Spread
| English | Chinese | Pinyin |
|---|---|---|
| range | 全距 | quán jù |
| quartiles | 四分位数 | sì fēn wèi shù |
| interquartile range | 四分位距 | sì fēn wèi jù |
| box plot | 箱线图 | xiāng xiàn tú |
| standard deviation | 标准差 | biāo zhǔn chà |
Two classes, same mean, different worlds
- Class A scores 58, 60, 62. Class B scores 20, 60, 100. Both average 60.
- The average is the same and the situations are not, which is why spread matters as much as centre.
- Every statistics report in this module is marked on reporting both.
Two classes both average 60. Why report the spread as well?
58, 60, 62 and 20, 60, 100 share a mean and share nothing else.
Range and quartiles
- The range 全距 is largest minus smallest. One number, and one outlier destroys it.
- The quartiles 四分位数 cut the ordered data into four. The interquartile range 四分位距, written IQR, is the middle 50%.
- The IQR ignores the extremes deliberately, which makes it the honest spread for skewed data.
Why is the interquartile range often preferred to the range?
The range is defined entirely by the two most extreme values, which are the least typical.
The box plot
- A box plot 箱线图 draws minimum, lower quartile, median, upper quartile and maximum.
- The box is the IQR; the line inside it is the median; the whiskers reach the extremes.
- Two box plots side by side compare centre and spread at once, which is why they appear in every project.
Two data sets, same median
The box shows the middle half; the whiskers show how far the extremes reach.
Standard deviation
- The standard deviation 标准差 is the typical distance of a value from the mean.
- Squaring makes every distance positive and weights the far ones more; the square root brings it back to the original units.
- A larger standard deviation is a wider spread, in the same units as the data.
For the values 58, 60, 62 the deviations are −2, 0, 2. Give the standard deviation to 2 decimal places.
√((4 + 0 + 4)/3) = √2.67 = 1.63, in marks — the same unit as the data.
The two classes.
Class A: 58, 60, 62. Deviations $-2, 0, 2$, so $\sigma = \sqrt{\frac{4+0+4}{3}} = 1.63$.
Class B: 20, 60, 100. Deviations $-40, 0, 40$, so $\sigma = \sqrt{\frac{1600+0+1600}{3}} = 32.7$.
Same mean, spreads differing by a factor of twenty. Reporting the mean alone would hide the entire difference between the two classes.
Two classes both average 60, with standard deviations 1.6 and 33. Write one sentence reporting this properly.
Example: "Both classes averaged 60, but one was tightly grouped (sd 1.6) while the other ranged widely (sd 33)."
Report a centre and a spread together, always. "Mean 60, standard deviation 1.6" and "mean 60, standard deviation 33" describe completely different situations, and a report giving only the first number has told the reader almost nothing.
The standard deviation has the same units as the original data.
The square root undoes the squaring. The variance, its square, does not share the units.
Standard deviation is in the same units as the data; the variance, its square, is not. An answer in "marks squared" is a variance labelled as a standard deviation, and it is a lost mark that looks like a rounding error.