Univariate Descriptive Statistics Chapter 2

Lecture Overview Tabular and Graphical Techniques Distributions Measures of Central Tendency Measures of Dispersion

Tabular and Graphical Techniques Frequency Tables –Ungrouped –Grouped Histograms Cumulative Frequency Histogram

Frequency Tables BinFrequency

Histograms Note: sometimes percent is on the Y axis rather than frequency

Cumulative Frequency Histograms

Key Concepts Choosing Intervals (i.e., choosing your “bins”) Rules from the textbook (pages 38 – 39) Commonly Used Examples from GIS –Equal Interval –Quantiles (e.g., quartiles and quintiles) –Natural Breaks –Standard Deviation

Rules For Bin Sizes Note: This is very relevant for GIS Rule 1: Use intervals with simple bounds Rule 2: Respect natural breakpoints Rule 3: Intervals should not overlap Rule 4: Intervals should be the same width Rule 5: Select an appropriate number of classes

The Effect of Classification Equal Interval –Splits data into user-specified number of classes of equal width –Each class has a different number of observations

The Effect of Classification Quantiles –Data divided so that there are an equal number of observations are in each class –Some classes can have quite narrow intervals

The Effect of Classification Natural Breaks –Splits data into classes based on natural breaks represented in the data histogram

The Effect of Classification Standard Deviation –Mean + or – Std. Deviation(s)

Key Concepts Making sense of your histograms using distributions –Rectangular –Unimodal –Bimodal –Multimodal –Skew (positive and negative)

Bimodal Distribution

Multimodal Distribution

Skew An asymmetrical distribution

Measures of Central Tendency Measures of central tendency –Measures of the location of the middle or the center of a distribution –Mean, median, mode, midrange

Definitions Midrange Mode Median –Quantiles Mean

Definitions Sample Mean Population Mean

Description of Mean Mean – Most commonly used measure of central tendency Average of all observations The sum of all the scores divided by the number of scores Note: Assuming that each observation is equally significant

Symbols n: the number of observations N: the number of elements in the whole population Σ: this (capital sigma) is the symbol for sum i: the starting point of a series of numbers X: one element in our dataset, usually has a subscript (e.g., i, min, max) : the sample mean : the population mean

Summation Notation: Components indicates we are taking a sum refers to where the sum of terms begins refers to where the sum of terms ends indicates what we are summing up

Mathematical Notation of Mean The mathematical notation used most often in this course is the summation notation The Greek letter capital sigma is used as a shorthand way of indicating that a sum is to be taken: The expression is equivalent to:

A summation will often be written leaving out the upper and/or lower limits of the summation, assuming that all of the terms available are to be summed Summation Notation: Simplification

Equation for Mean Sample mean: Population mean:

Example Mean Calculations Example I –Data: 8, 4, 2, 6, 10 Example II –Sample: 10 trees randomly selected from Battle Park –Diameter (inches): 9.8, 10.2, 10.1, 14.5, 17.5, 13.9, 20.0, 15.5, 7.8, 24.5

Example Mean Calculations Example III Monthly mean temperature (°F) at Chapel Hill, NC (2001). Annual mean temperature (°F)

Mean annual precipitation (mm) Mean annual temperature (°F) (°F) Mean (mm) Mean Examples IV & V Chapel Hill, NC ( )

Advantage –Sensitive to any change in the value of any observation Disadvantage –Very sensitive to outliers Explanation of Mean #Tree Height (m) #Tree Height (m) Mean = 6.19 mwithout #8 Mean = 8.10 mwith #8

Measures of Dispersion Used to describe the data dispersion/spread/variation/deviation numerically Usually used in conjunction with measures of central tendency

Measures of variation score # of obs score Low variationHigh variation Groups have equal means and equal n, but one varies more than the other

Definitions Range Mean Deviation Variance Standard Deviation Coefficient of Variation Pearson’s

Symbols s 2 : the sample variance σ 2 : the population variance s: the sample standard deviation σ : the population standard deviation

Sample Variance and Standard Deviation Note: as with the mean there are both sample and population standard deviations & variances VarianceStandard Deviation

Next Class Read chapter 3 Work on the homework Come with questions Bring your laptop