A population is the full set of units you want to describe. A sample contains the units actually observed. If a school wants to know how long all its students take to get there, the students in a single classroom are only a sample. They can reveal patterns in that classroom. The way they were selected determines what the results can say about the whole school.
A statistic describes the observed sample
A population mean is a parameter. The mean calculated from a sample is a statistic and can serve as an estimate of that parameter. Random selection can reduce selection bias, while collecting more answers from a single convenient class can leave the same bias in place.
How the selection changes what you can conclude
Start with a list of students from every year group. A simple random sample draws names from the full list. A stratified sample draws separately within each year, so that smaller year groups can be represented. If students answer through a voluntary online form, those with unusually long or short journeys may be more inclined to respond. The sample size alone cannot tell you whether this happened.
Read frequencies before calculating spread
Suppose six recorded travel times in minutes are 10, 10, 20, 20, 20 and 40. The absolute frequencies for 10, 20 and 40 minutes are 2, 3 and 1. Their relative frequencies are two sixths, three sixths and one sixth. Half the observations are 20 minutes.
| Time (min) | Count | Relative frequency | Cumulative count |
|---|---|---|---|
| 10 | 2 | 33.3% | 2 |
| 20 | 3 | 50% | 5 |
| 40 | 1 | 16.7% | 6 |
From individual values to intervals
For the six journeys, the cumulative count at 20 minutes is 5. Divide by 6 to find that about 83.3% lasted at most 20 minutes. The count strictly below 20 minutes is only 2. With dozens of distinct durations, use intervals. For example, the interval from 10 up to but excluding 20 contains two journeys, while the next interval from 20 up to but excluding 30 contains three. A histogram groups numerical values into intervals. With unequal interval widths, bar area represents frequency, so the height must account for width.
One dataset gives several measures of dispersion
The range is 30 minutes, from 10 to 40. The total of the observations is 120 minutes and the mean is 20 minutes. Each deviation is its observation minus the mean. The deviations are −10, −10, 0, 0, 0 and 20. They add to zero, so their ordinary average does not measure spread. Taking absolute values first gives a mean absolute deviation of about 6.67 minutes.
If these six observations are the whole population of interest, divide the sum of squared deviations by 6 to obtain a variance of 100 square minutes. Its square root is a standard deviation of 10 minutes. If using the six as a sample to estimate population variance, the usual denominator is 5. That gives a sample variance of 120 square minutes and a sample standard deviation of about 10.95 minutes.
Two classes can have the same mean and range
Compare the observed times with 10, 10, 10, 10, 40 and 40 minutes. Both lists have mean 20 and range 30, but the second has no journey near 20. Its squared deviations sum to 1200, giving a population variance of 200 and standard deviation of about 14.14 minutes. The first list had a standard deviation of 10. This comparison shows why the range, which uses only the extremes, can miss a difference in how the observations are distributed.
