Percentages and uncertainty in surveys

A percentage is simply a fraction expressed out of 100: $54{,}2\% = 54{,}2/100 = 0{,}542$. To go from a percentage to the absolute value, multiply by the total — out of 24 students, $54{,}2\%$ is $24\cdot 0{,}542 \approx 13$ students. Conversely, to go from a count to a percentage one uses

$\text{percentage} = \frac{\text{count}}{\text{total}}\cdot 100.$

An important, and often overlooked, result is that a percentage calculated on a finite sample of $N$ individuals is necessarily affected by uncertainty: the smaller $N$ is, the more the estimate can vary if the survey were repeated on a different group. Elementary statistics provides a formula to quantify this uncertainty.

Principle — Uncertainty on a proportion

$\ev{\text{err}(p) = \sqrt{\dfrac{p(1-p)}{N}}}$ where $p$ is the proportion (a number between 0 and 1, not a percentage) and $N$ is the sample size.

Key formula

The larger the sample $N$, the smaller the error on the percentage: the uncertainty decreases as $1/\sqrt{N}$.

The $1/\sqrt{N}$ dependence explains why halving the uncertainty of a survey requires quadrupling the number of respondents. It is also why serious surveys involve thousands of people: only then does the error drop to a few percentage points.

Consider a class of 24 students asked: “Do you prefer dogs or cats?” 13 answer dogs and 11 answer cats. The proportion in favour of dogs is $p = 13/24 = 0{,}542$, i.e. $54{,}2\%$, with uncertainty

$\text{err}(p) = \sqrt{\frac{0{,}542\cdot 0{,}458}{24}} \approx \sqrt{\frac{0{,}248}{24}} \approx 0{,}102,$

i.e. $10{,}2\%$. Dogs win with $(54{,}2 \pm 10{,}2)\%$, an interval $[\,44{,}1\%\;;\;64{,}3\%\,]$: for such a small sample the uncertainty is enormous, and with only 24 students it is not statistically clear whether the majority really prefers dogs.

Sanity check with small samples

A few checks on the formula:

• 3 reds out of 5 total: $p=0{,}6$, err $\approx 22\%$.
• 2 greens out of 3 total: $p=0{,}67$, err $\approx 27\%$.
• 8 in favour out of 12 total: $p=0{,}67$, err $\approx 14\%$.

Below 20 samples the error on a proportion is of the order of $20\%$ or more: a practical rule worth remembering when reading election polls in the newspapers.

Collegamenti

Argomenti: Grandezze fisiche e misura Concetti: Cifre significative e incertezza Competenze: Propagazione degli errori

Esercizi collegati: Esercizio svolto — Area di un rettangolo misurato · Esercizio svolto — Cane o gatto: il sondaggio della classe · Problema — Ranking di incertezze relative