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Publication Number: FHWAHRT05042
Date: October 2005 
Safety Effects of Differential Speed LimitsPDF Version (960 KB)
PDF files can be viewed with the Acrobat® Reader® Alternative TextFigure 1. Map. Speed Limits Throughout the 1990s on Rural Interstate Highways. The map of the United States shows shaded areas for the nine states that participated in the study. Three states (Arizona, Iowa and North Carolina) maintained a uniform speed limit, while three other states (Washington, Indiana and Illinois) maintained a differential speed limit. Two states (Idaho and Arkansas) changed from a uniform to a differential speed limit, while Virginia changed from a differential to a uniform speed limit. Figure 2. Chart. Data Analysis Process Flowchart. The flow chart starts with data coming in. Then all the data undergoes an analysis of variance (ANOVA) to determine if the change between uniform and differential speed limits is significant. If a difference greater than the 5 percent confidence level is detected, then number of data groups N is set up for the post hoc analysis that uses the SPSS software. Levine's test is performed first as a screening procedure to determine if the groups have equal variances. Tukev's test is then used if the results show equal variances, while Dunnett's test is used for samples with unequal variances. Figure 3. Equation. Crash Rate. The equations reads: Crash Rate is equal to the product of 100,000,000 and the Annual Crash Frequency divided by the product of 365, the ADT, and the Section Length. Figure 4. Chart. Fundamental Steps of the Empirical Bayes Approach. Just comparing before and after crashes might not show the results of treatment. The Empirical Bayes Approach adds to the analysis of a before/after test because it also considers a reference population. Under reference population, the first box shows data for a representative sample with years 1 to Y (before) and Y plus 1 to Y plus Z (after). The next box is the multivariate model with the estimate of the expected frequency of crashes E(m) where m is crash frequency, and the variance of the frequency of crashes VAR(m), for both before and after annual data. Under treated entities the crash counts (K) appear with covariate values for years before and after. This box is tied to the estimates of crash frequency (m) for before treatment and the predictions of m for the after period. Both the estimates and the variances for the reference population are tied to the estimate box for treated entities while the variance for the reference population is also tied to the predictions for the treated entities. This model helps evaluate how treatment affected safety by predicting what the expected crash frequency would have been in the after period had there been no treatment and then comparing the "would have been" value to the actual number of crashes in the after period. Figure 5. Equation. Crash Models for Virginia. The equation reads: The expected value of (m) is equal to the product of 0.022, the length raised to the power 0.622, and the ADT raised to the power of 0.548. Figure 6. Equation. Crash Models for Washington The equation reads: The expected value of (m) is equal to the product of 0.531, the length raised to the power 0.440, and the ADT raised to the power of 0.340. Figure 7. Chart. Comparison of Crash Estimation Models for Virginia and Washington State Based on 19911993 Data. The graph shows Virginia and Washington crash estimation comparisons. The horizontal axis is average daily traffic (ADT) ranging from 0 to 40,000 in 5000 increments. The vertical axis is crashes estimated by the crash estimation model ranging from 0 to 40 in increments of 5. The Virginia plot is blue diamonds ranging from 2 crashes at 1000 ADT to 15 crashes at 35,000 ADT. The Washington plot is magenta squares ranging from 11crashes at 1000 ADT to 39 crashes at 35,000 ADT. This plot confirms that a different number of crashes at a given site with a given volume can be predicted Figure 8. Equation. Expected Frequency of Crashes. The equation reads: The expected value of (m) is equal to the product of alpha, the length raised to the power beta subscript 1, and the ADT raised to the power beta subscript 2. Figure 9. Chart. Plot of Goodness of Fit for the Crash Estimation Method Versus ADT. This plot of cumulative residuals (the difference between actual crash counts and model estimates) with Virginia data visually depicts whether the model equation used yielded appropriate results. The horizontal axis is AADT ranging from 2925.3215 to 57,833.333. The vertical axis is cumulative