Spread
Introduced| English | Meaning |
|---|---|
| quartiles/ˈkwɔːtaɪlz/ | One of the three boundary values that divide ordered data into four parts of approximately equal size. |
| range/reɪndʒ/ | range |
| interquartile range/ˌɪntəˈkwɔːtaɪl reɪndʒ/ | The upper quartile minus the lower quartile, measuring the spread of the middle half of ordered data. |
| box plot/bɒks plɒt/ | A display of a distribution using its median and quartiles, with whiskers drawn according to a stated rule. |
| standard deviation/ˈstændəd ˌdiːvɪˈeɪʃn/ | A measure of spread equal to the square root of the variance. |
One average hides two score patterns
- The lists 58, 60, 62 and 20, 60, 100 both have mean 60. Their different spread affects what that shared mean tells you.
- A spread summary adds information without explaining the cause of a difference. Keep measurement units and population scope the same when comparing.
Two classes both average 60. Why report the spread as well?
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58, 60, 62 and 20, 60, 100 share a mean and share nothing else.
State how the quartiles are calculated
- The range 全距 is maximum minus minimum. The quartiles 四分位数 mark positions in ordered data; interquartile range 四分位距 is $Q_3-Q_1$.
- Sheet 2.7 uses medians of ordered halves, excluding the overall median for an odd count. Software can use other conventions; specify yours before comparing results.
Why is the interquartile range often preferred to the range?
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The range is defined entirely by the two most extreme values, which are the least typical.
A box plot uses a defined whisker rule
- A box plot 箱线图 shows quartiles and median. Under this sheet's rule, whiskers extend to minimum and maximum.
- Other plots use outlier fences and separately plotted values. A five-number summary does not reveal every observation, count or explanation for an extreme.
Two data sets, same median
The box shows the middle half; the whiskers show how far the extremes reach.
Distinguish population and sample spread
- Population standard deviation 标准差 is $\sigma=\sqrt{\sum(x-\mu)^2/n}$. For the stated sample-estimate convention use $s=\sqrt{\sum(x-\bar x)^2/(n-1)}$.
- Squared deviations make spread nonnegative; taking the square root restores the original units. Variance has squared units and is not the same quantity.
Treat 58, 60, 62 as the complete population. Find its population standard deviation to 2 decimal places.
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Use the population divisor 3: √((4 + 0 + 4)/3) = √(8/3) ≈ 1.63, in the original units.
Treat the supplied three-value lists as complete populations. For 58, 60, 62, $\sigma_A=\sqrt{(4+0+4)/3}\approx1.63$. For 20, 60, 100, $\sigma_B=\sqrt{(1600+0+1600)/3}\approx32.7$. The second population has twenty times the standard deviation under the same convention.
Two classes both average 60, with standard deviations 1.6 and 33. Write one sentence reporting this properly.
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Example: Both groups have mean score 60, but the group with standard deviation 33 is more spread around the mean than the group with standard deviation 1.6.
Choose an informative pair of summaries. Report a centre and a spread where they answer the comparison, along with units, n and the calculation convention. Sheet 2.7 examines how an extreme changes range without changing its supplied middle-half span.
The standard deviation has the same units as the original data.
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The square root undoes the squaring. The variance, its square, does not share the units.
Do not discard an extreme value merely because it changes a summary. Check its origin and report any justified correction. A sample standard deviation and population standard deviation can differ even for the same printed list.