NMMS – Mathematics (SAT)
Topic: Data Handling
Goal:
Understand how to interpret and manipulate data presented in tables, charts, and graphs.
Apply basic statistical measures (mean, median, mode, range, proportion, percentage) to solve typical SAT‑style questions.
1. Simple Explanation
Data handling is all about reading data that is already collected and analyzing it with simple mathematical tools.
In the SAT you’ll see:
| Data form | Typical representation | What you need to do |
|---|---|---|
| Quantitative | Tables, line graphs, bar charts, scatter plots | Compute averages, percentages, differences |
| Categorical | Pie charts, bar charts | Find the largest/smallest category, proportions |
| Mixed | Combination of above | Combine techniques (e.g., find mean of a group then compare to a whole) |
The key is to transform the numbers into meaningful information, not just crunch them blindly.
2. Key Points
| Concept | Formula / Idea | When to use |
|---|---|---|
| Mean (Average) | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) | Overall tendency of a set of numbers |
| Median | Middle value after sorting | Central value when data is skewed |
| Mode | Most frequent value | When you need the “typical” value |
| Range | (\displaystyle \text{Range} = \max(x_i) - \min(x_i)) | Spread of data |
| Proportion | (\displaystyle \frac{\text{part}}{\text{whole}}) | Parts of a whole |
| Percentage | (\displaystyle %\text{part} = \frac{\text{part}}{\text{whole}} \times 100%) | Convert proportion to percent |
| Percentage Change | (\displaystyle \frac{\text{new} - \text{old}}{\text{old}}\times100%) | Growth/decline |
| Ratio | (\displaystyle \frac{a}{b}) | Compare two quantities |
| Weighted Average | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | When items have different importance |
| Frequency Distribution | Count of occurrences per category | Summarize categorical data |
| Data Interpretation Tips | • Read all captions & units. • Check for “average”, “median”, “mode” terms. • Look for “per 100”, “per 1000” etc. |
Before calculations |
3. Worked Example
Problem
A class of 30 students takes the SAT Math test.
The score distribution is given below:
| Score | Number of Students |
|---|---|
| 200 | 3 |
| 220 | 5 |
| 240 | 7 |
| 260 | 6 |
| 280 | 4 |
| 300 | 5 |
| 320 | 0 |
| 340 | 0 |
| 360 | 0 |
| 380 | 0 |
- What is the mean score?
- What is the median score?
- What percentage of students scored above 260?
- If the average score increased to 260 in the next test, what is the percentage increase from the previous mean?
Solution
1. Mean
[ \bar{x} = \frac{\sum (\text{score} \times \text{frequency})}{\text{total students}} ]
[ \begin{aligned} \text{Numerator} &= 200(3) + 220(5) + 240(7) + 260(6) + 280(4) + 300(5)\ &= 600 + 1100 + 1680 + 1560 + 1120 + 1500 = 7460\ \text{Denominator} &= 30 \end{aligned} ]
[ \bar{x} = \frac{7460}{30} \approx 248.7 ]
Mean ≈ 249.
2. Median
Sort all scores (already sorted). With 30 students, median is the average of the 15th and 16th scores.
Cumulative frequencies:
| Score | Cumulative |
|---|---|
| 200 | 3 |
| 220 | 8 |
| 240 | 15 |
| 260 | 21 |
| 280 | 25 |
| 300 | 30 |
The 15th score is 240 (cumulative reaches 15 at 240).
The 16th score is the first score after 240 → 260.
[ \text{Median} = \frac{240 + 260}{2} = 250 ]
Median = 250.
3. Percentage of students scoring above 260
Students scoring >260:
- 280 (4) + 300 (5) = 9 students.
[ %\text{above 260} = \frac{9}{30}\times100% = 30% ]
30 % scored above 260.
4. Percentage increase to new average (260)
Old mean = 248.7, New mean = 260.
[ \text{Percentage increase} = \frac{260-248.7}{248.7}\times100% = \frac{11.3}{248.7}\times100% \approx 4.54% ]
≈ 4.5 % increase.
4. Quick Revision Summary
- Mean – add all values, divide by count.
- Median – middle value after sorting (average of two middle values if even count).
- **Mode