Twenty people at a company work day and night shifts. Five volunteer commute times in a daytime chat. Do their answers describe all twenty? A sample is part of a defined population. Size and an attractive chart do not repair the selection method. Use fractions and coordinate axes.
First define whom we study
The population is all twenty workers on a specified date. The sample comprises those whose answers were obtained. The list used for selection is the sampling frame: if it contains only day staff, night staff are already excluded.
Volunteering adds self-selection: people who care most may respond first. More answers from the same incomplete chat do not remove that bias. A simple random sample from a complete list gives each group of the chosen size an equal chance; it does not guarantee every realized sample perfectly resembles the population. Record nonresponse too. OpenStax
Data, frequency and proportion
These are invented teaching data from five answers rounded to minutes: 10, 10, 11, 11, 11. Time is quantitative; day/night shift is categorical. Units and rounding must agree; a missing answer is not automatically zero.
| Time, min | Responses | Proportion |
|---|---|---|
| 10 | 2 | 2/5=0.4 |
| 11 | 3 | 3/5=0.6 |
| Total | 5 | 1 |
Frequency nᵢ counts observations; relative frequency fᵢ is that count divided by n responses:
f_(i) = (n_(i))/(n), (∑)_(i)n_(i) = n, (∑)_(i)f_(i) = 1.
Thus 60% of these five reported 11 minutes. Without a justified selection process, do not extend 60% to all workers.
Same numbers, different impression
From zero: proportional lengths
n=5; 2+3=5
Cropped: misleading comparison
n=5; 2+3=5
Both panels show counts 2 and 3, a difference of one response. With a zero baseline the height ratio is 3/2=1.5. Cropping at 1.5 makes visible heights 0.5 and 1.5, a ratio of 3, exaggerating the impression. For bars whose lengths are compared, a zero baseline preserves proportions; any cropping must be explicit.
A bar chart compares categories or separate values; a histogram groups numeric observations into intervals. For unequal histogram widths, height must be frequency divided by width, so area represents frequency. A line suits time order; a pie chart shows parts of one whole. Do not connect arbitrary categories as though they form a time path. Graph types, histogram density, NIST.
Before concluding, check population, selection, missing data, units, denominator and scale. Even an association between shift and commute time does not itself prove the shift caused the difference: home location or transport may differ too.