Tags: moving average, smoothing, velocity from position data, normal distribution, Gaussian, z-score, confidence interval, population standard deviation, sample size, margin of error, ENGR 216, PHYS 216, Texas A&M
This section covers how to smooth noisy experimental data with moving averages, how the normal (Gaussian) distribution relates measured populations to probabilities, and how to construct confidence intervals and determine sample sizes. The core skill is moving between z-scores and probabilities using a z-table, then applying that to real measurement scenarios.
Moving average
A smoothing technique that replaces each data point with the average of itself and its neighbours. A three-point moving average at index i uses points (i−1), i, and (i+1).
Velocity from smoothed data
After smoothing position data, velocity at a point is estimated as the slope between adjacent smoothed values: v = Δx_smooth / Δt.
Normal distribution (Gaussian distribution)
A symmetric, bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ). Approximately 68% of values fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.
Z-score
The number of standard deviations a value lies from the mean: z = (x − μ) / σ. Converts any normal distribution to the standard normal (μ = 0, σ = 1).
Confidence interval
A range around a sample mean within which the true population mean is expected to fall, at a stated confidence level. For a known population standard deviation: x̄ ± z*(σ/√N).
Margin of error (E)
The half-width of a confidence interval: E = z*(σ/√N). Determines how precise your estimate of the mean is.
A three-point moving average at index i is:
x̃ᵢ = (xᵢ₋₁ + xᵢ + xᵢ₊₁) / 3
This can only be computed for interior points (not the first or last data point). After smoothing, velocity is the central difference of the smoothed values divided by the time step.
Worked example (Problem 6):
Time (s): 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 Position (m): 3, 5, 7, 13, 15, 19, 21, 24, 27, 29
Step 1: Compute three-point moving averages for the interior points.
x̃(0.2) = (3 + 5 + 7)/3 = 5.0
x̃(0.3) = (5 + 7 + 13)/3 = 8.333
x̃(0.4) = (7 + 13 + 15)/3 = 11.667
x̃(0.5) = (13 + 15 + 19)/3 = 15.667
x̃(0.6) = (15 + 19 + 21)/3 = 18.333
x̃(0.7) = (19 + 21 + 24)/3 = 21.333
x̃(0.8) = (21 + 24 + 27)/3 = 24.0
x̃(0.9) = (24 + 27 + 29)/3 = 26.667
Step 2: Velocity at t = 0.5 s using central difference of smoothed values.
v(0.5) = [x̃(0.6) − x̃(0.4)] / (0.6 − 0.4)
v(0.5) = (18.333 − 11.667) / 0.2 = 6.667 / 0.2 = 33.3 m/s
The standard normal table gives the probability that a value falls below a given z-score (cumulative probability from −∞ to z). To find the probability between two values:
P(a < X < b) = P(Z < z_b) − P(Z < z_a)
where z_a = (a − μ)/σ and z_b = (b − μ)/σ.
Worked example (Problem 7):
75.4% of watermelons have radii between 15.0 cm and 22.0 cm. Population mean μ = 18.5 cm. Find σ.
The distribution is symmetric about 18.5 cm. Note that 15.0 is 3.5 cm below the mean and 22.0 is 3.5 cm above the mean, so the interval is symmetric.
For a symmetric interval: P(−z < Z < z) = 0.754, which means each tail has probability (1 − 0.754)/2 = 0.123.
So P(Z < z) = 1 − 0.123 = 0.877. From the z-table, z ≈ 1.16.
Since z = (22.0 − 18.5)/σ = 3.5/σ:
1.16 = 3.5/σ
σ = 3.5/1.16 ≈ 3.02 cm
Worked example (Problem 8):
Air filter catches 90% of particles. Mean diameter μ = 0.5 microns, σ = 0.2 microns. What is the largest diameter that passes through?
The filter catches the largest 90% of particles (larger particles are easier to catch). The 10% that pass through are the smallest. So we need the 10th percentile of the distribution.
From z-table: P(Z < z) = 0.10 gives z ≈ −1.28.
Largest passing diameter = μ + z·σ = 0.5 + (−1.28)(0.2) = 0.5 − 0.256 = 0.244 microns
(Note: the physical interpretation here depends on the exam's framing. If the filter catches the smallest 90%, then you would want the 90th percentile instead, giving 0.5 + 1.28 × 0.2 = 0.756 microns. Read the problem statement carefully.)
A confidence interval for the population mean, when σ is known:
CI = x̄ ± z* × (σ / √N)
Common z* values:
90% confidence: z* = 1.645
95% confidence: z* = 1.960
99% confidence: z* = 2.576
Problem 9 setup (Zinc concentration):
For a 99% CI: Upper bound = x̄ + 2.576 × (σ/√N), Lower bound = x̄ − 2.576 × (σ/√N).
Plug in the given average concentration, known σ, and sample size N.
To achieve a desired margin of error E at a given confidence level:
N = (z* × σ / E)²
Always round up to the next whole number, since you cannot take a fractional measurement.
Problem 10 setup:
Given σ and a desired margin of error E at a stated confidence level, substitute into the formula above. For example, if σ = 5, E = 1, and 95% confidence:
N = (1.960 × 5 / 1)² = (9.8)² = 96.04
Round up: N = 97
Three-point moving average: x̃ᵢ = (xᵢ₋₁ + xᵢ + xᵢ₊₁) / 3
Central difference velocity: v(tᵢ) = [x̃(tᵢ₊₁) − x̃(tᵢ₋₁)] / (tᵢ₊₁ − tᵢ₋₁)
Z-score: z = (x − μ) / σ
Confidence interval (known σ): CI = x̄ ± z* × (σ / √N)
Required sample size: N = (z* × σ / E)²
⚠️ Moving averages can only be computed for interior points. The first and last data points in a set have no three-point average.
⚠️ When a problem says "75.4% fall between a and b," check whether the interval is symmetric about the mean. If it is, use the symmetric z-table shortcut. If not, you need to work with two separate z-values.
⚠️ Always round sample size N up. N = 96.04 means you need 97 measurements, not 96.
⚠️ Know the common z* values by heart: 1.645 (90%), 1.960 (95%), 2.576 (99%). These appear in nearly every confidence interval problem.
⚠️ Read filter/threshold problems carefully. "Catches 90% of particles" could mean it catches the largest 90% or the smallest 90%, depending on context. The direction changes which z-score you use.
Q: You have 25 measurements with a sample mean of 48.3 and a known population standard deviation of 6.0. What is the 95% confidence interval?
A: CI = 48.3 ± 1.960 × (6.0/√25) = 48.3 ± 1.960 × 1.2 = 48.3 ± 2.352. The interval is (45.95, 50.65).
Q: A five-point moving average at index i uses which data points?
A: Points at indices i−2, i−1, i, i+1, and i+2. Their sum is divided by 5.
Q: If 95% of a population falls between 20 and 40, and the mean is 30, what is the standard deviation?
A: The interval is symmetric (both bounds are 10 units from the mean). For 95%, z* = 1.96. So 1.96 = 10/σ, giving σ = 10/1.96 ≈ 5.10.
Q: How many samples do you need to estimate a mean within ±0.5 units at 99% confidence, if σ = 3?
A: N = (2.576 × 3 / 0.5)² = (15.456)² = 238.9, so N = 239.
three-point moving average, smoothing, central difference, numerical differentiation, velocity from position, Gaussian distribution, normal distribution, bell curve, z-score, z-table, standard normal, cumulative probability, confidence interval, confidence level, margin of error, sample size determination, population standard deviation, known sigma, ENGR 216, PHYS 216, Texas A&M, experimental physics lab