2/23/2015 GEOGRAPHY 204: STATISTICAL PROBLEM SOLVING IN GEOGRAPHY THE NORMAL DISTRIBUTION THE NORMAL DISTRIBUTION

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1 Fall 2015: Lembo GEOGRAPHY 204: STATISTICAL PROBLEM SOLVING IN GEOGRAPHY Most generally applied probability distribution Enables assumptions about data properties Basis for sampling theory and statistical inference Continuous distribution on x-axis Symmetrical = no skew Central value is peak Most frequently occurring value Mean, median and mode Bell-shaped Intermediate kurtosis Tails lowest frequency Useful in making probability statements about actual outcomes in many situations Ex. if a location s precipitation amount over a number years is assumed to be normally distributed, the probability that the location receives a given amount of precipitation can be calculated Provides theoretical basis for sampling and statistical inference 1

2 The way in which area is distributed under the normal curve is the basis for making probability estimates Total area = 100% For any point on x-axis, determine percentage of values under curve above and below the value Ex. 50% right or left of mean A value has.50 probability of falling above mean Area under curve Integral calculus Table of normal values Software To use table of normal values, data must be standardized standard score (or normal deviate): the number of standard deviations that separate a particular value from the mean of the distribution May be positive, above the mean, or negative, below the mean Ex. standard score = 1 One standard deviation greater than the mean standard score = -1 one standard deviation below the mean The larger the standard score, the greater the difference of the value from the mean Standard score and probability Table interpretation Probability of a value falling between the mean and standard score location for a set of normally distributed data Multiply probability by 100, the percentage of all values that lie between the mean and standard score Table provides information to determine the area under the normal curve for any interval Standard score = 1, table value =.3413, 34% between mean and value Table of normal values 2

3 Remember: 1 standard deviation = 68% of all values 2 standard deviations = 95% of all values Table of normal values: standardized scale Specific set of values: convert original units of measurement (X) to standardized scale without units of measurement (Z). Z-value (or Z-score): positive or negative Number of standard deviations from the mean Assumption: original data must be normally distributed There are techniques to assess normality (later) Example: Washington, DC 40-year precipitation Probability questions: Given a particular value, what is the corresponding probability? or Given a particular probability value, what is the corresponding data value? 3

4 What is the probability of Washington s annual precipitation exceeding 48 inches? Data normally distributed Mean = inches, SD = 7.5 inches Three step process 1. Calculate standard score 2. Determine probability from table 3. Evaluate probability value EXAM 1 RESULTS Spring 10 Fall 09 Spring 13 SPRING 12 Mean: 65.6 Median: 72 St.Dev: 20 Skew: -.71 High: 88% Low: 25% Mean: 57.8 Median: 56.6 St.Dev: 15.9 Skew: -.17 High: 81% Low: 25% Mean: 71.5 Median: 71 St.Dev: 12 Skew: -.17 High: 92% Low: 41% Mean: 67 Median: 68 St.Dev: 13 Skew: -.5 High: 91% Low: 35% Fall10 Mean: 65.3 Median: 66 St.Dev: 20 Skew: -.01 High: 96% Low: 36% Spring 2011 Mean: 63.1 Median: 66 St.Dev: 16 Skew: -.73 High: 90% Low: 18% 4

5 WASHINGTON, DC PRECIPITATION Table =.3577 (nearly 36%) 36 years of 100, precipitation in Washington, DC falls between and 48 inches 3. Meaning? 14 years in 100 Precip. exceeds 48 inches WASHINGTON, DC PRECIPITATION What amount of annual precipitation in Washington, DC is likely exceeded with a probability of.90? In other words, 9 out of 10 years Steps different 1. Determine probability from the normal table WASHINGTON, DC PRECIPITATION Step 2. Calculate standard score.40 = Z-score 1.28 (probability =.3997) Below mean = Precipitation that occurs at least 90% of the time (9 years out of 10)is 1.28 standard deviations below the mean Step 3. Calculate the precipitation value Very useful probability distribution for geography! 5

6 PROBABILITY MAPPING Apply this technique to multiple locations across a region probability map or probability surface Assumption: variable must be continuously distributed across space Not population, number of murders, traffic accidents, etc.! Natural or physical variables Climatology, meteorology, etc. Ex. atmospheric particulates, ozone levels, winter storms, ph levels of acid deposition PROBABILITY MAPPING: EXAMPLE Annual precipitation for selected United States cities Calculate mean and standard deviation for each Determine the amount of annual precipitation exceeded in 9 out of 10 years Assign probability values to each location and connect with isolines Why not just look at annual precipitation values? Probability considers central tendency and variability at each location PROBABILITY MAPPING: EXAMPLE Two cities A and B each average 50 inches of precipitation very different variability City A less annual variability minimum expected slightly lower than mean (43.6 inches) City B more annual variability minimum expected much less than mean (24.4 inches) 6

7 PROBABILITY MAPPING: EXAMPLE Map minimum annual precipitation expected in 9 out of 10 years Spatial pattern differs versus annual average precipitation Provides additional information: risk of floods and/or droughts EXAMPLE: ANNUAL HEATING DEGREE DAYS Heating degree day (HDD)- surrogate for the amount of heating energy needed to produce comfortable indoor warmth in cool/cold climates Widely used measure of yearly heating energy demand Simplest calculation 65 F (base temperature) average daily temperature = HDD Negative value no heat required EXAMPLE: ANNUAL HEATING DEGREE DAYS Use 40-year records for level-one stations in conterminous US Total HDDs for year for over 100 stations Calculate means and standard deviations Calculate the number of HDDs expected to be exceeded 1 year out of 10 rare occurrence once per decade 7

8 EXAMPLE: ANNUAL HEATING DEGREE DAYS Lexington, KY HDD exceeded 10% of time EXAMPLE: ANNUAL HEATING DEGREE DAYS 1 out of 10 years Connect threshold HDD with isolines pattern? East-west lines mean Dips south Change probability 1 out of 20 years (.05) 3 out of 10 years (.30) Key consideration: precision of isoline placement is directly affected by density of the point pattern EXAMPLE: LAST SPRING FROST PATTERN SE UNITED STATES Isochronal probability map showing likelihood of a late spring frost occurring at any particular location for any particular date we select What is the probability of the last spring frost occurring on or after April 1 st for each of the 76 stations? 8

9 EXAMPLE: LAST SPRING FROST PATTERN SE UNITED STATES Ex. Waycross, GA EXAMPLE: LAST SPRING FROST PATTERN SE UNITED STATES Waycross 16.6% Pattern? Factors Regional? Local? 9