Chapter 3 An Introduction to Statistical Problem Solving in Geography

Summarized by Lana Hesler

Learning ObjectivesUnderstand the basic descriptive

measures of central tendencyUnderstand the basic descriptive

measures of dispersionUnderstand the concept of relative

variabilityDetermine the value of measuring

shape or relative positionRealize potential effects of location

data on descriptive statistics

Summarizing Data SetsMeasures 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 StatisticsMode

◦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 MeanWeighted 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 VariabilityDefined as the amount of spread in a set of

variablesSpread 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 VarianceStandard 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 Distribution68-95-99.7 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 StatisticsBoundary delineation

◦Idea that a location of boundaries can affect various descriptive statistics

http://proceedings.esri.com/library/userconf/cahinvrug09/papers/user-presentations/watershed_boundary_delineation.pdf

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.

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

http://www.agriculture.purdue.edu/ssmc/Frames/SSMC_newsletter11_2006.pdf

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.

Spatial Data and Descriptive Statistics (cont.)Spatial Aggregation

◦Idea that different spatial levels, or scales, can vary the descriptive statistics

http://www.nationalatlas.gov/articles/people/a_unemployment.html#one

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.

Lesson ReviewMedian, 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