14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace

14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace

PTOE Exam Focus: Spot speed studies are fundamental to Domain 5. Candidates must understand study design protocols, correct for radar/LiDAR cosine error ($v_{\text{true}} = \frac{v_{\text{measured}}}{\cos\theta}$), construct and interpret cumulative frequency distribution curves, determine the 85th percentile speed ($V_{85}$) for speed zone establishment, calculate the 10-mph pace and percentage in pace, and mathematically relate Time Mean Speed ($\bar{v}_t$) to Space Mean Speed ($\bar{v}_s$) via Wardrop's formulation.

---

1. Objectives & Field Protocols of Spot Speed Studies

A spot speed study records the instantaneous speeds of individual vehicles traversing a specific, localized roadway cross-section. Its primary engineering applications include:

• Establishing and updating regulatory speed limits (speed zoning per MUTCD).
• Assessing speed dispersion and crash risk correlation.
• Evaluating the effectiveness of traffic calming countermeasures.
• Sizing traffic signal change and clearance intervals.
• Verifying design speed consistency and stopping sight distance adequacy.

Field Sampling Protocols:

1. Free-Flowing Conditions: Only sample vehicles with time headways $\ge 4\text{ to }5\text{ seconds}$. Do not sample following vehicles trapped in platoons.
2. Unbiased Selection: Sample every $n$-th vehicle or random free-flow vehicles across all lanes. Do not preferentially target high-speed outliers or sports cars.
3. Ideal Geometry & Environmental Conditions: Conduct studies on straight, level tangents away from driveways, signals, and intersections during dry, daylight off-peak hours.
4. Unobtrusive Observer Placement: Conceal the observer and equipment to prevent driver braking upon spotting surveillance.

---

2. Speed Measurement Technologies & Cosine Error Correction

Common spot speed instrumentation includes:

• Radar / LiDAR Guns: Handheld Doppler radar ($24.15\text{ GHz}$ K-band or $34.7\text{ GHz}$ Ka-band) and infrared LiDAR laser pulses ($904\text{ nm}$). Handheld units must be aimed along the direction of travel.
• Pneumatic Road Tubes: Dual rubber tubes placed across the lane separated by a fixed distance ($d = 10\text{ to }16\text{ ft}$). An electronic counter measures pulse time difference $\Delta t$, yielding $v = d / \Delta t$.
• Video Analytics & Computer Vision: High-definition cameras calibrated with roadway ground control points tracking vehicle bounding boxes.

Cosine Error Mechanics & Correction

When a radar or LiDAR operator stands off the roadway shoulder, the line-of-sight vector forms an angle $\theta$ with the vehicle's true trajectory. Because Doppler radar measures only the radial velocity component along the line of sight:

$$v_{\text{measured}} = v_{\text{true}} \cdot \cos\theta$$

                         Roadway Centerline (True Vehicle Trajectory)
  ====================================[ Vehicle: v_true ]========================>
                     ^               /
                      \             /  Line of Sight Vector
                       \           /   (Measured Velocity: v_measured)
                        \ theta   /
                         \       /
                          \     /
                           [ Radar Gun Operator ] (Offset on Shoulder)

Critical Cosine Error Rule:

Cosine error always causes the device to underestimate true speed (since $\cos\theta \le 1.0$). To recover the true vehicular velocity:

$$v_{\text{true}} = \frac{v_{\text{measured}}}{\cos\theta}$$

• If $\theta \le 10^\circ$: $\cos(10^\circ) = 0.9848$, error is $< 1.5\%$ (acceptable without manual correction in routine screening).
• If $\theta > 10^\circ$ (e.g., $\theta = 25^\circ$ to $30^\circ$): $\cos(25^\circ) = 0.9063$, error exceeds $9.4\%$ and must be mathematically corrected.

---

3. Cumulative Speed Distribution & Percentile Metrics

Spot speed observations are grouped into frequency bins (typically $2\text{ mph}$ or $5\text{ mph}$ intervals) to generate a Cumulative Speed Distribution Curve (an S-shaped ogive curve plotted as Cumulative Percentage on the vertical axis versus Speed on the horizontal axis).

  Cumulative
  Percent (%)
     100 ^                                            .---'
         |                                        _.-'
      85 |------------------------------------.-' |  <-- 85th Percentile Speed (V_85)
         |                                 .-'    |
      50 |-----------------------------.-'        |  <-- Median Speed (V_50)
         |                          .-'           |
      15 |-----------------------.-' |            |  <-- 15th Percentile Speed (V_15)
         |                   _.-'    |            |
       0 +------------------+--------+------------+------------------------->
         0                 V_15     V_50         V_85              Speed (mph)
                            |<-- 10-mph Pace ---->|

Critical Percentile Benchmarks:

1. 85th Percentile Speed ($V_{85}$): The speed at or below which $85\%$ of free-flowing vehicles travel. It represents the speed that reasonable, prudent drivers adopt under prevailing roadway conditions. MUTCD Section 2B.35 mandates that posted regulatory speed limits be established within $5\text{ mph}$ of the 85th percentile speed.
2. 50th Percentile Speed ($V_{50}$ / Median): The speed that divides the distribution into two equal halves ($50\%$ drive slower, $50\%$ drive faster).
3. 15th Percentile Speed ($V_{15}$): The speed at or below which only $15\%$ of vehicles travel; used as an engineering guideline for establishing minimum speed limits or identifying slow-moving hazards.

---

4. The 10-mph Pace & Modal Speed

Definition of the 10-mph Pace

The 10-mph Pace is the specific $10\text{ mph}$ speed increment (e.g., $42\text{ to }52\text{ mph}$) that contains the highest percentage of sampled vehicles across the entire distribution.

