Describe before inferring
Plot raw values and look for drift, outliers and distinct groups before fitting a model. A mean can hide systematic changes. Observations taken close together may not be independent.
Opening Lilerno
lilernoEngineeringlilernoEngineeringOpening Lilerno
Chapter 09 · Guided self-study
Intro + external course studyDistinguish a measured effect from noise and quantify its uncertainty.
Build the idea
An original introduction, worked example and practice, followed by a curated reading sequence. The external courses supply deeper teaching and problem sets.
This guide is an orientation, not a complete university module. Practical work needs suitable facilities, safety review and supervision.
Start with these prerequisites
Shared science foundations
Statistics turns variation into evidence, but only after the experiment and assumptions are understood.
Module lens
Describe raw observations first, then state the assumptions behind a summary or uncertainty estimate.
Plot raw values and look for drift, outliers and distinct groups before fitting a model. A mean can hide systematic changes. Observations taken close together may not be independent.
Sample standard deviation measures scatter using n − 1 in the variance denominator. Under appropriate assumptions, standard error of the mean is s/√n. Neither automatically includes instrument bias; confidence intervals require an explicit model.
See the relationship
Change one quantity, watch the graph respond, then explain the result in your own words.
Predict → change → explain
Does a larger sample reduce the spread of individual observations?
Axes: Sample size n → Spread (mm). Bounds may rescale when inputs change.
s = 2 mm; estimated standard error = 1 mm.
Independent observations with a supplied sample standard deviation s. s/√n estimates standard error, not bias or a complete uncertainty budget.
Original Lilerno illustration. Inputs are illustrative; this is not experimental evidence or a design rating.
| Series | Sample size n | Spread (mm) |
|---|---|---|
| Standard error | 2 | 1.414 |
| Standard error | 3.225 | 1.114 |
| Standard error | 4.45 | 0.9481 |
| Standard error | 5.675 | 0.8396 |
| Standard error | 6.9 | 0.7614 |
| Standard error | 8.125 | 0.7016 |
| Standard error | 9.35 | 0.6541 |
| Standard error | 10.57 | 0.615 |
| Standard error | 11.8 | 0.5822 |
| Standard error | 13.03 | 0.5542 |
| Standard error | 14.25 | 0.5298 |
| Standard error | 15.47 | 0.5084 |
| Standard error | 16.7 | 0.4894 |
| Standard error | 17.93 | 0.4724 |
| Standard error | 19.15 | 0.457 |
| Standard error | 20.38 | 0.4431 |
| Standard error | 21.6 | 0.4303 |
| Standard error | 22.82 | 0.4186 |
| Standard error | 24.05 | 0.4078 |
| Standard error | 25.27 | 0.3978 |
| Standard error | 26.5 | 0.3885 |
| Standard error | 27.73 | 0.3798 |
| Standard error | 28.95 | 0.3717 |
| Standard error | 30.18 | 0.3641 |
| Standard error | 31.4 | 0.3569 |
| Standard error | 32.63 | 0.3502 |
| Standard error | 33.85 | 0.3438 |
| Standard error | 35.08 | 0.3377 |
| Standard error | 36.3 | 0.332 |
| Standard error | 37.52 | 0.3265 |
| Standard error | 38.75 | 0.3213 |
| Standard error | 39.98 | 0.3163 |
| Standard error | 41.2 | 0.3116 |
| Standard error | 42.42 | 0.3071 |
| Standard error | 43.65 | 0.3027 |
| Standard error | 44.88 | 0.2986 |
| Standard error | 46.1 | 0.2946 |
| Standard error | 47.33 | 0.2907 |
| Standard error | 48.55 | 0.287 |
| Standard error | 49.77 | 0.2835 |
| Standard error | 51 | 0.2801 |
| Standard error | 52.23 | 0.2768 |
| Standard error | 53.45 | 0.2736 |
| Standard error | 54.67 | 0.2705 |
| Standard error | 55.9 | 0.2675 |
| Standard error | 57.13 | 0.2646 |
| Standard error | 58.35 | 0.2618 |
| Standard error | 59.58 | 0.2591 |
| Standard error | 60.8 | 0.2565 |
| Standard error | 62.02 | 0.2539 |
| Standard error | 63.25 | 0.2515 |
| Standard error | 64.47 | 0.2491 |
| Standard error | 65.7 | 0.2467 |
| Standard error | 66.92 | 0.2445 |
| Standard error | 68.15 | 0.2423 |
| Standard error | 69.38 | 0.2401 |
| Standard error | 70.6 | 0.238 |
| Standard error | 71.83 | 0.236 |
| Standard error | 73.05 | 0.234 |
| Standard error | 74.28 | 0.2321 |
| Standard error | 75.5 | 0.2302 |
| Standard error | 76.72 | 0.2283 |
| Standard error | 77.95 | 0.2265 |
| Standard error | 79.17 | 0.2248 |
| Standard error | 80.4 | 0.223 |
| Standard error | 81.63 | 0.2214 |
| Standard error | 82.85 | 0.2197 |
| Standard error | 84.08 | 0.2181 |
| Standard error | 85.3 | 0.2165 |
| Standard error | 86.53 | 0.215 |
| Standard error | 87.75 | 0.2135 |
| Standard error | 88.97 | 0.212 |
| Standard error | 90.2 | 0.2106 |
| Standard error | 91.42 | 0.2092 |
| Standard error | 92.65 | 0.2078 |
| Standard error | 93.88 | 0.2064 |
| Standard error | 95.1 | 0.2051 |
| Standard error | 96.33 | 0.2038 |
| Standard error | 97.55 | 0.2025 |
| Standard error | 98.78 | 0.2012 |
| Standard error | 100 | 0.2 |
| Observation scatter s | 2 | 2 |
