Optional: uncertainty, log scales and plotting code
Error bars can show very different things, so each must be named. On a log axis, equal distances mean equal multiplication factors.
These W6 pages (pp. 34-36) are optional extensions that preview later weeks.
Every interval needs a name (p. 34). For n=25, sample mean 70 and sample SD s=10:
| Quantity | Calculation | Meaning |
|---|---|---|
| Sample SD | s = 10 grade points | Spread among individual grades |
| Standard error | SE = s/√n = 10/√25 = 2 | Estimated uncertainty of the mean |
| Approx. 95% interval | 70 ± 2.064×2 = [65.87, 74.13] | t interval for the population mean (24 degrees of freedom) |
An error bar of ±1 SD and a 95% confidence interval answer different questions and have very different widths, so the caption must say which is drawn.
Logarithmic position (p. 36). Base-10 log position: 1→0, 10→1, 100→2, 1,000→3. From 10 to 100 and from 100 to 1,000 the ratio is 10 each time and the log distance is 1 each time: equal log distances represent equal multiplication factors. Zero and negative values have no finite log position, so a log axis cannot show them.
From counts to a chart in Python (p. 35). Every plotted number comes from the table and the denominators stay in the labels:
import matplotlib.pyplot as plt
passed = [60, 18]
total = [100, 20]
rates = [100 * a / b for a, b in zip(passed, total)] # [60.0, 90.0]
fig, ax = plt.subplots()
ax.bar(['Full-time (60/100)', 'Part-time (18/20)'], rates)
ax.set_ylim(0, 100) # bars start at zero
ax.set_ylabel('Observed pass rate (%)')
ax.set_title('Illustrative pass rates by study mode')
First bar: 100×60/100 = 60. Second bar: 100×18/20 = 90.
Source: W6 PDF pp. 1-37. Most numbers in the lecture are illustrative or constructed for teaching; pp. 26-27 reuse the Week 5 synthetic registration counts. The Week 6 lab manual (L6) practises Week 5 EDA reasoning and is worked through section by section in Lab work.
Practice problems
Supplementary study questions on the lecture concepts — not verbatim exam questions. Each answer counts toward spaced review.
1. Conceptual understanding
Definitions and distinctions.
n=25 and s=10. What is the standard error of the mean?
On a base-10 log axis, what do the equal distances 10→100 and 100→1,000 represent?
2. Application
Apply the concept to a case or calculation.
An error bar shows ±1 SD. What does it describe?
n=25, mean 70, SE 2 and t=2.064. What is the approximate 95% interval?
3. Interpretation and limitations
Evaluate assumptions and justify conclusions.
Why can a log axis not show a value of 0?
In the lecture's matplotlib code, what does ax.set_ylim(0, 100) ensure?
Explain it in writing
Self-checked, not marked. Write a few sentences, then compare with the example answer. Your writing is saved as you type.
Show an example answer
SE = 10/√25 = 2 grade points. The SD (10) describes how individual grades vary; the SE describes how uncertain the estimated mean is.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?
Show an example answer
Both steps multiply by 10, and the log10 positions are 1, 2 and 3. Equal log distances mean equal multiplication factors.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?