residuals ranging from minus 80 to 120. The key shows the yellow line as minus 2 standard deviations, the magenta line as plus 2 standard deviations and the residuals are the blue line. The negative standard deviation plots from minus 20 to minus 70 while the positive standard deviation plots from 20 to 70. If the cumulative residuals oscillate around 0 within the range of two standard deviations, a goodness of fit plot results. This graph shows only one spike of the cumulative residual of 110 at 10,333.333 AADT so the figure shows a good quality of fit. Figure 10. Chart. Plot of Goodness of Fit for the Crash Estimation Model Versus Length. This plot of cumulative residuals (the difference between actual crash counts and model estimates) with Virginia data visually depicts whether the model equation used yielded appropriate results. The horizontal axis is length ranging from 1.05 to 8.53. The vertical axis is cumulative residuals ranging from minus 80 to 120. The key shows the yellow line as plus 2 standard deviations, the blue line as minus 2 standard deviations and the residuals are the magenta line. The negative standard deviation plots from minus 20 to minus 70 while the positive standard deviation plots from 20 to 70. If the cumulative residuals oscillate around 0 within the range of two standard deviations, a goodness of fit plot results. This graph shows only two spikes of the cumulative residual of 1.05 length at 59 and 5.17 length at 110 exceeding the standard deviation so the figure shows a good quality of fi Figure 11. Equation. Alternative Crash Estimate Model. The equation reads: The expected value of (m) is equal to the product of alpha subscript y, the Length raised to the power beta subscript 1, and the ADT raised to the power beta subscript 2. Figure 12. Equation. CEM for Before Years. The equation reads: C subscript i,y is equal to the quotient of the expected value of (m) subscript i,y over the expected value of (m) subscript i,1. Figure 13. Equation. Expected Crash Frequency m for Period 1. The equation reads: m subscript i,1 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y over the sum of the quotient k over the expected value of m subscript i,1 and the summation of C subscript i,y from y equals 1 to Y Figure 14. Equation. Variance of Expected Crash Frequency m for Period 1. The equation reads: the variance of m subscript i,1 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the squared sum of the quotient k over the expected value of m subscript i,1 and the summation of C subscript i,y from y equals 1 to Y, which is equal to the quotient of m subscript i,1 over the sum of the quotient of k over the expected value of m subscript i,1 and the summation of C subscript i,y from y equals 1 to Y. Figure 15. Equation. Expected Crash Frequency m for Period y. The equation reads: m subscript i,y is equal to the product of C subscript i,y and m subscript i,1 Figure 16. Equation. Variance of Expected Crash Frequency m for Period y. The equation reads: the variance of m subscript i,y is equal to the product of the square of C subscript i,y and the variance of m subscript i,1. Figure 17. Equation. "Would have been" crashes had there been no speed limit changes. The equation reads: pi is equal to the summation of pi subscript i. Figure 18. Equation. "Actual crashes" given that the speed limit did change. The equation reads: lambda is equal to the summation of lambda subscript i Figure 19. Equation. The difference between would have been and actual crashes. The equation reads: delta is equal to pi minus lambda Figure 20. Equation. Variance for δ The equation reads: the variance of delta is equal to the sum of the variance of pi and the variance of lambda, which is equal to the sum of the summation of the variance of pi subscript i and the summation of the variance of lambda subscript i Figure 21. Equation. Confidence intervals for d. The equation reads: empirical confidences bounds for delta are equal to delta plus or minus the product of two and the square root of the variance of delta. Figure 22. Equation. Reduction in the Expected Number of Crashes. The equation reads: theta is equal to the quotient of lambda over pi. Figure 23. Equation. Ratio of Actual to "Would Have Been" Crashes. The equation reads: theta is equal to the quotient of the quotient of lambda over pi over the sum of 1 and the quotient of the variance of pi over pi squared. Figure 24. Equation. Variance of Ratio of Actual to "Would Have Been" Crashes. The equation reads: the variance of theta is equal to the quotient of the product of theta squared and the sum of the quotient of the