Geography 360:

Exam I Explanation

Note: the point designations indicate total possible points. In many cases partial points were awarded.

1) a) Two points were awarded for each correctly identified data category. Accepted answers for soil included: nominal-soil type; weak ordinal-potential soil erosion categories; strong ordinal-ranking of productivity potential according to several soil features; ratio-percent clay. For city population: nominal-race; weak ordinal-income classes; strong ordinal-ranking preferred neighborhoods based on population characteristics; ratio-family size. Desertification: nominal-presence/absence of vegetation; weak ordinal-classifying herding practices by predominant animal and their impact on vegetation; strong ordinal-assigning desertification risk potential values to areas based on several variable scales; ratio-annual precipitation.

b) One point awarded for correctly matching category to measure of central tendency. Mode can be calculated for any of the examples, median for examples of ordinal (strong and weak), and ratio examples, mean for examples of ratio data.

2) a) Figure similar to following received two points.

b) two points for each correct formula, two for calculations. mean is 6 {(1+3+7+8+11)/5}, standard deviation is 4 {square root of (5²+3²+1²+2²+5²)/(5-1)= square root of (25+9+1+4+25)/4= square root of 64/4= square root of 16 = 4)}

c) one points for recognizing negative skewness (no calculation necessary).

d) one point for recognizing kurtosis is positive.

e) two points for each correct formula. one point for each correct calculation. (Skewness start: second moments of deviation (3³+2³+....)/(5)(4³)= (27+8+....)/324; kurtosis start: ((3⁴+2⁴+....)/(5)(4⁴))-3 = (81+16+....)/1280.)

3) 5 points for each drawing relative to how the approximated the statistics. Blue curves below show sufficient understanding of skewness and kurtosis.

4) a) 3 points for correct formula. 2 points for Mean center is (5, 4) { mean X = (3+4+5+6+7)/5; mean Y = (5+4+2+4+5)/5.} 2 points for recognizing that mean center location to northeast of city center suggests that fire protection services are concentrated in the region of the city.

b) 3 points for recognizing that new fire station would shift mean center towards the new station.

5) 3 points for each section.

a) Nature reserve would either argue for consideration of populated city boundaries as opposed to actual boundaries or measurement of weighted mean center based on population. Would most likely provide support for current distribution of stations.

b) Weighted mean center based on population in this case supports current distribution, whereas based on economic value would support case for new station.

c) Weighted mean based on travel time from nearest station would support case for new station.

6) a) 2 points for correct formula. 4 points for standard distance calculation. S_D= square root of 3.2 {square root of (((3-5)2+(5-4)2+(4-5)2+(4-4)2+(5-5)2+(2-4)2+(6-5)2+(4-4)2+(7-5)2+(5-4)2)/5) = square root of ((4+1+1+0+0+4+1+0+4+1)/5) = square root of (16/5) = square root of 3.2.} Nearest perfect square is 4, so estimate of standard distance is 2. 2 points if noted that 3 stations are within one standard distance of mean center, 2 are within one standard distance from city center.

b) Standard distance would increase with new station. (2 points)

7) a) Mean and median calculated for each data set allow for comparison of central tendency and can show growth, if higher in 1970, or decline, if lower. If change in mean is significantly higher that change in median, you might conclude that highest productivity increased more than productivity in general, or in this case that technological innovation provided potential for greater increases in productivity in some instances than others. Further research on highest productivity would be required in order to more accurately determine the cause. Mode is important if the data exhibits more than one mode (distinct peaks in the frequency distribution), suggesting that more than one productivity type occurs within the county. The data available does not provide information necessary to establish cause of different modes if present.

b) Calculation of Standard deviation provides some indication of the variation in production. If standard deviation is a small fraction of the mean, productivity differs relatively little among farmers. If on the other hand standard deviation is equal to or greater than the mean, variation may be considered relatively large. Similarly, the smaller the standard deviation, the more representative is the mean. Direct comparison of the standard deviation for each year provides little information, but dividing each standard deviation by the respective mean provides a measure of relative variation (coefficient of variation). A higher coefficient of variation in 1970 would suggest that variation among farmers has increased, whereas if lower it suggests that productivity levels are more similar with the adoption of technological innovations.

c) Skewness (symmetry) and kurtosis (bunching) provide additional information about the shape of the frequency distribution and may be used in direct comparisons as they are relative values. Negative skewness in either year suggests that conditions exist (whether climatic, economic, social etc. we do not know at this point) which may cause much lower productivity than the majority of farmers. A positive skewness implies the opposite. If skewness for the 1970s data is greater than that of the 1930s data, it suggests that innovations made higher productivity, and possibly extreme high values, more possible. In the opposite case, it may indicate that innovations have made wheat production more susceptible to pest, soil, or climatic conditions which can severely lower yields. A positive kurtosis in either year suggests a high number of farmers with similar productivity levels, whereas negative kurtosis indicates a more diverse (spread out) distribution of productivity levels. If 1970s kurtosis is higher, innovations appear to facilitate more uniform productivity among farmers, whereas lower kurtosis suggests that production using the innovations is more subject to differentiation in its application from farmer to farmer (again, the exact cause-be it response to climate, soil, social or economic factors, must be obtained from other information).

Additional credit was provided for more exact definitions of each statistic in the answer to question 7.