分布、変換とばらつき
導入済み| English | 意味 |
|---|---|
| standard deviation/ˈstændəd ˌdiːvɪˈeɪʃn/ | 標準偏差 |
| frequency/ˈfriːkwənsi/ | 頻度 |
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:
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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:
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(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 ____.
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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.
このケースに当てはまる答えはどれか？
平均値、中央値、範囲および標準偏差を定性的に比較する
Two distributions with equal means must have equal spread.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.