variance of lambda over lambda squared and the variance of pi over pi squared all over the squared sum of one and the quotient of the variance of pi over pi. Figure 25. Equation. Confidence Intervals for q. The equation reads: theta is equal to theta plus or minus the square root of the product of two and the variance of theta. Figure 26. Chart. Mean Speed for All Vehicles. The graph shows the mean speeds for vehicles from five states: Iowa, Illinois, Indiana, Idaho and Virginia. The horizontal axis is years ranging from 1991 to 2001. The vertical axis is mean speed ranging from 58 to 72 miles per hour. Idaho and Iowa show the largest increases from 64 to 70 miles per hour. This plot shows gaps in data for all time periods and except for Virginia, speeds appear to be increasing over time. Figure 27. Chart. 85^{th} Percentile Speeds and Median Speeds. The graph shows the 85th percentile speed trend for vehicles from five states: Iowa, Illinois, Indiana, Idaho and Virginia. In the first plot, the horizontal axis is years ranging from 1991 to 2001 in twoyear increments. The vertical axis is 85th percentile speed ranging from 65 to 80 miles per hour. Idaho with speed limit changes in both 1996 and 1998 shows the steepest speed trend from 70 to 79 miles per hour. Figure 28. Chart. Median Speed Trends. The graph shows the median speed trend for vehicles from five states: Iowa, Illinois, Indiana, Idaho and Virginia. The horizontal axis is years ranging from 1991 to 2001 in oneyear increments. The vertical axis is Median Speed ranging from 55 to 70 miles per hour. The data plots are spotty, but Virginia shows the only deceasing speed trend. Figure 29. Chart. Speed Variance and Noncompliance Rates. The graph shows statistical analyses for the speed variance for vehicles from five states: Iowa, Illinois, Indiana, Idaho and Virginia. In the plot, the horizontal axis is years ranging from 1991 to 2001. The vertical axis is speed variance ranging from 0 to 100 miles per hour. The data plots are spotty, but increasing speed trends are not usually significant. Figure 30. Chart. Noncompliance Rates. The graph shows statistical analyses for the speed variance for vehicles from five states: Iowa, Illinois, Indiana, Idaho and Virginia. In the plot, the horizontal axis is years from 1991 to 2001 in twoyear increments. The vertical axis is noncompliance speed ranging from 0 to 100. The data show that changes in speed from differential or uniform speed limits were not supported by the study. Figure 31. Chart. Total Crash Rates. The graph shows total crash rates from five states: Arizona, North Carolina, Idaho, Arkansas and Virginia. The horizontal axis is years from 1991 to 2000. The vertical axis is total crash rates (crashes per 100 million VMT), ranging from 0 to 120. Most states average from 20 to 60 crashes except for Idaho that ranges from 60 to 120. The note indicates that Idaho changed its speed limits in both 1996 and 1998. Despite maintaining uniform speed limits, North Carolina showed a significant total crash rate increase. None of the other states that changed speed limits showed a change in the total crash rate. Figure 32. Chart. Total Truckinvolved Crash Rates in Virginia Interstate Highways. The graph shows the state truck crash rates as well as rates for five different interstate segments. The horizontal axis is years ranging from 1991 to 1999. The vertical axis is crash rates ranging from 0 to 20. Data is missing for 1994. The low is 5 crashes in 1992 on I85 and the high is 19 in 1997 on 177. No single interstate segment dominated the statewide average although truck crash rates increased over time. Figure 33. Chart. Relationship between the Poisson and Negative Binomial Distributions for Crash Frequencies. This chart illustrates the relationship between the Poisson probability distribution and the negative binomial distribution where i equals a particular site, y equals a particular year and the distribution of the number of actual crashes is K subscript i, y while expected crashes are m subscript i, y. The horizontal headers in the chart are years from 1991 through 1996 with both actual and expected formulae listed. The vertical columns are sites from 1 though 6. The 1991 column contains entries for six sites, while the remaining years list only two sites. An arrow to the right of the 1996 column and pointing down gives the Poisson formula as K subscript 1y. An arrow pointing down below the 1991 column gives the formula K subscript i1 as the negative binomial. A longer arrow to its right points to the gamma distribution of these assumptions as m subscript i1. Figure 