Phân tích dữ liệu: độ phân tán, trung bình có trọng số và xác suất điều kiện
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| weighted average/ˈweɪtɪd ˈævrɪdʒ/ | trung bình có trọng số |
| overlap/ˌəʊvəˈlæp/ | The items that belong to two groups at the same time. |
The average that needs weights
- Class A (20 students) averaged 70; class B (30 students) averaged 80. What is the average of all 50 students? Not 75 — the larger class counts more.
- GRE data analysis tests averages, spread, distributions and probability with the right reference group.
Weighted averages 加权平均
- Total ÷ count: $\dfrac{20 \times 70 + 30 \times 80}{50} = \dfrac{1400 + 2400}{50} = 76$.
- The combined average lies between the two, closer to the larger group.
10 students averaged 60 and 30 students averaged 80. What is the overall average?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$(600 + 2400) \div 40 = 75$.
A combined average lies closer to the average of the ____ group.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
It carries more weight.
Spread and distribution
- Range, interquartile range and standard deviation measure spread. Adding a constant to every value leaves them unchanged; multiplying scales them.
- In a normal distribution, about 68% lies within 1 SD and 95% within 2 SD of the mean.
Data A: 10, 10, 10, 10. Data B: 0, 10, 10, 20. Same mean (10); B has the larger standard deviation because its values are further from the mean.
Adding 10 to every value in a data set leaves its standard deviation unchanged.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Shifting all values moves the mean by 10 but does not change how spread out they are.
Probability with overlaps 重叠部分
- $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. Of 100 students, 60 study French, 50 Spanish, 20 both: French or Spanish $= 90$.
- Conditional: "Of those who study Spanish, what fraction also study French?" $= \tfrac{20}{50}$ — the reference total is 50.
Which total?
Choose the correct reference total for a probability.
Of 200 people, 120 like tea, 90 like coffee and 40 like both. How many like tea or coffee?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$120 + 90 - 40 = 170$.
Same group: of those who like coffee, what fraction also like tea?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The reference total is the 90 coffee drinkers.
Common traps
- Averaging averages without weights.
- Using the grand total instead of the conditioning group.
Recap
- Weighted average = total ÷ count; it sits nearer the larger group.
- Subtract the overlap for "or"; divide by the conditioning group for "given".
- overlap: The items that belong to two groups at the same time.