| Observation scatter s | 3.225 | 2 |
| Observation scatter s | 4.45 | 2 |
| Observation scatter s | 5.675 | 2 |
| Observation scatter s | 6.9 | 2 |
| Observation scatter s | 8.125 | 2 |
| Observation scatter s | 9.35 | 2 |
| Observation scatter s | 10.57 | 2 |
| Observation scatter s | 11.8 | 2 |
| Observation scatter s | 13.03 | 2 |
| Observation scatter s | 14.25 | 2 |
| Observation scatter s | 15.47 | 2 |
| Observation scatter s | 16.7 | 2 |
| Observation scatter s | 17.93 | 2 |
| Observation scatter s | 19.15 | 2 |
| Observation scatter s | 20.38 | 2 |
| Observation scatter s | 21.6 | 2 |
| Observation scatter s | 22.82 | 2 |
| Observation scatter s | 24.05 | 2 |
| Observation scatter s | 25.27 | 2 |
| Observation scatter s | 26.5 | 2 |
| Observation scatter s | 27.73 | 2 |
| Observation scatter s | 28.95 | 2 |
| Observation scatter s | 30.18 | 2 |
| Observation scatter s | 31.4 | 2 |
| Observation scatter s | 32.63 | 2 |
| Observation scatter s | 33.85 | 2 |
| Observation scatter s | 35.08 | 2 |
| Observation scatter s | 36.3 | 2 |
| Observation scatter s | 37.52 | 2 |
| Observation scatter s | 38.75 | 2 |
| Observation scatter s | 39.98 | 2 |
| Observation scatter s | 41.2 | 2 |
| Observation scatter s | 42.42 | 2 |
| Observation scatter s | 43.65 | 2 |
| Observation scatter s | 44.88 | 2 |
| Observation scatter s | 46.1 | 2 |
| Observation scatter s | 47.33 | 2 |
| Observation scatter s | 48.55 | 2 |
| Observation scatter s | 49.77 | 2 |
| Observation scatter s | 51 | 2 |
| Observation scatter s | 52.23 | 2 |
| Observation scatter s | 53.45 | 2 |
| Observation scatter s | 54.67 | 2 |
| Observation scatter s | 55.9 | 2 |
| Observation scatter s | 57.13 | 2 |
| Observation scatter s | 58.35 | 2 |
| Observation scatter s | 59.58 | 2 |
| Observation scatter s | 60.8 | 2 |
| Observation scatter s | 62.02 | 2 |
| Observation scatter s | 63.25 | 2 |
| Observation scatter s | 64.47 | 2 |
| Observation scatter s | 65.7 | 2 |
| Observation scatter s | 66.92 | 2 |
| Observation scatter s | 68.15 | 2 |
| Observation scatter s | 69.38 | 2 |
| Observation scatter s | 70.6 | 2 |
| Observation scatter s | 71.83 | 2 |
| Observation scatter s | 73.05 | 2 |
| Observation scatter s | 74.28 | 2 |
| Observation scatter s | 75.5 | 2 |
| Observation scatter s | 76.72 | 2 |
| Observation scatter s | 77.95 | 2 |
| Observation scatter s | 79.17 | 2 |
| Observation scatter s | 80.4 | 2 |
| Observation scatter s | 81.63 | 2 |
| Observation scatter s | 82.85 | 2 |
| Observation scatter s | 84.08 | 2 |
| Observation scatter s | 85.3 | 2 |
| Observation scatter s | 86.53 | 2 |
| Observation scatter s | 87.75 | 2 |
| Observation scatter s | 88.97 | 2 |
| Observation scatter s | 90.2 | 2 |
| Observation scatter s | 91.42 | 2 |
| Observation scatter s | 92.65 | 2 |
| Observation scatter s | 93.88 | 2 |
| Observation scatter s | 95.1 | 2 |
| Observation scatter s | 96.33 | 2 |
| Observation scatter s | 97.55 | 2 |
| Observation scatter s | 98.78 | 2 |
| Observation scatter s | 100 | 2 |
| Chosen n | 4 | 1 |
Save your place when you finish reading.
Original Lilerno example
Find the sample standard deviation of illustrative readings 9, 10 and 11 mm.
Mean = 10 mm.
Squared deviations sum to 1 + 0 + 1 = 2 mm².
Divide by n − 1 = 2, then take the square root: s = 1 mm.
Test the model
Open learning, traceable sources
Work in this order. These links open the publisher’s material; free access does not always permit republication.
NIST / SEMATECH
Start with: Exploratory Data Analysis → Process Modeling → Experimental Design
Inspect assumptions before choosing a distribution, regression or experimental design.
MIT OpenCourseWare · Fall 2014
Start with: Probability and statistics materials
Reproduce one small calculation from raw observations.
Recall, then record
Close the explanation and answer these in your own words. Return tomorrow, then again later in the week.
What does independence mean here?
How do scatter and bias differ?
What does a confidence interval depend on?
Record a small repeatable measurement study. Keep raw data, a plot, descriptive statistics and limitations together.
Self-reported tasks, not an assessment of mastery or university credit. Reading a page does not complete a chapter.
Loading local record…
Open learning, traceable sources
Original Lilerno lessons and diagrams, supported by these references. Free access does not always permit republication. Links open the publisher’s material.
Measurement-process characterization, exploratory analysis, process modelling, quality and experimental design.
Linked reference. NIST-authored federal material and third-party contributions can have different rights; verify item notices before reuse.
Open source ↗Rights / publisher record ↗Programming, numerical solutions, linear algebra, and probability. Source assignments use MATLAB.
CC BY-NC-SA, except marked exceptions; external study only.
Open source ↗Rights / publisher record ↗