An outlier (not outliner, which is a cosmetic term!) is a data item that is outside of the data included in the box and whiskers plot. It's determined by the interquartile range (IQR).
The data is first ordered: 97, 98, 100, 103, 105, 110, 112, 117, 129, 241. The median is the average of 105 and 110=215/2=107.5 (Q2 or quartile 2 value). So the data splits into two groups, the first group below 107.5 and the second group above 107.5:
- 97, 98, 100, 103, 105
- 110, 112, 117, 129, 241
Now we find the median for each of these: 100 and 117 (Q1 and Q3). The box contains the data between 100 and 117, and Q2=107.5 is the separator within the box. The IQR is 117-100=17 and 1.5*17=25.5, (1.5 is a standard multiplier to determine "fences" which "keep out" the outliers), which is going to tell us where the outliers are. Anything above Q3+25.5=142.5 is an outlier, so 241 is an outlier; and anything below Q1-25.5=74.5 is an outlier, so there are no values that low. Values outside the IQR, which are not outliers, are whiskers (97, 98 and 129).
(We can keep everything in this case to whole numbers without affecting the result, so Q2 can be 108; and 75 and 143 are "fences" instead of 74.5 and 142.5.)