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Short answer
Mean square displacement (MSD) is the average of the squared distances a particle has moved between two positions a fixed time interval apart. For a tracked trajectory it is computed at each lag, and for simple Brownian motion in two dimensions the curve is a straight line, MSD = 4Dt, from which the diffusion coefficient D is the fitted slope divided by 4.
Definition and scope
MSD is a standard statistic for describing how fast and in what manner a particle moves, and it is widely used for diffusivity in single-particle tracking and microrheology (Didier and Zhang, 2015). The word “displacement” matters: it is the straight-line change in position between two time points, not the length of the path travelled in between. A particle that wanders widely and returns to its starting point has a small displacement and a long path.
The “time” on an MSD plot is the lag, the interval between the two positions being compared, not the clock time of the experiment. Taking the square root of an MSD returns a quantity in distance units: an MSD of 10,000 nm² corresponds to a typical displacement of roughly 100 nm, since √10,000 = 100.
How is MSD calculated from a trajectory?
Start with N recorded positions at a constant frame interval Δt. The time-averaged MSD at lag n (a whole number of frames, n = 1 to N−1) is the mean of the squared position differences over every pair of points n frames apart (Michalet, 2010):
MSD(nΔt) = 1/(N−n) × Σ (ri+n − ri)², summed from i = 1 to N−n, with the lag time given by t = nΔt.
How do I calculate displacement?
Displacement is the vector difference between two positions, ri+n − ri. Its square is the sum of the squared coordinate differences (x and y in a 2D track). Only these squared displacements are averaged; the path between the two points plays no part.
A small arithmetic illustration
Take four positions along one axis, in micrometres: 0, 1, 3, 2 (N = 4). At lag 1 the displacements are 1, 2 and −1, so the squares are 1, 4 and 1 and the MSD is 6/3 = 2 µm². At lag 2 the displacements are 3 and 1, giving (9 + 1)/2 = 5 µm². At lag 3 there is a single displacement of 2, giving 4 µm². Note how the number of averaged pairs falls as the lag grows, which is why the long-lag end of an MSD curve is the least reliable part.
From the curve to a diffusion coefficient
For two-dimensional Brownian motion in an isotropic medium, the expected MSD with localization error is ρ(t) = ε + 4Dt, where ε = 4σ² and σ is the localization uncertainty per coordinate. Fitting a line a + bt to the first points of the curve gives D = b/4 (Michalet, 2010). The positive intercept is not noise to be ignored: “the localization uncertainty introduces a positive offset in the MSD curve”, so a non-zero intercept at zero lag reflects measurement error as well as motion (Michalet, 2010; Bitesize Bio). The factor 4 belongs to this two-dimensional case, and the model leaves out a small correction for finite camera exposure time that the paper sets aside.
Reading the shape of an MSD curve
| Curve shape | What it indicates | Example |
|---|---|---|
| Straight line | Undirected diffusion | Unbound GFP in the cytoplasm |
| Curve rising faster than a line | Directed motion | Dynein moving along microtubules |
| Rises, then levels off (plateau) | Confinement; the plateau reflects the size of the confining region | A centromere in the nucleus |
Source for the table: Bitesize Bio (checked 2026-10-29). Whether a curve is “straight enough” is a statistical question: for Gaussian motions with stationary increments, the distribution of the estimated MSD and how quickly it converges depend on the diffusion exponent (Didier and Zhang, 2015), so the exponent should be checked rather than assumed.
How many MSD points to fit, and how long a trajectory
The best number of fitted points depends only on the reduced localization error x = σ²/(DΔt) and on N (Michalet, 2010):
- When x is much smaller than 1, the best estimate of D comes from the first two points of the MSD curve.
- For larger x, fit more points; the empirical minimum is pmin = E(2 + 2.7·x0.5), with p smaller than N, checked for N = 10, 100 and 1000.
- An unweighted fit over pmin points reaches the same minimum relative error as weighted fits.
- Reaching an accuracy of about 10% requires trajectories of about 1000 data points, and the optimum number of fitted points does not depend on how a very long trajectory is segmented (Ernst and Köhler, 2013).
Where it matters for labs
MSD analysis underlies single-particle tracking and microrheology, so it turns up wherever a lab follows labelled molecules or beads through a sample. Imaging choices set the quantities in the formulas above: the frame interval fixes Δt and localization precision fixes σ. Related reading on this site includes super-resolution microscopy, TIRF microscopy, FRAP and the general linear regression guide, which covers the line fitting step.
Checklist for reporting an MSD analysis
- Frame interval Δt and number of positions N per trajectory.
- Localization precision σ, and how the intercept was treated.
- Range of lags fitted (the number of points p) and the reason for it.
- Whether the curve was checked for a linear, rising or plateau shape before a single D was quoted.
- Whether trajectories were segmented, and the segment length.
Common misunderstandings
- Displacement is not path length. MSD compares two positions, never the route between them.
- The x-axis is lag time, not experiment time.
- MSD is not MSE. Mean squared error is a different statistic that averages squared differences between predicted and observed values; MSD averages squared changes in position over a time lag.
- A zero intercept is not guaranteed. Localization error adds a positive offset.
FAQ
How is MSD calculated?
Average the squared position differences over all pairs of points a given number of frames apart, repeat for each lag, then fit the early part of the curve; in 2D, D is the slope divided by 4.
How do I calculate displacement?
Subtract the earlier position from the later one; the displacement is that vector difference, and it ignores the path taken.
How is MSE calculated?
MSE stands for mean squared error, the average of squared differences between predicted and observed values. It is a different quantity from MSD, although both average squared differences.
How do you calculate RMS speed?
RMS speed is the root-mean-square speed: square each speed, average, and take the square root. It is a speed statistic, not an MSD, though the square root of an MSD does give a distance.
Sources
- Michalet X. Mean square displacement analysis of single-particle trajectories with localization error: Brownian motion in an isotropic medium. Phys Rev E 2010;82(4):041914. pmc.ncbi.nlm.nih.gov/articles/PMC3055791 (checked 2026-10-29).
- Ernst D, Köhler J. Measuring a diffusion coefficient by single-particle tracking: statistical analysis of experimental mean squared displacement curves. Phys Chem Chem Phys 2013;15(3):845-849. epub.uni-bayreuth.de/4771 (abstract, checked 2026-10-29).
- Didier G, Zhang K. The asymptotic distribution of the pathwise mean squared displacement in single particle tracking experiments. arXiv 1507.06567. arxiv.org/abs/1507.06567v2 (abstract, checked 2026-10-29).
- Bitesize Bio. How does it move? Interpreting motion of an object with the mean squared displacement. bitesizebio.com/10951 (checked 2026-10-29).
