---
title: "Roughness parameters from a surface trace"
subject: "Metrology"
chapter: 10
key: "semester_3/metrology:roughness-ra-rq-rz-calculation"
source: https://www.bctengineer.com/q/semester_3/metrology/roughness-ra-rq-rz-calculation
site: BCT Engineering
---

# Roughness parameters from a surface trace

Metrology · Chapter 10: Measurement of Surface Finish

- Asked in: year not given, 6 marks
- Video: [Surface Roughness Parameters: Ra, Ry, Rz, RMS (हिंदी में)](https://www.youtube.com/watch?v=WnKXj61YKKA) — Mr Mechanical Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/metrology/roughness-ra-rq-rz-calculation
- Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-10

## Question

Define Ra, Rq (RMS) and Rz/Ry. The heights of ten equally spaced ordinates of a surface profile from the mean line are (in micrometres): +2.1, -1.8, +3.2, -2.4, +1.5, -3.0, +2.6, -1.2, +1.9, -2.8. Calculate Ra, Rq and the maximum peak-to-valley height. Convert Ra to micro-inches.

## Short answer

Ra = arithmetic mean of absolute deviations; Rq = RMS; Rz is peak-to-valley height. From given ordinates: Ra = 2.25 µm (88.6 µ in), Rq = 2.34 µm, maximum peak-to-valley height = 6.2 µm.

## Answer

#### Definitions
- **Ra (centre-line average, CLA)**: the arithmetic mean of the absolute deviations of the profile from the mean line over the sampling length:
$$
R_a = \frac{1}{n}\sum_{i=1}^{n}|y_i|
$$
- **Rq (RMS)**: the root mean square deviation, $R_q = \sqrt{\dfrac{1}{n}\sum y_i^2}$. It is about 1.11 times Ra for a sine-like profile.
- **Rz / Ry (peak to valley)**: the vertical distance between the highest peak and the lowest valley within the sampling length (Rz in ISO is the mean of five such heights in five sampling lengths).

#### Calculation
| $i$ | $y_i$ ($\mu$m) | $\lvert y_i\rvert$ | $y_i^2$ |
|---|---|---|---|
| 1 | +2.1 | 2.1 | 4.41 |
| 2 | -1.8 | 1.8 | 3.24 |
| 3 | +3.2 | 3.2 | 10.24 |
| 4 | -2.4 | 2.4 | 5.76 |
| 5 | +1.5 | 1.5 | 2.25 |
| 6 | -3.0 | 3.0 | 9.00 |
| 7 | +2.6 | 2.6 | 6.76 |
| 8 | -1.2 | 1.2 | 1.44 |
| 9 | +1.9 | 1.9 | 3.61 |
| 10 | -2.8 | 2.8 | 7.84 |
| Sum | | 22.5 | 54.55 |

$$
R_a = \frac{22.5}{10} = 2.25\ \mu\text{m}
$$

$$
R_q = \sqrt{\frac{54.55}{10}} = \sqrt{5.455} = 2.34\ \mu\text{m}
$$

Highest peak $= +3.2$, deepest valley $= -3.0$:

$$
R_y = 3.2 - (-3.0) = 6.2\ \mu\text{m}
$$

In micro-inches, 1 $\mu$m $= 39.37\ \mu$in: $R_a = 2.25 \times 39.37 = 88.6\ \mu$in.

**Answer: $R_a$ = 2.25 $\mu$m (88.6 $\mu$in), $R_q$ = 2.34 $\mu$m, $R_y$ = 6.2 $\mu$m.**

## Check yourself

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

1. Ra is defined as:
   A. The arithmetic mean of the absolute deviations of the profile from the mean line
   B. The root mean square of the deviations
   C. The distance from highest peak to lowest valley
   D. The sum of the signed deviations
   Answer: A. Ra = (1/n) sum |yi|, also called centre-line average. (Key point: Ra definition)

2. For the ten ordinates, sum |y| = 22.5 micrometre. What is Ra?
   A. 22.5 micrometre
   B. 2.25 micrometre
   C. 0.225 micrometre
   D. 2.34 micrometre
   Answer: B. Ra = 22.5/10 = 2.25 micrometre. (Key point: Ra value)

3. Which is correct for Rq with sum y^2 = 54.55?
   A. Rq = 54.55/10 = 5.455 micrometre
   B. Rq = 54.55/22.5 = 2.42 micrometre
   C. Rq = sqrt(54.55/10) = 2.34 micrometre
   D. Rq = sqrt(54.55) = 7.39 micrometre
   Answer: C. Rq is the root mean square: square root of the mean of squares, 2.34 micrometre. (Key point: Rq value)

4. What are Ry and Ra in micro-inches (1 micrometre = 39.37 micro-inch)?
   A. Ry = 0.2 micrometre (3.2 - 3.0); Ra = 88.6 micro-inch
   B. Ry = 6.2 micrometre; Ra = 0.057 micro-inch
   C. Ry = 3.2 micrometre; Ra = 22.5 micro-inch
   D. Ry = 6.2 micrometre (3.2 + 3.0); Ra = 88.6 micro-inch
   Answer: D. Ry = highest peak minus deepest valley = 3.2 - (-3.0) = 6.2; Ra = 2.25 x 39.37 = 88.6 micro-inch. (Key point: Ry and units)

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## Related in other subjects

- [Surface roughness parameters Ra, Rq](https://www.bctengineer.com/question.md?key=semester_4%2Fmanufacturing_and_production_processes%3Asurface-roughness-ra-rq-numerical) — Manufacturing and Production Processes, chapter 3