at which it was measured Time x axis Variable y axis Examples stock price unemployment rate daily temperature Great for identifying changing patterns over time What to look for Trend Seasonal variations Major deviations 8 31 06 Lecture 2 2 24 Example Number of Suicides in USA 1900 1970 8 31 06 Lecture 2 2 25 Call Center Daily Call Volume in Sep 2002 Time Plot of of Calls for Agent By Date in September 70000 60000 of Calls for Agent 50000 40000 30000 20000 10000 0 0 8 31 06 5 10 15 Date in September Lecture 2 2 20 25 30 26 Outliers Observations that lie outside the overall pattern of a distribution Possible reasons error in data entry most likely reason Equipment failure Human error Missing value code extraordinary individuals Jordan s salary 8 31 06 Lecture 2 2 27 Handling Outliers Detect it using graphical and numerical methods Check the data to make sure correct entry Reducing influence of outlier delete the observation BE CAREFUL Use transformations robust methods 8 31 06 Lecture 2 2 28 Call Center Daily Call Volume in Sep 2002 Time Plot of of Calls for Agent By Date in September 70000 60000 of Calls for Agent 50000 40000 30000 20000 10000 0 0 8 31 06 5 10 15 Date in September Lecture 2 2 20 25 30 29 Take Home Message Graphical tools for quantitative data Examine distributions Histograms Time plots Overall pattern Shape Symmetric or skewed How many modes Bell shaped Outliers 8 31 06 Lecture 2 2 30