34. Chart. Comparison of Poisson Distribution and Actual Crash Distribution. This figure compares the actual crash frequency distribution with the Poisson probability distribution for one site on Interstate 85 in Virginia between mileposts 19.52 and 24.73 between 1991 and 1999. The horizontal axis from 1 through 9 corresponds to years and the vertical axis from 0 to 10 equals crashes. The actual and statistic plots correlate well with ranges from 0 in 1991 to 4 or 5 in 1999. Figure 35. Chart. Comparison of Negative Binomial Distribution and Actual Crash Distribution (Probability Density Function). This figure compares the negative binomial distribution and the actual crash rate distribution (rather than total number of crashes because of the variation in section lengths). The horizontal axis is the negative binomial distribution ranging from 10 through 210. The vertical axis is crash rates from 0 to 50 in increments of five. The statistic peak is 21 while the actual is 30 at 45 on the horizontal axis. The plots correlate well as a good fit for these data. Figure 36. Equation. Crash Frequency for Year 1 as Base Year. The equation reads: the quotient of m subscript i,y over m subscript i,1 is equal to the quotient of the expected value of m subscript i,y over the expected value of m subscript i,1 which is equal to C superscript one subscript i,y. Figure 37. Equation. Crash Frequency for Year 3 as Base Year. The equation reads: the quotient of m subscript i,y over m subscript i,3 is equal to the quotient of the expected value of m subscript i,y over the expected value of m subscript i,3 which is equal to C superscript 3 subscript i,y. Figure 38. Equation Expected Value of Crash Count for Year 1. The equation reads: m subscript i,1 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the sum of the quotient of k over the expected value of m subscript i,1 and the summation of C superscript 1 subscript i,y from y equals 1 to Y. Figure 39. Equation Variance of Expected Value of Crash Count for Year 1. The equation reads: the variance of m subscript i,1 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the squared sum of the quotient of k over the expected value of m subscript i,1 and the summation of C superscript 1 subscript i,y from y equals 1 to Y which is equal to the quotient of m subscript i,1 over the sum of the quotient of k over the expected value of m subscript i,1 and the summation of C superscript 1 subscript i,y from y equals 1 to Y. Figure 40. Equation Estimation of Estimated Values of Crash Counts for Year 1. The equation reads: m subscript i,y is equal to the product of C superscript one subscript i,y and m subscript i,1 which is equal to the product of the quotient of the expected value of m subscript i,y over the expected value of m subscript i,1 and m subscript i,1. Figure 41. Equation Variance of Estimation of Estimated Values of Crash Counts for Year 1. The equation reads: the variance of m subscript i,y is equal to the product of the square of C superscript one subscript i,y and m subscript i,1 which is equal to the product of the squared quotient of the expected value of m subscript i,y over the expected value of m subscript i,1 and m subscript i,1. Figure 42. Equation Expected Value of Crash Count for Year 3 The equation reads: m subscript i,3 is equal to the product of C subscript i,3 and m subscript i,1 which is equal to the product of the quotient of the expected value of m subscript i,3 over the expected value of m subscript i,1 and m subscript i,1, which is equal to the product of the quotient of the expected value of m subscript i,3 over the expected value of m subscript i,1 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the sum of the quotient of k over the expected value of m subscript i,1 and the summation of C superscript 1 subscript i,y from y equals 1 to Y, which is equal to the product of the quotient of the expected value of m subscript i,3 over the expected value of m subscript i,1 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the sum of the quotient of k over the expected value of m subscript i,1 and the summation of the quotient of the expected value of m subscript i,y over the expected value of m subscript i,1 from y equals 1 to Y.Which is equal to the product of the expected value of m subscript i,3 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the product of the expected value of m subscript i,1 and the sum of the quotient of k over the expected value of m subscript i,1 and the summation of the quotient of the expected value of m subscript i,y over the expected value of m subscript i,1 from y equals 1 to Y.Which is equal to