---
title: "Spot speed frequency data: speed limits and modal speed"
subject: "Transportation Engineering II"
chapter: 1
key: "semester_7/transportation_engineering_ii:spot-speed-frequency-table-limits"
source: https://www.bctengineer.com/q/semester_7/transportation_engineering_ii/spot-speed-frequency-table-limits
site: BCT Engineering
---

# Spot speed frequency data: speed limits and modal speed

Transportation Engineering II · Chapter 1: Traffic Engineering

- Asked in: 2074 Asoj, 8 marks
- Video: [Upper limit speed, lower speed, Design speed, Modal speed | Transportation numerical](https://www.youtube.com/watch?v=8QVwBDuo628) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/transportation_engineering_ii/spot-speed-frequency-table-limits
- Whole chapter: https://www.bctengineer.com/md/semester_7/transportation_engineering_ii/chapter-1

## Question

Spot speed observation at a particular link provides the following data, calculate maximum speed limit, minimum speed limit, design speed and modal speed for regulation of traffic.

| Speed range (kmph) | Frequency |
|---|---|
| 6-10 | 1 |
| 10-14 | 4 |
| 14-18 | 7 |
| 18-22 | 20 |
| 22-26 | 44 |
| 26-30 | 80 |
| 30-34 | 82 |
| 34-38 | 79 |
| 38-42 | 49 |
| 42-46 | 36 |
| 46-50 | 26 |
| 50-54 | 9 |
| 54-58 | 10 |
| 58-62 | 3 |

## Short answer

From the cumulative frequency table: 85th percentile maximum speed limit = 43.8 km/h, 15th percentile minimum limit = 25.2 km/h, 98th percentile design speed = 55.6 km/h. Modal speed = 31.6 km/h.

## Answer

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 $N = 450$.

| Speed class (km/h) | Frequency | Cumulative frequency | Cumulative % |
|---|---|---|---|
| 6-10 | 1 | 1 | 0.2 |
| 10-14 | 4 | 5 | 1.1 |
| 14-18 | 7 | 12 | 2.7 |
| 18-22 | 20 | 32 | 7.1 |
| 22-26 | 44 | 76 | 16.9 |
| 26-30 | 80 | 156 | 34.7 |
| 30-34 | 82 | 238 | 52.9 |
| 34-38 | 79 | 317 | 70.4 |
| 38-42 | 49 | 366 | 81.3 |
| 42-46 | 36 | 402 | 89.3 |
| 46-50 | 26 | 428 | 95.1 |
| 50-54 | 9 | 437 | 97.1 |
| 54-58 | 10 | 447 | 99.3 |
| 58-62 | 3 | 450 | 100.0 |

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.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which percentile speeds are used for the maximum speed limit, minimum speed limit and design speed respectively?
   A. 85th, 15th, 98th
   B. 98th, 85th, 15th
   C. 50th, 15th, 85th
   D. 85th, 98th, 15th
   Answer: A. The 85th percentile is the upper limit, the 15th the lower limit, and the 98th the design speed. (Key point: Percentile speeds for regulation)

2. With N = 450, in which class does the 85th percentile speed lie (cumulative frequencies 366 at 38-42 and 402 at 42-46)?
   A. 38-42
   B. 34-38
   C. 42-46
   D. 46-50
   Answer: C. 0.85 x 450 = 382.5, which is above 366 and below 402, so it falls in the 42-46 class. (Key point: Locating the percentile class)

3. What is the 85th percentile speed from v = 42 + (382.5 - 366)/36 x 4?
   A. 42.5 km/h
   B. 43.8 km/h
   C. 45.0 km/h
   D. 46.1 km/h
   Answer: B. (16.5/36) x 4 = 1.83, so v85 = 43.8 km/h. (Key point: Interpolation formula)

4. What is the modal speed using the class 30-34 (f1 = 82, f0 = 80, f2 = 79)?
   A. 30.8 km/h
   B. 32.0 km/h
   C. 33.2 km/h
   D. 31.6 km/h
   Answer: D. 30 + (82-80)/(164-80-79) x 4 = 30 + (2/5)(4) = 31.6 km/h. (Key point: Modal speed)

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