Spot speed frequency data: speed limits and modal speed

Speed limits for regulation are read from the cumulative frequency distribution (standard speed-study practice): the 85th percentile speed is the maximum (upper) speed limit, the 15th percentile speed is the minimum (lower) speed limit, and the 98th percentile speed is the design speed. The modal speed is the speed with the highest frequency.

Total observations $N = 450$.

By interpolation within the class, $v_p = L + \dfrac{(pN/100 - F_{prev})}{f}\,h$:

Maximum speed limit (85th percentile)

$$v_{85} = 42 + \frac{382.5 - 366}{36}(4) = 43.8\ \text{km/h}$$

Minimum speed limit (15th percentile)

$$v_{15} = 22 + \frac{67.5 - 32}{44}(4) = 25.2\ \text{km/h}$$

Design speed (98th percentile)

$$v_{98} = 54 + \frac{441.0 - 437}{10}(4) = 55.6\ \text{km/h}$$

Modal speed

The modal class is 30-34 (f = 82), with the class before $f_0 = 80$ and the class after $f_2 = 79$:

$$v_{mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h = 30 + \frac{82-80}{2(82)-80-79}(4) = 31.6\ \text{km/h}$$

Answer: maximum speed limit = 43.8 km/h, minimum speed limit = 25.2 km/h, design speed = 55.6 km/h, modal speed = 31.6 km/h. (In practice the limits are rounded to 45, 25 and 55 km/h.)