---
title: "Time/space mean speed and density from spot speeds"
subject: "Transportation Engineering II"
chapter: 1
key: "semester_7/transportation_engineering_ii:spot-speed-20-values-time-space"
source: https://www.bctengineer.com/q/semester_7/transportation_engineering_ii/spot-speed-20-values-time-space
site: BCT Engineering
---

# Time/space mean speed and density from spot speeds

Transportation Engineering II · Chapter 1: Traffic Engineering

- Asked in: 2066 Bhadra, 3+3+4 marks
- Video: [Time mean speed & Space mean speed](https://www.youtube.com/watch?v=IZYP6SQ-rhc) — DAA ETech
- 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-20-values-time-space
- Whole chapter: https://www.bctengineer.com/md/semester_7/transportation_engineering_ii/chapter-1

## Question

The following speed data were collected during a two-minute segment of a spot speed study (speed in kmph).

45, 55, 48, 52, 60, 48, 60, 42, 52, 65, 64, 63, 58, 56, 68, 54, 68, 64, 66, 70

Calculate: (i) The time mean speed; (ii) The space mean speed. What will be the average density of the above traffic stream if the mean headway is 4.5 sec?

## Short answer

Time mean speed is the arithmetic mean, 57.90 km/h. Space mean speed is the harmonic mean, 56.71 km/h. With mean headway 4.5 s, flow is 800 veh/h, giving density 14.1 veh/km.

## Answer

#### (i) Time mean speed
The arithmetic mean of the 20 spot speeds:

$$
\bar v_t = \frac{\sum v_i}{n} = \frac{1158}{20} = 57.90\ \text{km/h}
$$

#### (ii) Space mean speed
The harmonic mean of the spot speeds:

$$
\bar v_s = \frac{n}{\sum 1/v_i} = \frac{20}{0.35265} = 56.71\ \text{km/h}
$$

#### (iii) Average density
Mean headway 4.5 s gives the flow $q = 3600/4.5 = 800$ veh/h. From $q = k\,\bar v_s$:

$$
k = \frac{q}{\bar v_s} = \frac{800}{56.71} = 14.11\ \text{veh/km}
$$

**Answer: time mean speed = 57.90 km/h; space mean speed = 56.71 km/h; density = 14.1 veh/km.**

## Check yourself

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

1. Which mean gives the space mean speed from spot speeds?
   A. Arithmetic mean
   B. Harmonic mean
   C. Median
   D. Geometric mean
   Answer: B. Space mean speed = n / sum(1/v_i), the harmonic mean. (Key point: Space mean speed)

2. What is the time mean speed (sum = 1158 km/h, n = 20)?
   A. 56.71 km/h
   B. 60.0 km/h
   C. 57.90 km/h
   D. 55.0 km/h
   Answer: C. 1158/20 = 57.90 km/h. (Key point: Time mean speed)

3. A mean headway of 4.5 s gives what flow?
   A. 800 veh/h
   B. 450 veh/h
   C. 1600 veh/h
   D. 4.5 veh/h
   Answer: A. q = 3600/4.5 = 800 veh/h. (Key point: Flow from headway)

4. Density k = q / (space mean speed) is:
   A. 13.8 veh/km
   B. 14.1 veh/km
   C. 16.0 veh/km
   D. 800 veh/km
   Answer: B. k = 800/56.71 = 14.1 veh/km; using the time mean would wrongly give 13.8. (Key point: Density)

## Related in Transportation Engineering II

- [Time and space mean speed from 20 spot speeds](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aspot-speed-20-observations-travel-time) — Transportation Engineering II, chapter 1
- [Spot speed frequency data: speed limits and modal speed](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aspot-speed-frequency-table-limits) — Transportation Engineering II, chapter 1
- [Time and space mean speed from 14 travel times](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Atime-space-mean-speed-14-vehicles) — Transportation Engineering II, chapter 1
- [Spot speed study methods and rotary limitations](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aspot-speed-methods-rotary-advantages) — Transportation Engineering II, chapter 1

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