the product of the expected value of m subscript i,3 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the sum of k and the summation of the expected value of m subscript i,y from y equals 1 to Y.Which is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the sum of the quotient of k over the expected value of m subscript i,3 and the summation of the quotient of the expected value of m subscript i,y over the expected value of m subscript i,3 from y equals 1 to Y, which is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the sum of the quotient of k over the expected value of m subscript i,3 and the summation of C superscript 3 subscript i,y from y equals 1 to Y. The variance of m subscript i,3 is equal to the product of the square of C superscript 1 subscript i,3 and the variance of m subscript i,1, which is equal to the product of the square of the quotient of the expected value of m subscript i,3 over the expected value of m subscript i,1 and the variance of m subscript i,1. Which is equal to the product of the square of the quotient of the expected value of m subscript i,3 over the expected value of m subscript i,1 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the square of the sum of the quotient of k over the expected value of m subscript i,1 and the summation of C superscript 1 subscript i,y from y equals 1 to Y, which is equal to the product of the square of the quotient of the expected value of m subscript i,3 over the expected value of m subscript i,1 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the square of the sum of the quotient of k over the expected value of m subscript i,1 and the summation of the quotient of the expected value of m subscript i,y over the expected value of m subscript i,1 from y equals 1 to Y, which is equal to the product of the square of the expected value of m subscript i,3 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the product of the square of the expected value of m subscript i,y and the square of the sum of the quotient of k over the expected value of m subscript i,1 and the summation of the quotient of the expected value of m subscript i,y over the expected value of m subscript i,1. Which is equal to the product of the square of the expected value of m subscript i,3 and the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the square of the sum of k and the summation of the expected value of m subscript i,y from y equals 1 to Y. Which is equal to quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the square of the sum the quotient of k over the expected value of m subscript i,3 and the summation of the quotient of the expected value of m subscript i,y over the expected value of m subscript i,3. Which is equal to quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the square of the sum the quotient of k over the expected value of m subscript i,3 and the summation of C superscript 3 subscript i,y from y equals 1 to Y. Figure 43. Equation Expected Value of Crash Count for Year 3. The equation reads: m subscript i,3 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the sum of the quotient of k over the expected value of m subscript i,3 and the summation of C superscript 3 subscript i,y from y equals 1 to Y. Figure 44. Equation Expected Value of Crash Count for Year 3. The equation reads: the variance of m subscript i,3 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y all over the squared sum of the quotient of k over the expected value of m subscript i,1 and the summation of C superscript 3 subscript i,y from y equals 1 to Y, which is equal to the quotient of m subscript i,1 over the sum of the quotient of k over the expected value of m subscript i,1 and the summation of C superscript 3 subscript i,y from y equals 1 to Y. Figure 45. Equation Statistically Significant Difference in Mean Speeds. The equation reads: Is the absolute value of U subscript x minus U subscript y greater than the product of 1.96 and the square root of the sum of the quotient of S squared subscript x over n subscript x and the quotient S squared subscript y over n subscript y? Figure 46. Chart. Histogram Based on Random Numbers. This figure plots values for two simulated hypothetical sites with 1,365 speeds generated between a predefined range. Each bin is 1 mile per hour in width and the horizontal axis ranges from 30 to 90 bins. The vertical axis is frequency ranging from minus 5 to 40. Both X and Y plots represent sites X and Y and start at 35 to 39 bins with 0 frequency, quickly rising to a range of 20 to 35 for the remainder of the plot. The standard deviation is 14.31, which would be lower in a normal