Determine mean/median from a frequency table and analyze how changes affect center/spread.
Find the mean or median from a frequency table and update the statistics when one value changes.
核心知识
To find the mean from a frequency table, multiply each value by its frequency, sum those products, and divide by the total count. To track how changes to the data affect center and spread, trace the arithmetic — don't just guess.
深入理解
Frequency tables compress data. Instead of listing 30 individual values, the table gives you each distinct value and how many times it appears. To compute the mean:
For the median, add up frequencies to find
The SAT's harder questions add a twist: what happens if you change the data? Typical scenarios:
- Add a value above the mean → mean increases.
- Remove a value below the median → median may shift up.
- Replace a value with a larger one → mean increases by
n e w − o l d 𝑛
Track the change, not the full recalculation. If the original mean is 80 with
分步讲解
- Multiply each value by its frequency to get the products.
- Sum all products to get the total sum.
- Divide by the total frequency to find the mean.
- For median: count through cumulative frequencies to find the middle position.
- If a value is added, removed, or changed, adjust the sum and count accordingly and recompute.
常见误解
- Dividing the sum of the values column by the number of rows instead of by the total frequency — you must weight by how often each value appears.
- Assuming that adding a value equal to the current mean doesn't affect the mean — it doesn't change the mean's value, but students often overthink this.
- Thinking a change to one value always shifts the median — if the changed value stays in the same half of the data, the median may not move at all.
示例解析
A frequency table shows quiz scores for 20 students:
| Score | Frequency |
|---|---|
| 6 | 3 |
| 7 | 5 |
| 8 | 7 |
| 9 | 4 |
| 10 | 1 |
If one student who scored 6 retakes the quiz and scores 10, what is the new mean score?
选择一个答案查看解析