INTRODUCTORY STATISTICS Chapter 2 DESCRIPTIVE STATISTICS Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics CHAPTER 2 DESCRIPTIVE STATISTICS 2 1 Stem and Leaf Graphs Stemplots Line Graphs and Bar Graphs 2 2 Histograms Frequency Polygons and Time Series Graphs 2 3 Measures of the Location of the Data 2 4 Box Plots 2 5 Measures of the Center of the Data 2 6 Skewness and the Mean Median and Mode 2 7 Measures of the Spread of the Data 2 8 Descriptive Statistics Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics CHAPTER OBJECTIVES By the end of this chapter the student should be able to Display data graphically and interpret graphs stemplots histograms and box plots Recognize describe and calculate the measures of location of data quartiles and percentiles Recognize describe and calculate the measures of the center of data mean median and mode Recognize describe and calculate the measures of the spread of data variance standard deviation and range Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics 2 1 STEM AND LEAF GRAPHS STEMPLOTS LINE GRAPHS AND BAR GRAPHS Stem and Leaf Graphs To create the plot divide each observation of data into a stem and a leaf The leaf consists of a final significant digit For example 23 has stem two and leaf three The number 432 has stem 43 and leaf two Write the stems in a vertical line from smallest to largest Draw a vertical line to the right of the stems Then write the leaves in increasing order next to their corresponding stem Example Create a stem and leaf graph for the following data set Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics LINE GRAPHS Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics BAR GRAPHS Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics 2 2 HISTOGRAMS FREQUENCY POLYGONS AND TIME SERIES GRAPHS Histograms A histogram consists of contiguous adjoining boxes It has both a horizontal axis and a vertical axis The horizontal axis is labeled with what the data represents for instance distance from your home to school The vertical axis is labeled either frequency or relative frequency or percent frequency or probability The graph will have the same shape with either label The histogram like the stemplot can give you the shape of the data the center and the spread of the data To construct a histogram first decide how many bars or intervals also called classes represent the data Example Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics FREQUENCY POLYGONS Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics FREQUENCY POLYGONS Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics TIME SERIES GRAPH Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics 2 3 MEASURES OF THE LOCATION OF THE DATA Median The median is a number that measures the center of the data You can think of the median as the middle value but it does not actually have to be one of the observed values It is a number that separates ordered data into halves Half the values are the same number or smaller than the median and half the values are the same number or larger For example consider the following data 1 11 5 6 7 2 4 8 9 10 6 8 8 3 2 2 10 1 Ordered from smallest to largest 1 1 2 2 4 6 6 8 7 2 8 8 3 9 10 10 11 5 Since there are 14 observations the median is between the seventh value 6 8 and the eighth value 7 2 To find the median add the two values together and divide by two Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics QUARTILES Quartiles are numbers that separate the data into quarters Quartiles may or may not be part of the data To find the quartiles first find the median or second quartile The first quartile Q1 is the middle value of the lower half of the data and the third quartile Q3 is the middle value or median of the upper half of the data Examples The interquartile range is a number that indicates the spread of the middle half or the middle 50 of the data It is the difference between the third quartile Q3 and the first quartile Q1 IQR Q3 Q1 The IQR can help to determine potential outliers A value is suspected to be a potential outlier if it is less than 1 5 IQR below the first quartile or more than 1 5 IQR above the third quartile Potential outliers always require further investigation Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics OUTLIERS A potential outlier is a data point that is significantly different from the other data points These special data points may be errors or some kind of abnormality or they may be a key to understanding the data Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics INTERPRETING PERCENTILES QUARTILES AND MEDIAN A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest Percentages of data values are less than or equal to the pth percentile For example 15 of data values are less than or equal to the 15th percentile Low percentiles always correspond to lower data values High percentiles always correspond to higher data values Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics PERCENTILES A Formula for Finding the kth Percentile k the kth percentile It may or may not be part of the data i the index ranking or position of a data value n the total number of data Order the data from smallest to largest Calculate i k n 1 100 If i is a positive integer then the kth percentile is the data value in the ith position in the ordered set of data If i is not a positive integer then round i up and round i down to the nearest integers Average the two data values in these two positions in the ordered data set Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics PERCENTILES Prepared by the College of Coastal Georgia for OpenStax Introductory Statistics PERCENTILES Formula for Finding the Percentile of a Value in a Data Set Order the data from smallest to largest x the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile y the number of data values equal to the data value for which you want to find the percentile n the total number of data Calculate x 0 5y n 100 Then