distribution. Figure 47. Equation. Formula to determine confidence intervals associated with mean speed. The equation reads: the mean of x plus or minus the quotient of the product of z and s over the square root of n. Figure 48. Equation. Example of formula in figure 47. The equation reads: 50 plus or minus the quotient of the product of 1.96 and 3 over the square root of 200, which is equal to the range 49.58 to 50.42 miles per hour. Figure 49. Equation. Confidence interval for 85^{th} percentile speed. The equation reads: 53.11 plus or minus the quotient of the product of 1.96 and 3 over the square root of 200, which is equal to the range 52.69 to 53.53 miles per hour. Figure 50. Equation. Binomial distribution. The equation reads: p plus or minus the product of z and the square root of the quotient of the product of p and one minus p over n, which is equal to 0.85 plus or minus the product of 1.96 and the square root of the quotient of the product of 0.85 and one minus 0.85 over 200, which is equal to the range 0.8005 to 0.8995. Figure 51. Chart. Arizona Total Crash Rate Versus ADT. This histogram shows crash rates versus ADT in Arizona where no speed limit change was instituted. The horizontal axis is ADT ranging from 0 to 64,000. The vertical axis is total crash rate ranging from 0.0 to 90.0. The lowest rates are zero at 42,000, 48,000 and 54,000 ADT with the highest rate of 79 at 6,000 ADT. The lower ADTs show a higher crash rate. This graph shows as the ADT increases, the total crash rate decreases. Figure 52. Chart. Virginia Total Crash Rate Versus ADT (Differential Speed Limits in Place). This histogram shows crash rates versus ADT in Virginia with differential speed limits in place. The horizontal axis is ADT ranging from 0 to 36,000. The vertical axis is total crash rate ranging from 0.0 to 90.0. The lowest crash rates of about 35 occur at 22,000 and 27,000 ADT. The highest crash rates of 80 occur at the lowest (4,000) and highest (32,000) ADTs. No trends can be determined from this graph. Figure 53. Chart. Virginia Total Crash Rate Versus ADT (Uniform Speed Limits in Place). This histogram shows crash rates versus ADT in Virginia with uniform speed limits in place. The horizontal axis is ADT ranging from 0 to 36,000. The vertical axis is total crash rate ranging from 0.0 to 160.0. The lowest crash rates of about 40 occur between 12,000 and 24,000 ADT. The highest crash rates ranging from 100 to 140 cluster at 28,000 to 32,000 ADT. No trends can be determined from this graph. Figure 54. Equation. Crash estimate model. The equation reads: the expected value of m is equal to the product of 0.02242775, the length raised to the power of 0.62225, and the ADT raised to the power of 0.5480. Figure 55. Equation. Mean of estimate for 1991. The equation reads: the expected value of m subscript 1,1991 is equal to the product of 0.02242775, 7.16 raised to the power of 0.62225, and 4000 raised to the power of 0.5480, which is equal to 7.191. Figure 56. Equation. Mean of estimate for 1992. The equation reads: the expected value of m subscript 1,1992 is equal to the product of 0.02242775, 7.16 raised to the power of 0.62225, and 4250 raised to the power of 0.5480, which is equal to 7.434.. Figure 57. Equation. Mean of estimate for 1993 The equation reads: the expected value of m subscript 1,1993 is equal to the product of 0.02242775, 7.16 raised to the power of 0.62225, and 4300 raised to the power of 0.5480, which is equal to 7.481. Figure 58. Equation. Calculation for ratio before year y. The equation reads: C subscript i,y is equal to the quotient of the expected value of m subscript i,y over the expected value of m subscript i,1. Figure 59. Equation. Ratio before year 1991 The equation reads: C subscript 1,1991 is equal to the quotient of the expected value of m subscript 1,1991 over the expected value of m subscript 1,1991, which is equal to 1. Figure 60. Equation. Ratio before year 1992 The equation reads: C subscript 1,1992 is equal to the quotient of the expected value of m subscript 1,1992 over the expected value of m subscript 1,1991, which is equal to 1.033782. Figure 61. Equation. Ratio before year 1993 The equation reads: C subscript 1,1993 is equal to the quotient of the expected value of m subscript 1,1993 over the expected value of m subscript 1,1991, which is equal to 1.040429. Figure 62. Equation. Expected crash counts. The equation reads: m subscript i,1 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y over the sum of the quotient k over the expected value of m subscript i,1 and the summation of C subscript i,y from y equals 1 to Y. Figure 63. Equation Variance