Chapter 2 Review•What type of graph is this? What kind of variable?!!•How would you describe the shape?Chapter 2 ReviewWhat type of graph is this? What kind of variable?!!What does this graph say about allowing or forbidding?Chapter 3: Numerical Summaries of Center and VariationReminder: Describing DistributionsThe things to always describe when considering a distribution!!•Shape - ex. unimodal, symmetric!•Center - the “typical” value!•Spread - how spread out is the data (variation)!!!What values do we use to measure center and spread?Center - The Typical ValueMean is the arithmetic averageMedian is the midpoint of ranked valuesMode is the most frequently observed valueMean•Mean is the arithmetic average.!!Below are the final scores of 5 previous Stats 10 students:!79, 82, 94, 83, 92 !!The mean of this data set is:79 + 82 + 94 + 83 + 925=4305= 86Median•Median is the midpoint of ranked values!Below are the final scores of 5 previous Stats 10 students:!79, 82, 94, 83, 92!In order to find the median we must first put the values in increasing order:!79, 82, 83, 92, 94!The median will be the value in the middle of the data set:!79, 82, 83, 92, 94Median•Let's add an outlier to the example data set: !3, 79, 82, 83, 92, 94!Notice we now have a even number of values so finding the median requires an extra small step:!3, 79, 82, 83, 92, 94 = (82+83)/2 = 82.5 The new mean will be:!outliers have huge effect on the meanMedian Example• What is the median?!309, 278, 5, 311, 320, 321, 309, 355, 355!5,278,309,309,311,320,321,355,355will have 50% above and belowMean vs. MedianImagine a town with 10 people, 9 of whom have 0 sheep and the 10th person has 1,000 sheep. The mean in this town 100 sheep, while the median is 0 sheep. Which is the typical value of sheep that a person owns in this town?!!!!!!!The long tail of skewed distributions and outliers affect the value of the mean by much more than the median, i.e. the median is resistant to outliers.!median is a better representationMean vs. Median!!!!•If the distribution is symmetric, center is the mean!•Symmetric: mean = median!•If the distribution is skewed or has outliers, center is the median!•Right-skewed: mean > median !•Left-skewed: mean < medianSpread•The standard deviation is described by the square root of the variance. The average distance of a value from the mean.!!!•The interquartile range (IQR) is the third quartile minus the first quartile:!IQR = Q3 – Q1!•The range is the maximum value minus the minimum value:!Range = Max - Min!Standard deviation•The standard deviation is described by the square root of the variance. The average distance of a value from the mean.!!!!•It represents a typical distance from the mean of observations.!• means to take each data point, subtract the mean, and then square that difference, this is called a deviation. !•The (sigma) means to add up all the all the deviations.!Steps to Finding the Standard Deviation1. Find the mean of your data.!2. Subtract the mean from each data point and then square those differences!3. Add all the squared values from #2!4. Divide the value from #3 by the number of your data points minus one. This gives you variance.!5. Take the square root of the variance to get standard deviation.!Standard Deviation Example•Let's again consider the final scores of 5 Stats 10 students. 79, 82, 94, 83, 92!1. We found the mean to be, x = 86!2. Now subtract the mean from each value and square the difference:!(79 – 86) = 49 !(82 – 86) = 16 !(94 – 86) = 64 !(83 – 86) =9 !(92 – 86) = 36!22222_Standard Deviation Example3. Sum all the the squared values from #2 !49 + 16 + 64 + 9 + 36 = 174!!4. Divide the #3 by the number of data points minus one!174/(5 – 1) = 174/4 = 43.5!!The variance is 43.5!!!5. Take the square root of the variance to get standard deviation!Standard Deviation Example•What is the standard deviation? !1, 5, 5, 1, 8!mean = 4(1-4)^2=9(5-4)^2=1(5-4)^2=1(1-4)^2=9(8-4)^2=16(9+1+1+9+16)=3636/(5-1)=9sqrt(9)=3S=3Comparing Standard Deviations•Which has the smallest, which has the largest standard deviation?Finding Q1, Q3, and IQR•Quartiles!•Q1 is the first quartile, 25% of the data are below this point !•Q2 is the median, 50% of the data are below this point!•Q3 is the third quartile, 75% of the data are below this point!•In order to find Q1 and Q3 we must first put the values in increasing order:!79, 82, 83, 92, 94!!!!•IQR is the difference between Q3 and Q1, i.e. the middle 50% of the data, so IQR = 93 – 80.5 = 12.5!Q1= 25% below, 75% aboveQ2= 50% above and belowQ3= 75% below, 25% aboveQ1= find median of the lower 50%Q3= find median of the upper 50%IQR Example•What is the IQR?!!1, 5, 5, 1, 7, 3, 1, 5, 9, 7, 8!1,1,1,3,5,5,5,7,7,8,9Median = 5Q1 = 1Q3 = 7IQR = 7-1 = 6Range•Again, consider our five Stat 10 scores: !!79, 82, 83, 92, 94!!•Range is simply max – min, so: !!Range = 94 – 79 = 15!!!•Range is a poor measure of spread because it is not resistant to outliers and generally doesn't tell us where most of the data is located.!Thinking About Variation•Since statistics is about variation, spread is an important fundamental concept of statistics!•Measures of spread help us talk about what we don't know!•When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation are small!•When the data values are scattered far from the center, the IQR and standard deviation are large!Which Center and Spread are best?•Use the mean and standard deviation when the distribution is mound shaped (unimodal, symmetric)!•Use the median and IQR when the distribution is skewed left or skewed right.!•If the distribution is not unimodal, it may be better to split the data!•Neither the mean nor the median represent typical values!•Investigate further into possible separate sub-populations!•Present graphs and statistics of sub-populations separately!Reminder: Describing Distributions•The things to always describe when considering a distribution!•Shape - ex. unimodal, symmetric!•Center - the“typical” value!•Use the mean as the typical value for symmetric distribution!•Use the median as the typical value for skewed distribution!•Spread - how spread out is the data!•Use the standard deviation as the spread for symmetric distribution!•Use the interquartile range (IQR) as the spread for skewed distribution!Choose the Numerical