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Chapter 3 An Introduction to Statistical Problem Solving in Geography Summarized by Lana Hesler Learning Objectives Understand the basic descriptive measures of central tendency Understand the basic descriptive measures of dispersion Understand the concept of relative variability Determine the value of measuring shape or relative position Realize potential effects of location data on descriptive statistics Summarizing Data Sets Measures of central tendency ◦ Numbers that represent the center or typical value of a frequency distribution Includes mode, median, and mean Measures of dispersion ◦ Numbers that depict the amount of spread or variability in a data set Includes range, interquartile range, standard deviation, variance, and coefficient variation Summarizing Data Sets (cont.) Measures of shape or relative position ◦ Numbers that further describe the nature or shape of a frequency distribution Includes skewness – symmetry of a distribution Includes kurtosis – degree of flatness or peakedness in a distribution Descriptive Statistics Mode ◦ Value that occurs most frequently Median ◦ Middle value from a set of ranked observations. Value with equal number of data units above and below it. Mean ◦ The arithmetic average of the values Graphics provided by: http://www.transtutors.com/statistics-homework-help/numerical-measures Weighted Mean Weighted Mean defined ◦ Arithmetic average calculated from class intervals and class frequencies Assumptions ◦ Without information to the contrary, data are distributed evenly within the interval ◦ Best summary representation of the values in each interval is the class midpoint Class midpoint – value that is exactly midway between extreme values that identify the class interval http://www.transtutors.com/statistics-homework-help/numerical-measures/weighted-mean.aspx Relative Variability Defined as the amount of spread in a set of variables Spread can be measured in different ways ◦ Simplest measure of variability is the range - difference between largest and smallest value ◦ Quantiles are used to define intervals, portions, or percentiles ◦ Interquartile range – data is divided into 4 equal portions. Difference between 25th and 75th percentile is the interquartile range http://www.mathsisfun.com/definitions/range-statistics-.html http://faculty.uncfsu.edu/dwallace/lesson%205.pdf Standard Deviation and Variance Standard Deviation ◦ The least squares property of the mean carries over into the most common measure of variability or dispersion Variance ◦ The square of the standard deviation Formula provided by: http://en.wikipedia.org/wiki/Standard_deviation Standard Normal Distribution 68-95-99.7 rule ◦ http://www.oswego.edu/~srp/stats/z.htm http://www.oswego.edu/~srp/stats/z.htm Measures of Shape or Relative Position Skewness ◦ measures the degree of symmetry in a frequency distribution ◦ determines the extent to which the values are evenly or unevenly distributed on either side of the mean Kurtosis ◦ measures flatness or peakedness of a data set Graphics provided by: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm Spatial Data and Descriptive Statistics Boundary delineation ◦ Idea that a location of boundaries can affect various descriptive statistics For example: The watershed area is highlighted in yellow as the area that will be covered in this watershed study. The other colored areas are watershed areas that will not be covered. http://proceedings.esri.com/library/userconf/cahinvrug09/papers/user-presentations/watershed_boundary_delineation.pdf Spatial Data and Descriptive Statistics (cont.) Modifiable areal units ◦ Idea that using alternative subdivision or regionlization schemes within the same overall study area can influence descriptive statistics For example: These Aggregated Districts have modifiable areas of study. The study area has been modified several times in order to show the east-west aggregation of Indiana’s crop aggregation in figure C and then north south in figure D. http://www.agriculture.purdue.edu/ssmc/Frames/SSMC_newsletter11_2006.pdf Spatial Data and Descriptive Statistics (cont.) Spatial Aggregation ◦ Idea that different spatial levels, or scales, can vary the descriptive statistics For example: The first image shows the unemployment statistics based on region. The second image shows the unemployment statistics based on state. The same information is given in two different graphs based on the scale the data is portrayed. http://www.nationalatlas.gov/articles/people/a_unemployment.html#one Lesson Review Median, mode, and mean are used to measure central tendency Measures of dispersion is determined based on relative variability, standard deviation, and variance Boundary delineation, modifiable areal units, and spatial aggregation are all measurements of shape or relative position