of the Expected Crash County for Year 1. The equation reads: the variance of m subscript i,1 is equal to the quotient of the sum of k and the summation of K subscript i,y from y equals 1 to Y over the squared sum of the quotient k over the expected value of m subscript i,1 and the summation of C subscript i,y from y equals 1 to Y, which is equal to the quotient of m subscript i,1 over the sum of the quotient k over the expected value of m subscript i,1 and the summation of C subscript i,y from y equals 1 to Y. Figure 64. Equation Expected Crash Counts. The equation reads: m subscript i,y is equal to the product of C subscript i,y and m subscript i,1. Figure 65. Equation Variance of Expected Crash Counts. The equation reads: the expected value of m subscript i,y is equal to the product of the square of C subscript i,y and the variance of m subscript i,1. Figure 66. Equation. Application for 1991, The equation reads: m subscript 1,1991 is equal to the quotient of the sum of 5.9 and 26 over the sum of the quotient of 5.9 over 7.191 and 3.074211, which is equal to 8.190599 Figure 67. Equation. Application for variance 1991. The equation reads: the variance of m subscript 1,1991 is equal to the quotient of 8.190599 over the sum of the quotient of 5.9 over 7.191 and 3.074211, which is equal to 2.103007 Figure 68. Equation. Application for 1992. The equation reads: m subscript 1,1992 is equal to the product of C subscript 1,1992 and m subscript 1,1991, which is equal to the product of 1.033782 and 8.190599, which is equal to 8.467292. Figure 69. Equation. Application for variance 1992. The equation reads: the variance of m subscript 1,1992 is equal to the product of the square of C subscript 1,1992 and the variance of m subscript 1,1991, which is equal to the product of 1.033782 squared and 2.103007, which is equal to 2.247493 Figure 70. Equation. Application for 1993. The equation reads: m subscript 1,1993 is equal to the product of C subscript 1,1993 and m subscript 1,1991, which is equal to the product of 1.040429 and 8.190599, which is equal to 8.521739. Figure 71. Equation. Application for variance for 1993. The equation reads: the variance of m subscript 1,1993 is equal to the product of the square of C subscript 1,1993 and the variance of m subscript 1,1991, which is equal to the product of 1.040429 squared and 2.103007, which is equal to 2.27649. Figure 72. Equation. Computation of E(m_{1,1995}). The equation reads: the expected value of m subscript 1,1995 is equal to the product of 0.02242775, 7.16 raised to the power of 0.62225, and 4500 raised to the power of 0.5480, which is equal to 7.670. Figure 73. Equation. Computation of E(m_{1,1995}). The equation reads: the expected value of m subscript 1,1996 is equal to the product of 0.02242775, 7.16 raised to the power of 0.62225, and 4650 raised to the power of 0.5480, which is equal to 7.809. Figure 74. Equation. Computation of C_{1,1995}. The equation reads: C subscript 1,1995 is equal to the quotient of the expected value of m subscript 1,1995 over the expected value of m subscript 1,1991, which is equal to 1.066677. Figure 75. Equation. Computation of C_{1,1996}. The equation reads: C subscript 1,1996 is equal to the quotient of the expected value of m subscript 1,1996 over the expected value of m subscript 1,1991, which is equal to 1.086018 Figure 76. Equation. Expected Crash Counts, Year y The equation reads: m subscript i,y is equal to the product of C subscript i,y and m subscript i,1. Figure 77. Equation. Variance of Expected Crash Counts, Year y. The equation reads: the variance of m subscript i,y is equal to the product of the square of C subscript i,y and the variance of m subscript i,1. Figure 78. Equation. Expected Crash Counts, Year 1995. The equation reads: m subscript 1,1995 is equal to the product of C subscript 1,1995 and m subscript 1,1991 which is equal to the product of 1.066677 and 8.190599, which is equal to 8.73672. Figure 79. Equation. Variance of Expected Crash Counts, Year 1995. The equation reads: the variance of m subscript 1,1995 is equal to the product of the square of C subscript 1,1995 and the variance of m subscript 1,1991, which is equal to the product of1.066677 squared and 2.103007, which is equal to 2.392799. Figure 80. Equation. Expected Crash Counts, Year 1996. The equation reads: m subscript 1,1996 is equal to the product of C subscript 1,1996 and m subscript 1,1991, which is equal to the product of 1.086018 and 8.190599, which is equal to 8.895134. Figure 81. Equation. Variance of Expected Crash Counts, Year 1996. The equation reads: the variance of m subscript 1,1996 is equal