2/2/2015 GEOGRAPHY 204: STATISTICAL PROBLEM SOLVING IN GEOGRAPHY MEASURES OF CENTRAL TENDENCY CHAPTER 3: DESCRIPTIVE STATISTICS AND GRAPHICS

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1 Spring 2015: Lembo GEOGRAPHY 204: STATISTICAL PROBLEM SOLVING IN GEOGRAPHY CHAPTER 3: DESCRIPTIVE STATISTICS AND GRAPHICS Descriptive statistics concise and easily understood summary of data set characteristics Measures of central tendency: numbers that represent the center or typical value of a frequency distribution Mode, median, and mean Measures of dispersion: numbers that depict the amount of spread or variability in a data set Range, interquartile range, standard deviation, variance and coefficient of variation Measures of shape or relative position: numbers that further describe the nature or shape of a frequency distribution Skewness amount of symmetry of a distribution Kurtosis degree of flatness or peakedness in a distribution MEASURES OF CENTRAL TENDENCY Each measure has advantages and disadvantages Appropriate selection based on geographic situation Mode, median and mean Example: DC precipitation 1

2 MEASURES OF CENTRAL TENDENCY Mode - the value that occurs most frequently in a set of ungrouped data values Available for all levels of measurement but Nominal data category containing largest number of observations Ex. Religious affiliation: Catholic 15, Protestant 4, Judaism 2, Other 3 Ordinal modal class - category with the largest number of observations Ex. Senior 7, Junior 10, Sophomore 5, Freshmen 2 Interval-ratio grouped in classes - crude mode midpoint of model class interval Ex. Credit Hour Equivalent 0 to 30, 31 to 60, 61 to 90, 91 to 120 Mid point = 75.5 MEASURES OF CENTRAL TENDENCY Which measure is more appropriate? D.C. precipitation, number of children Interval/ratio data? Rare to have exact same value. Number of children Frequency However, better than 2.2 children (mean) MEASURES OF CENTRAL TENDENCY Grouped D.C. precipitation data Crude mode 37.5 inches 2

3 GRAPHICS: GROUPED DATA Graphic summaries of grouped data vertical (Y) axis frequency of values horizontal (X) axis range of data Information absolute frequencies (counts) or relative frequencies (percentages or probabilities) Histogram frequency of values is series of vertical bars GRAPHICS: GROUPED DATA Frequency polygon similar to histogram however vertical position is a point GRAPHICS: GROUPED DATA ogive - cumulative frequency diagram aggregates frequencies from class to class and display cumulative frequencies at each position number of values less than or equal to each value or class 60% below 41 inches 3

4 MEASURES OF CENTRAL TENDENCY Median middle value from a set of ranked observations Calculated from ordinal, interval or ratio data Value with equal number of data units above and below Odd number of observations unique value Even number of observations midpoint of two middle values Ex. D.C. precipitation 40 years Rank 20 = in., Rank 21 = in. Median = in. Unlike mean, not affected by extreme values! MEASURES OF CENTRAL TENDENCY Mean (i.e., average or arithmetic mean) the sum of a set of values divided by the number of observations Appropriate for interval/ratio data Most widely used measure MEASURES OF CENTRAL TENDENCY Population mean μ (Greek mu) Fixed value Grouped data weighted mean Access to summarized data only Calculated from class intervals and class frequencies 4

5 MEASURES OF CENTRAL TENDENCY Weighted mean Assumptions Even distribution in classes Class midpoint best summary Arithmetic mean = in. Weighted mean = in. PROPER MEASURE OF CENTRAL TENDENCY Most common mean However, affected by any change in data set value Not always true for mode and median Important element in inferential statistics Frequency distribution impacts measures... unimodal, symmetrical PROPER MEASURE OF CENTRAL TENDENCY bimodal or multimodal one or more modes Mean/median Not representative Modes are best A and B Outliers extreme or atypical values Impacts mean most heavily 5

6 Range difference between highest and lowest values in interval-ratio data set Potentially misleading...includes extremes Ex. Washington DC precipitation ( ) = Clustering? Most observations between 32 and 38 inches Quantiles - data divided into equal portions or percentiles Quartiles (fourths), quintiles (fifths), deciles (tenths) Median = 50 th percentile interquartile range - difference between 25 th percentile and 75 th percentile middle half of data Ex. Washington DC precipitation = % of observations. Boxplot (box-and-whiskers) graphical representation of dispersion Extension of interquartile range Comparison 40 year annual precipitation for Buffalo, NY, St. Louis, MO, and San Diego, CA. Consistency? Less Consistency? 6

7 average deviation (mean deviation) - mean of the set of individual deviations Deviation difference between the mean and each value Absolute value of each individual deviation Sum of the deviation about mean is always zero! least squares property of the mean the sum of squared deviations about a mean is less than the sum of squared deviations about any other number Linear regression analysis later! standard deviation - most common measure of variability or dispersion Squaring removes the problem of negative deviations the square root of the value is taken to reverse the effect of squaring n>30, n-1 and N nearly the same, thus s and σ close n<30, σ underestimated, n-1 corrects Sample lower case, italicized s Population - sigma variance square of standard deviation Measure of average squared deviation of a set of values around the mean Important measure in inferential statistics ANOVA analysis of variance Confidence intervals reliability of an estimate 7

