Phân phối, phép biến đổi và độ phân tán
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| mean/miːn/ | The sum of a set of values divided by the number of values. |
| outlier/ˈaʊtlaɪə/ | An observation unusually far from the rest of the data or from the pattern expected under a model. |
| standard deviation/ˈstændəd ˌdiːvɪˈeɪʃn/ | Độ lệch chuẩn |
One huge salary
- Ten employees earn between 40 and 60 thousand dollars; then the company adds the director's salary of 900 thousand. What happens to the mean 平均数 and the median?
- SAT statistics questions are mostly about reasoning, not long calculations: how measures respond to changes in the data.
Centre and spread
- Mean: the total divided by the count; pulled by outliers 离群值. Median: the middle value; resistant to outliers.
- Range: max minus min. Standard deviation 标准差: roughly the typical distance from the mean — larger when values are spread out.
Adding a 900 salary moves the mean from 50 to about 127 but the median only from 50 to 51
What is the median of 7, 3, 9, 1, 5, 11?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Ordered: 1, 3, 5, 7, 9, 11; middle two are 5 and 7, so the median is 6.
The ____ is the measure of centre that is least affected by an outlier.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The median depends only on the middle position.
Effects of changes
- Adding an outlier: the mean moves a lot, the median a little, range and standard deviation grow. The director's salary drags the mean far above every other salary.
- Adding 5 to every value: mean and median rise by 5; range and standard deviation stay the same. Multiplying every value by 2 doubles all four.
Data 2, 4, 4, 6, 9: mean $= 25 \div 5 = 5$, median $= 4$, range $= 7$. Add 10 to each: mean 15, median 14, range still 7.
What changes?
Predict how a change to a data set affects centre and spread.
The data 12, 14, 15, 15, 18 gains one new value, 60. Which statement is true?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The outlier pulls the mean up a lot; the median moves only from 15 to 15.
Adding 4 points to every student's score leaves the standard deviation unchanged.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Shifting every value by the same amount moves the mean by 4 but does not change the spread.
Frequency tables
- In a frequency table, the value and the count are different columns. "Score 3, frequency 5" means five students scored 3.
- To find the mean: multiply each value by its frequency, add, and divide by the total frequency.
A table shows scores: 1 (frequency 2), 2 (frequency 3), 3 (frequency 5). What is the mean score?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 5) \div 10 = 23 \div 10 = 2.3$.
Traps
- Averaging the frequencies instead of the values.
- Spread from shifting: adding a constant moves the data but does not change the standard deviation.
Recap
- Mean is sensitive to outliers; median is resistant. Range and standard deviation measure spread.
- Adding a constant shifts centre only; multiplying scales centre and spread.