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Intro Stat Notes Four

by: Qihua Wu

Intro Stat Notes Four TMATH 110 C

Qihua Wu
University of Washington Tacoma
Intro Stat Applications

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About this Document

This week focuses on box plots, including percentile, outliers, modified box plots, and distributions of box plots.
Intro Stat Applications
Class Notes
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This 4 page Class Notes was uploaded by Qihua Wu on Sunday October 25, 2015. The Class Notes belongs to TMATH 110 C at University of Washington Tacoma taught by KENNEDY,MAUREEN C. in Fall 2015. Since its upload, it has received 12 views. For similar materials see Intro Stat Applications in Math at University of Washington Tacoma.


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Date Created: 10/25/15
Percentiles describe the location of the data in the data set It separates the data in 100 groups every increment of percentile is 1 To nd out the percentile of x nd the number of values that are LESS than x divide by the total number of data then multiply 100 When we nd out the percentile of x let39s say the percentile is 30 this value represents that 30 of the data are below it and 70 of the data are above it Finding the value that is at the percentile of x There is no best way to nd the values The one we are using in class is 1 Sort the data 2 Calculate L by using k100 multiply n L is the location of the percentile k is the percentile that is given and n is the sample size a If you get a whole number for L then L is the Lth value L1th value then divide by 2 b If you get a decimal value always round up to the next whole number and the data value at this rounded up L is the value you are looking for Quartiles also describes the location of the data instead of dividing the data set into 100 groups quartile divide data set into 4 groups which means the rst quartile is equivalent to P25 and so on Quartiles is not sensitive to extreme values it only shifts the location In fact all quartiles care about is the location of the sorted data not its value Boxplot includes minimum value rst quartile second quartile median third quartile and maximum value also known as venumber summary How to construct a box plot 1 Sort the data and identify the ve numbers minimum Q1 Q2 Q3 and maximum 2 Construct either a horizontal or vertical scale that includes the range of the data whether horizontal or vertical is personal preference 3 At the corresponding values of Q1 and Q3 draw a box that extends between these two values and draw a vertical line inside the box at Q2 4 Extending the line outward from the box to both the minimum and maximum values Remember to Label your axis Modi ed box plot similar to boxplots except it excludes all the outliers from the box and use dots to represent the outliers making it easier to identify outliers with a modi ed box plot Outliers an observation that falls very far from the rest of the observations To identify an outlier 1 Find the IQR interquartile range by using Q3 Q1 2 A data value is an outlier if a It is greater than Q3 15 IQR b It is smaller than Q1 15 IQR When IQR increases so is the variability since IQR is the difference in values between Q3 and Q1 when Q3 and Q1 are very different then IQR would increase and the variability thus increases Reasons for outiers to appear 1 Quality Assurance or Quality Control a There could be an error in recording the data point go back and check the source data to make sure this is the real data value and the actual measured value is not caused by miscaibrated equipment 2 Misspeci ed sampling frame The measured individual is not a population of interest 3 Something else is happening new theories and experiments would arise To draw the modi ed box plot 1 Find the 5 number summary a Compared to the regular box plot Q1 Q2 and Q3 would be the same b Identify the outliers c The minimum and maximum values for the modi ed box plot would be the minimum and maximum values that are not outliers Draw it the same way as a regular box pot e Represent the outliers with a dot at the corresponding value 0 Labels Box plots help in identifying the distribution of the data Symmetric distribution symmetric and equal distances between all of the data in 5number summaries and the outliers Positive skew median is closer to 01 and there is a longer tail on the right side Negative skew median closer to Q3 and there is a longer tail on the left side 0 For skews both the median and tails need to correspond to one skew for example you can not have a positive skew when your median is closer to 01 but it has a longer tail on the left side 0 In real world when we look at distributions we have to look at both the histogram and boxplots since boxplot only uses ve numbers whereas histogram uses the entire data set zscores shows how many standard deviations a data is away from the mean when a data is not within 2 standard deviations from the mean it is then unusuaL Z unitless datamean standard deviation When standard deviation is 0 zvalue is irrelevant since there is no variability if the data is not the mean then it is unusual Empirical Rule Bellshaped distribution only use for normal distribution When the data is within positive or negative 1 standard deviation of mean 68 2 standard deviation of mean 95 3 standard deviation of mean 997 When we try to nd the percentile of a certain value we add or subtract the remaining of the data Ex If we have a positive 1 standard deviation from the mean we know when data is within positive or negative 1 standard deviation is 68 and this is a normal distribution then the percentile for the data positive 1 standard deviation from the mean is 68 100682 84 Chebyshev s Theorem only for symmetric distribution the range of the data within k standard deviations where k is greater than 1 of the mean is always at least 11kk P be the proportion of values P is greater than or equal to 11kk At least 75 of the data are within 2 standard deviation of mean and 89 within 3 standard deviation etc


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