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

# Time and space mean speed from 20 spot speeds

Transportation Engineering II · Chapter 1: Traffic Engineering

- Asked in: 2075 Chaitra, 8 marks
- Video: [time mean speed and space mean speed numerical| in nepali|transportation engineering II | chapter 1](https://www.youtube.com/watch?v=9r7iTP6xoEo) — pushpa thapa
- 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-observations-travel-time
- Whole chapter: https://www.bctengineer.com/md/semester_7/transportation_engineering_ii/chapter-1

## Question

In a field survey of spot speed measurement, the following twenty observations were taken. Find time mean speed, and space mean speed.

50, 40, 60, 54, 45, 31, 72, 58, 43, 52, 46, 56, 43, 65, 33, 69, 34, 51, 47, 41.

Also, assuming these vehicle speeds are fixed over a half km segment, calculate the corresponding travel times and show that the space mean speed calculated using travel times is equal to the point estimate.

## Short answer

Time mean speed = 49.50 km/h (arithmetic mean of 20 observations). Space mean speed = 46.95 km/h (harmonic mean). For the 0.5 km segment, mean travel time 38.34 s gives space mean speed 0.5/0.01065 h = 46.95 km/h, matching the harmonic mean.

## Answer

#### Time mean speed
The time mean speed is the arithmetic mean of the spot speeds:

$$
\bar v_t = \frac{\sum v_i}{n} = \frac{990}{20} = 49.50\ \text{km/h}
$$

#### Space mean speed
The space mean speed is the harmonic mean of the spot speeds:

$$
\bar v_s = \frac{n}{\sum (1/v_i)} = \frac{20}{0.42600} = 46.95\ \text{km/h}
$$

#### Travel times over 0.5 km
$t_i = 0.5/v_i \times 3600$ s:

| Vehicle | Speed (km/h) | Travel time (s) |
|---|---|---|
| 1 | 50 | 36.00 |
| 2 | 40 | 45.00 |
| 3 | 60 | 30.00 |
| 4 | 54 | 33.33 |
| 5 | 45 | 40.00 |
| 6 | 31 | 58.06 |
| 7 | 72 | 25.00 |
| 8 | 58 | 31.03 |
| 9 | 43 | 41.86 |
| 10 | 52 | 34.62 |
| 11 | 46 | 39.13 |
| 12 | 56 | 32.14 |
| 13 | 43 | 41.86 |
| 14 | 65 | 27.69 |
| 15 | 33 | 54.55 |
| 16 | 69 | 26.09 |
| 17 | 34 | 52.94 |
| 18 | 51 | 35.29 |
| 19 | 47 | 38.30 |
| 20 | 41 | 43.90 |

Mean travel time $\bar t = 766.80/20 = 38.340$ s $= 0.010650$ h.

$$
\bar v_s = \frac{L}{\bar t} = \frac{0.5}{0.010650} = 46.95\ \text{km/h}
$$

This equals the harmonic mean of the spot speeds (46.95 km/h), because $L/\bar t = L/\left(\frac{1}{n}\sum L/v_i\right) = n/\sum(1/v_i)$.

**Answer: time mean speed = 49.50 km/h; space mean speed = 46.95 km/h (same from travel times).** The space mean speed is lower than the time mean speed.

## Check yourself

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

1. What is time mean speed?
   A. Harmonic mean of spot speeds
   B. Arithmetic mean of spot speeds
   C. Median speed
   D. Distance over mean travel time
   Answer: B. Time mean speed is the arithmetic mean of spot speeds: 990/20 = 49.50 km/h. (Key point: Time mean speed)

2. What is space mean speed in terms of spot speeds?
   A. Arithmetic mean
   B. Harmonic mean, n / sum(1/v)
   C. Geometric mean
   D. Maximum speed
   Answer: B. It is the harmonic mean, giving 46.95 km/h here. (Key point: Space mean speed)

3. How do the two means compare for these data?
   A. Space mean is lower than time mean
   B. Space mean is higher than time mean
   C. They are equal
   D. Cannot be compared
   Answer: A. 46.95 km/h (space) is lower than 49.50 km/h (time). (Key point: Comparison)

4. Why does L / mean travel time equal the harmonic mean of spot speeds?
   A. Because mean time = (1/n) sum(L/vi), so L/t = n / sum(1/vi)
   B. Because all speeds are equal
   C. Because L cancels out of the speeds
   D. Because travel times are averaged geometrically
   Answer: A. Substituting ti = L/vi into the mean gives n / sum(1/vi), the harmonic mean. (Key point: Travel-time proof)

## Related in Transportation Engineering II

- [Time/space mean speed and density from spot speeds](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aspot-speed-20-values-time-space) — 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

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