Распределения, преобразования и разброс
Введено| English | Значение |
|---|---|
| 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/ | стандартное отклонение |
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?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
$(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 5) \div 10 = 23 \div 10 = 2.3$.
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.