STAT 110 1nd Edition Lecture 9 Outline of Previous Lecture I. Shapes of Distributions II. StemplotsIII. Measures of CenterOutline of Current LectureI. Measures of Spread II. Mean vs. MedianIII. Standard Deviation vs. Quartiles Current LectureI. Measures of Spread (Variability)a. Quartiles i. Quartile 1 (Q1): the median of the data points that are below the median (the point that is 1/4 of the way up in the data set)ii. Quartile 2 (Q2): the median of all of the data points (the point that is 1/2 of the way up in the data set)iii. Quartile 3 (Q3): the median of the data points that are above the median (the point that is 3/4 of the way up in the data set)b. A five number summary for a data set is a list containing (in order) the minimum value, the Q1 value, the Q2 value, the Q3 value, and the maximum value. It is a way to describe the center and spread at the same time. We can describe a five number summary in numerical form, or as a graphical representation. The graphical representation of a five number summary is called a boxplot. i. In a boxplot, the bottom of the box represents Q1, the line running through the box represents the median, and the top of the box represents Q3. The line running perpendicular to the box and extending from the bottom of the box represents the minimum, and the line running perpendicular to the box and extending from the top of the box represents the maximum. Boxplots do not show the mean.ii. Side by Side boxplots are used to compare two or more distributions c. Interquartile Range (IQR)i. IQR = Q3 - Q1These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.1. Formally, an outlier is an extreme observation if it is larger than 1.5 times larger than the IQR above Q3, or if it is smaller than 1.5 times smaller than the IQR below Q1. d. Sample Standard Deviation (s): standard deviation is the average distance of all observations from the mean. On average, the data points are all X units away from the mean; this X value is the standard deviation. i. Computing the sample standard deviation:1. Find the distance of each observation from the mean (these are called deviations)2. Square each deviation 3. Sum all of the squared deviations and then divide by n-1 (this is called the variance (s²))4. Take the square root of the variance to get your standard deviation (s)ii. Standard deviation (s) is used to describe the variability of a distribution only when you are using the mean to describe the center of the distribution.iii. s = 0 only when all of the data values are equal to the mean iv. As the observations become more spread out, the standard deviation increases (s becomes larger) II. Mean vs. Mediana. The mean is strongly influenced by extreme values and outliers; the median is not. i. If your distribution is symmetric, you should use the meanii. If your distribution is skewed, you should use the median III. Standard Deviation vs. Quartiles a. The standard deviation is strongly influenced by extreme values and outliers; quartiles are