Data analysis: spread, weighted averages and conditional probability
| English | 中文 | Pinyin |
|---|---|---|
| weighted average/ˈweɪtɪd ˈævrɪdʒ/ | 加权平均 | jiā quán píng jūn |
| mutually exclusive/ˈmjuːtʃuːəli eksˈkluːsɪv/ | 互斥的 | hù chì de |
A decision before an answer
- Three marked red counters out of four marked counters is 3/4, even when there are ten counters in the bag.
- Your goal: Distinguish mean, median, spread and a weighted average.
Read the relationship
- Mean is sum divided by count; median is the central ordered value or the mean of the middle two. Range is maximum minus minimum. An extreme value can alter mean and range without changing the median. Standard deviation measures spread about the mean; adding a constant changes the centre but not spread, while multiplying all values by c multiplies standard deviation by |c|.
- Interpret distributions and overlapping or conditional event counts.
Groups of 5 at mean 4 and 15 at mean 8 have combined mean:
Total 20+120=140 across 20 observations gives 7.
Use the defining rule
- Combine groups with their counts, not an unweighted mean of group means unless sizes are equal. A group of 10 with mean 6 contributes total 60; a group of 30 with mean 10 contributes total 300. The combined mean is (60+300)/40=9. A percentage of a subgroup uses that subgroup’s denominator.
- Use the correct reference total in table-based inference.
Add 10 to every observation. Standard deviation:
All deviations from the shifted mean are unchanged.
Check the conditions
- For equally likely discrete outcomes, probability is favourable count over total count. Conditional probability restricts the denominator to the given condition. Without replacement changes later counts. Overlapping events need subtraction of the overlap when counting a union. Independence must be given or justified; it is not the same as mutually exclusive.
- Use the correct reference total in table-based inference.
A bag holds four red counters, three marked, and six blue counters, one marked. There are four marked counters in total. P(red)=4/10, P(marked)=4/10, P(red given marked)=3/4. P(two red without replacement)=(4/10)(3/9)=2/15, not (4/10)². For data 1,2,3,4,20, mean=30/5=6 and median=3. Adding 5 to each shifts mean and median by 5 but leaves their pairwise deviations and standard deviation unchanged.
Three of four marked counters are red. P(red given marked)=____.
Restrict the denominator to marked counters.
Apply the task format
- Read displays for their axes, units and sample description before making inferences. A correlation does not isolate a cause. A convenience sample may not represent the target population. Basic counting can use ordered multiplication when roles differ, or divide duplicated arrangements when order does not matter. Keep the events and outcomes explicitly defined.
- Use the correct reference total in table-based inference.
A condition changes the reference population. Group means need weights, and no-replacement draws need changing denominators.
Which answer fits this case?
Distinguish mean, median, spread and a weighted average
Independent events are necessarily mutually exclusive.
Independent events can occur together; disjoint nonzero-probability events are not independent.
Keep the distinctions
- weighted average 加权平均 — An average that accounts for each group’s contribution or size.
- mutually exclusive 互斥的 — Events that cannot occur together in the same outcome.
- Distinguish mean, median, spread and a weighted average.
- Interpret distributions and overlapping or conditional event counts.
- Use the correct reference total in table-based inference.
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.