Errors — absolute, relative, percentage; systematic vs random

Systematic vs Random — the two flavours

Quantifying error — absolute, relative, percentage

Suppose you measure a quantity $n$ times and get $a_1, a_2, \dots, a_n$ .

Step 1 — Best estimate is the mean (WHY?)

$a_{\text{mean}} = \frac{1}{n}\sum_{i=1}^{n} a_i$

Step 2 — Absolute error of each reading

Step 3 — Mean absolute error (the reported uncertainty)

Step 4 — Relative (fractional) error

Worked examples

Worked example Example 1 — full pipeline

Measured period of a pendulum (s): $2.63,\;2.56,\;2.42,\;2.71,\;2.80$ .

Mean: $\dfrac{2.63+2.56+2.42+2.71+2.80}{5} = \dfrac{13.12}{5} = 2.624$ s. Why this step? Best estimate = mean (kills random scatter best).

Absolute errors: $|2.624-2.63|=0.006$ , $0.064$ , $0.204$ , $0.086$ , $0.176$ . Why? Distance of each reading from the best guess.

Mean absolute error: $\dfrac{0.006+0.064+0.204+0.086+0.176}{5}=\dfrac{0.536}{5}=0.107\approx 0.11$ s. Why? Typical deviation = our uncertainty.

Result: $T = 2.62 \pm 0.11$ s. (round error to match decimals)

Relative: $\dfrac{0.11}{2.62}=0.042$ . Percentage: $4.2\%$ . Why? So we can compare this with, say, a $4.2\%$ error in a mass measurement.

Forecast-then-Verify

Difference between systematic and random error in one line

Systematic = consistent bias in one direction (not removed by averaging); random = unpredictable scatter both ways (reduced by averaging).

Why does averaging reduce random but NOT systematic error?

Random errors are

\pm

and partly cancel when summed; systematic error is the same nudge every time, so it survives averaging.

Formula for absolute error of a single reading

\Delta a_i = |a_{\text{mean}} - a_i|

Formula for mean absolute error

\Delta a_{\text{mean}} = \frac{1}{n}\sum |\,a_{\text{mean}}-a_i\,|

Relative error formula and is it unitless?

\delta_r = \Delta a_{\text{mean}}/a_{\text{mean}}

; yes, dimensionless.

Percentage error formula

\delta_\% = (\Delta a_{\text{mean}}/a_{\text{mean}})\times 100\%

Accuracy vs precision

Accuracy = closeness to true value; precision = closeness of repeated readings to each other.

How do you fix a known systematic zero error?

Subtract (or add) the constant offset from every reading.

Why report uncertainty to only 1–2 significant figures?

The error is itself uncertain; extra digits are meaningless precision.

Best single estimate of the true value from n readings

The arithmetic mean.

Recall Feynman: explain to a 12-year-old

Imagine measuring your friend's height with a wonky ruler. If the ruler is missing its first centimetre, everyone you measure comes out 1 cm too short — that's a systematic mistake, the same goof every time, and measuring more friends won't fix it. But if your hand just shakes a little, sometimes you read a touch too high, sometimes too low — that's random. Measure many times and the too-highs and too-lows cancel out, so the average is good. "Error" is just how-much-you-might-be-off; "relative error" is that miss compared to the actual size — being 1 cm off matters a lot for an ant but not for an elephant.

Connections

Significant Figures and Rounding — how many digits your error allows you to keep.
Combination of Errors — propagating these errors through $+,-,\times,\div,$ powers.
Least Count and Vernier Calipers — instrumental source of systematic error.
Mean and Standard Deviation — statistical cousin of mean absolute error.
Dimensional Analysis — checking results once errors are stated.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, koi bhi measurement perfectly exact nahi hota — har reading ek "andaaza with confidence range" hota hai. Jab tum kehte ho rod 5.2 cm ka hai, matlab "roughly 5.2, thoda upar-neeche ho sakta hai." Wahi thoda upar-neeche wala part hi error hai. Physics tumse perfect hone ko nahi bolti, bas honestly batao ki kitne unsure ho.

Do flavour hote hain error ke. Systematic error matlab har baar same direction mein galti — jaise scale ka zero hi shift ho, to har reading 0.3 cm zyada aati hai. Ise average karne se fayda nahi, bias bach jaata hai; fix karne ke liye offset minus karo. Random error matlab kabhi thoda zyada kabhi thoda kam — haath ka kaampna, last digit ka andaaza. Isko bahut saari readings le kar average karne se kam kar sakte ho, kyunki plus aur minus ek dusre ko cancel kar dete hain. Yaad rakho: accuracy alag, precision alag — readings tightly cluster ho sakti hain (precise) par galat jagah (inaccurate).

Number mein error batane ke liye 3 step: pehle absolute error $\Delta a_i = |a_{mean}-a_i|$ (units ke saath, har reading kitni door hai), phir saari ka average — wahi tumhari uncertainty. Phir relative error $=\Delta a_{mean}/a_{mean}$ (divide karke, isliye unitless — taaki chhoti aur badi cheez ka error compare ho sake). Aur usko ×100 karo to percentage error mil gaya. Mnemonic: A-R-P — Absolute, Relative, Percent.

Exam tip: error ko sirf 1-2 significant figures tak likho, aur value ko bhi usi decimal place tak round karo. Jaise $T = 2.62 \pm 0.11$ s likhna sahi hai, $2.624 \pm 0.107$ nahi. Forecast-then-verify karo — pehle guess karo error 1% hai ya 10%, phir calculate karke check karo; isse intuition strong hoti hai.

Errors — absolute, relative, percentage; systematic vs random

WHY do errors exist at all?

Systematic vs Random — the two flavours

Quantifying error — absolute, relative, percentage

Step 1 — Best estimate is the mean (WHY?)

Step 2 — Absolute error of each reading

Step 3 — Mean absolute error (the reported uncertainty)

Step 4 — Relative (fractional) error

Step 5 — Percentage error

Worked examples

Forecast-then-Verify

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Measurement, Vectors & Kinematics

Connections