to the product of the square of C subscript 1,1996 and the variance of m subscript 1,1991, which is equal to the product of 1.086018 squared and 2.103007, which is equal to 2.2480358. Figure 82. Equation Total wouldhavebeen crashes for a particular site The equation reads: pi subscript i is equal to the summation of m subscript 1,y , which is equal to the sum of 8.73672, 8.895134, 9.051255, and 9.077637, which is equal to 35.76075 Figure 83. Equation Total actual crashes for a particular site. The equation reads: lambda subscript i is equal to the summation of K subscript 1,y, which is equal to the sum of 8, 7, 10, and 5, which is equal to 30. Figure 84. Equation Safety impact for a particular site. The equation reads: pi subscript i minus lambda subscript i is equal to 35.76075 minus 30, which is equal to 5.76 Figure 85. Equation. Ratio of Actual to "Would Have Been" Crashes. The equation reads: the quotient of lambda subscript i over pi subscript i is equal to the quotient of 30 over 35.76075, which is equal to 0.84, which is equal to 84 percent Figure 86. Chart. Cumulative Differences by Year at the Example Site. The graph shows the cumulative differences between actual and predicted crashes for four years at the example site. The blue line is cumulative actual crashes, the yellow line cumulative predicted crashes and the red line cumulative excess. The horizontal axis is years ranging from 1995 to 1999. The vertical axis is crashes ranging from 0 to 40. Actual crashes plot at 7 for 1995 and rise to 30 in 1999. Predicted crashes rise similarly but end at 36 in 1999 for a difference of 6 shown in the last cumulative excess plot point. An interpretation of the graph is that over the fouryear after period, the actual number of crashes at this site was 5.76 less than the predicted number that would have resulted without change to the speed limit. Figure 87. Equation. Total "Would Have Been" Crashes. The equation reads: pi is equal to the summation of pi subscript i, pi, which is equal to 13,365.91 Figure 88. Equation. Total Actual Crashes. The equation reads: lambda is equal to the summation of lambda subscript i, which is equal to 15,377. Figure 89. Equation. Safety Impact. The equation reads: delta is equal to pi minus lambda, which is equal to 13,365.91 minus 15,377, which is equal to negative 2,011.09. Figure 90. Equation. Variance of the Difference between "Would Have Been" Crashes and Actual Crashes. The equation reads: the variance of delta is equal to the sum of the variance of pi and the variance of lambda, which is equal to the sum of the summation of the variance of pi subscript i and the summation of the variance of lambda subscript i, which is equal to the sum of 4,246.91 and 15,377, which is equal to 19,623.91. Figure 91. Equation. Standard Deviation of the Difference between "Would Have Been" Crashes and Actual Crashes. The equation reads: the standard deviation of delta is equal to the square root of the variance of delta, which is equal to 140.0854. Figure 92. Equation. Computation of the index of effectiveness. The equation reads: theta is equal to the quotient of the quotient of lambda over pi all over the sum of 1 and the quotient of the variance of pi over pi squared.It follows that theta is equal to the quotient of the quotient of 15,377 over 13,365.91 all over the sum of 1 and the quotient of 4,246.913 over the square of 13,365.91.If follows that theta si equal to 1.150437, in other words, about a 15 percent increase. Figure 93. Equation. Variance of θ. The equation reads: the variance of theta is equal to the sum of the product of theta squared and the quotient of the variance of lambda over lambda squared and the quotient of the quotient of the variance of pi over pi squared all over the squared sum of 1 and the quotient of the variance of pi over pi squared. It follows that the variance of theta is equal to the sum of the product of 1.150437 squared and the quotient of 15,377 over 15,377 squared and the quotient of 4,246.913 over 13,365.91 squared all over the square of the sum of 1 and the quotient of 4,246.913 over 13,365.91 squared. It follows that the variance of theta is equal to 0.000118. Figure 94. Equation. Empirical confidence bounds. The equation reads: theta plus or minus the product of two and the square root of the variance of theta 1.15 plus or minus the product of two and the square root of 0.000118 in absolute units or, 15 percent plus or minus the product of two and the square root of 0.0118 percent is expressed as a percentage, or 15 percent plus or minus the product of 2 and 1.08 percent, or The range 12.9 percent to 17.2 percent. 