Distributions, transformations and spread
| English | 中文 | Pinyin |
|---|---|---|
| standard deviation/ˈstændəd ˌdiːvɪˈeɪʃn/ | 标准差 | biāo zhǔn chà |
| frequency/ˈfriːkwənsi/ | 频数 | pín shuò |
A decision before an answer
- Two groups can have the same mean but very different consistency. Centre alone does not describe spread.
- Your goal: Compare mean, median, range and standard deviation qualitatively.
Read the relationship
- The mean uses the sum of all observations divided by their count; the median is the middle of the ordered observations. Frequency means a value repeats: if 2 occurs three times and 8 once, the mean is (3·2+8)/4=3.5, not (2+8)/2. The horizontal positions on a display are values and the frequencies determine how many observations they represent.
- Predict the effect of adding or multiplying every data value.
Add 10 to every observation. Standard deviation:
All deviations from the shifted mean are unchanged.
Use the defining rule
- The range is maximum minus minimum; standard deviation describes typical distances from the mean. You need not memorise a full formula to compare simple equally centred distributions: data clustered close to the mean have smaller standard deviation than data moved further away symmetrically. An outlier can alter mean and spread substantially while leaving the median stable.
- Read frequency information without confusing values and counts.
2 occurs three times; 6 occurs once. Mean:
(2+2+2+6)/4=3.
Check the conditions
- Adding c to every observation raises both mean and median by c but leaves range and standard deviation unchanged, because pairwise distances do not change. Multiplying every observation by a positive k multiplies mean, median, range and standard deviation by k. More generally spread scales by |k|, while a negative multiplier reverses ordering.
- Read frequency information without confusing values and counts.
A={4,5,6} and B={0,5,10} both have mean and median 5, but B has greater spread. Adding 7 gives A+7={11,12,13}, mean 12 and the same standard deviation as A. Multiplying A by 2 gives {8,10,12}; its mean is 10 and its range and standard deviation double. Equal range alone does not guarantee equal standard deviation.
The range of 3, 7, 9 is ____.
Maximum minus minimum is 9-3.
Apply the task format
- When comparing plots, inspect the scale and the number of observations before judging apparent width. A graph stretched horizontally can look more variable without representing different values. A claim about equality of standard deviations needs the actual distributions, not merely equal ranges or equal means.
- Read frequency information without confusing values and counts.
Do not average distinct category labels when frequencies differ, or confuse a shifted distribution with a more spread-out one.
Which answer fits this case?
Compare mean, median, range and standard deviation qualitatively
Two distributions with equal means must have equal spread.
Centre and spread describe different features.
Keep the distinctions
- standard deviation 标准差 — A measure of distances of observations from their mean.
- frequency 频数 — The number of observations having a particular value or category.
- Compare mean, median, range and standard deviation qualitatively.
- Predict the effect of adding or multiplying every data value.
- Read frequency information without confusing values and counts.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.