8 Algebraic re-arrangement to improve computational efficiency Variance? Weighted standard deviation grouped data Same assumptions as weighted means Similar magnitude results 7.40 vs

9 STANDARD DEVIATION In probability theory and statistics, standard deviation is a measure of the variability or dispersion of a statistical population, a data set, or a probability distribution. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values. Normally distributed data 68% of observations within one standard deviation, 95% within two standard deviations, 99.7% within three standard deviations APPLICATION: CLASSIFICATION standard deviation breaks determines class breaks from mean and standard deviation Breaks rounded 1 std. dev. or 0.5 std. dev. Most effective normally distributed data Even number of classes mean is class break Above mean, below mean Odd number of classes middle category centered on mean CLASSIFICATION- STANDARD DEVIATION 9

10 CLASSIFICATION- STANDARD DEVIATION HDI Difference to to to 0.16 CLASSIFICATION Jenks method of natural breaks - determines class breaks by finding natural groupings in the overall distribution of values Iterative algorithm Comparing standard deviation and variance can be misleading Absolute measures their value depends on the size or magnitude of the units from which they are calculated Ex. large numbers (in the millions) large means, std. dev. and variance, small numbers small means, std. dev. and variance coefficient of variation (variability)- relative measure to resolve problem 10

11 coefficient of variation (variability) - CV MEASURES OF SHAPE OR RELATIVE POSITION Two additional relative measures describing nature or character of frequency distribution skewness - measures the degree of symmetry in a frequency distribution kurtosis - measures the flatness or peakedness of a data set Sum of individual deviations about the mean Numerator in variance expression 11

12 MEASURES OF SHAPE OR RELATIVE POSITION skewness third moment of a frequency distribution The denominator of this expression contains the cubed standard deviation Normalizing the value with ns, allows comparison of relative skewness in different frequency distributions If a value is greater than the mean, it s cubed deviation is positive, if it is less than the mean, the cubed deviation will be negative MEASURES OF SHAPE OR RELATIVE POSITION kurtosis measures fourth standardized moment of a frequency distribution Can be compared to normal probability distribution leptokurtic peaked > 3 platykurtic flat < 3 mesokurtic bell-shaped = 3 MEASURES OF SHAPE OR RELATIVE POSITION 12

13 MEASURES OF SHAPE OR RELATIVE POSITION Negative Kurtosis = platykurtic Positive Kurtosis = leptokurtic QUESTIONS TO ASK Which locations had the greatest relative variability? Which locations had the greatest skew? Which one is most leptokurtic? Why? ANSWERS... High skewness and kurtosis for San Diego relative to Buffalo and St. Louis While Buffalo and St. Louis have higher precipitation, they seldom deviates from the mean. 13

14 SPATIAL DATA AND DESCRIPTIVE STATISTICS Spatial/location-based data can affect value and magnitude of descriptive statistics Alteration of external boundary of study area Modification of internal (subarea) boundaries Change in level of spatial resolution by using a different scale or level of aggregation Absolute descriptive statistics should be evaluated comparatively only in relation to a particular study area IMPACT OF EXTERNAL BOUNDARY DELINEATION A: Inner City B: Entire County What happens to percent below poverty? IMPACT OF EXTERNAL BOUNDARY DELINEATION Number and distribution of people below the poverty level Three possible units... Impact of defined area on mean? Standard deviation? Variance? 14

15 BOUNDARY PROBLEM FOR POINT PATTERNS The standard distance or any other measure of dispersion can t be interpreted independent of the study area MODIFIABLE AREAL UNIT PROBLEM (MAUP) Modification of Internal Subarea Boundaries Grouping or zoning problem external boundaries fixed, internal areas/boundaries can be drawn multiple ways impacts descriptive statistics! How does the placement affect our views of racial segregation? MODIFIABLE AREAL UNIT PROBLEM (MAUP) Two considerations Placement of internal boundaries Geographic scale Secondary data often no choice Census tract, county, state units 15

16 MODIFIABLE AREAL UNIT PROBLEM (MAUP) A: Little interregional variation B: Significant inter-regional variation MODIFIABLE AREAL UNIT PROBLEM (MAUP) Agronomist 72 soil samples phosphorus (P) Promotes early growth, more rapid maturity potassium (K) Improves movement of water, nutrients and carbohydrates K deficiency stunts growth Ppm (parts per million) MODIFIABLE AREAL UNIT PROBLEM (MAUP) Case 1 East-West variation Case 2 North-South variation Case 3 Compromise Means same Variance different 16

17 MODIFIABLE AREAL UNIT PROBLEM (MAUP) MODIFIABLE AREAL UNIT PROBLEM (MAUP) Change in Scale or Level of Spatial Aggregation socioeconomic variables block, tract, county, planning region, etc. 3 scales 50 states 9 census divisions 4 census regions MODIFIABLE AREAL UNIT PROBLEM (MAUP) Scale increase magnitude of mean and standard deviation increase CV difficult widely variable state values, regional patterns less variable Skew positive larger states growing more quickly Kurtosis varied states leptokurtic peaked 17