Percentage of Vehicles in the 10-mph Pace

$$\text{Percent in Pace} = \left( \frac{\text{Number of Vehicles in 10-mph Pace}}{\text{Total Sample Size } N} \right) \times 100\%$$

• Uniform Flow (Ideal): $\ge 70\%$ of vehicles in pace. Indicates high driver consensus and low speed variance.
• Dispersed Flow (High Crash Risk): $< 50\%\text{ to }60\%$ in pace. Indicates wide speed disparities, causing frequent overtaking, tailgating, and elevated collision rates.

Modal Speed

The Modal Speed is the single speed value occurring with the greatest frequency (the peak of the speed frequency histogram).

---

5. Speed Dispersion, Skewness, & Standard Deviation

Standard Deviation ($s$)

Quantifies the dispersion or spread of speeds around the sample mean $\bar{v}$:

$$s = \sqrt{\frac{\sum_{i=1}^{N} (v_i - \bar{v})^2}{N - 1}}$$

For an approximately normal distribution, standard deviation can be estimated from percentiles:

$$s \approx \frac{V_{85} - V_{15}}{2 \cdot 1.036} \approx \frac{V_{85} - V_{15}}{2.07}$$

Skewness Index

Evaluates the asymmetry of the speed distribution:

$$\text{Skewness} = \frac{2(V_{85} - V_{50})}{V_{85} - V_{15}} - 1$$

• Zero Skewness ($0.0$): Perfectly symmetric normal distribution ($V_{85} - V_{50} = V_{50} - V_{15}$).
• Positive (Right) Skew ($> 0$): Mean and 85th percentile are pulled higher by a long tail of high-speed outliers.
• Negative (Left) Skew ($< 0$): Long tail of slow-moving vehicles.

---

6. Time Mean Speed ($\bar{v}_t$) vs Space Mean Speed ($\bar{v}_s$)

Traffic engineering strictly distinguishes between point-based arithmetic speed and length-based harmonic speed:

1. Time Mean Speed ($\bar{v}_t$)

The arithmetic average of instantaneous spot speeds measured at a single point:

$$\bar{v}_t = \frac{1}{N} \sum_{i=1}^{N} v_i$$

2. Space Mean Speed ($\bar{v}_s$)

The harmonic average of speeds over a roadway segment length $L$, equivalent to total vehicle-distance divided by total vehicle-travel-time:

$$\bar{v}_s = \frac{N}{\sum_{i=1}^{N} \frac{1}{v_i}} = \frac{L \cdot N}{\sum_{i=1}^{N} t_i}$$

Wardrop's Equilibrium Relation:

$$\bar{v}_t = \bar{v}_s + \frac{\sigma_s^2}{\bar{v}_s}$$

Where $\sigma_s^2$ is the variance of the space mean speed distribution.

Fundamental Principle: Because variance is non-negative ($\sigma_s^2 \ge 0$), Time Mean Speed is always greater than or equal to Space Mean Speed ($\bar{v}_t \ge \bar{v}_s$). They are equal only when all vehicles travel at identically equal speeds ($\sigma_s^2 = 0$).

Spot Speed Metrics, Mathematical Formulations, and Engineering Applications

7. Worked Calculation Examples

Example 1: Radar Cosine Error Correction

Problem: A technician positions a handheld radar gun on the outer highway shoulder, creating an angle of $\theta = 25^\circ$ between the radar line of sight and the center of the approaching travel lane. The radar unit displays a measured speed of $v_{\text{measured}} = 48.0\text{ mph}$. What is the true operating speed of the vehicle?

Step-by-Step Solution:

1. Identify Formula:

   $$v_{\text{true}} = \frac{v_{\text{measured}}}{\cos\theta}$$

2. Compute Trigonometric Cosine:

   $$\cos(25^\circ) = 0.9063$$

3. Calculate True Speed:

   $$v_{\text{true}} = \frac{48.0\text{ mph}}{0.9063} = 52.96\text{ mph} \approx 53.0\text{ mph}$$

Engineering Note: Failure to correct for the $25^\circ$ angle would result in underestimating the vehicle's speed by $5.0\text{ mph}$.

---

Example 2: Time Mean Speed vs. Space Mean Speed Calculation

Problem: Four vehicles traverse a $1.0\text{-mile}$ roadway segment with measured spot speeds of $30\text{ mph}$, $40\text{ mph}$, $50\text{ mph}$, and $60\text{ mph}$. Compute the Time Mean Speed ($\bar{v}_t$) and the Space Mean Speed ($\bar{v}_s$).

Step-by-Step Solution:

1. Time Mean Speed ($\bar{v}_t$, Arithmetic Mean):

   $$\bar{v}_t = \frac{30 + 40 + 50 + 60}{4} = \frac{180}{4} = 45.0\text{ mph}$$

2. Travel Times for Each Vehicle over $L = 1.0\text{ mile}$:

   • $t_1 = 1 / 30 = 0.03333\text{ hr}$
   • $t_2 = 1 / 40 = 0.02500\text{ hr}$
   • $t_3 = 1 / 50 = 0.02000\text{ hr}$
   • $t_4 = 1 / 60 = 0.01667\text{ hr}$
   • Total travel time $\sum t_i = 0.03333 + 0.02500 + 0.02000 + 0.01667 = 0.09500\text{ hr}$

3. Space Mean Speed ($\bar{v}_s$, Harmonic Mean):

   $$\bar{v}_s = \frac{N \cdot L}{\sum t_i} = \frac{4 \times 1.0}{0.09500} = 42.11\text{ mph} \approx 42.1\text{ mph}$$

Verification: $\bar{v}_t (45.0) > \bar{v}_s (42.1)$, satisfying Wardrop's principle.