Study Notes – Quantitative Aptitude
Topic: Tables (Frequency, Cumulative, Percent, etc.)
Why Tables?
Tables organize data so that you can see patterns, compare values, and compute statistics (mean, median, mode, percentiles) quickly – a skill frequently tested on the USMLE Step 1/2 and many entrance exams.
1. Simple Explanation
| Term | What it is | Why it matters |
|---|---|---|
| Tabular Data | Rows × Columns layout | Makes raw data readable |
| Frequency Table | Shows how many times each value/interval appears | Gives the distribution shape |
| Cumulative Frequency | Adds frequencies up to each class | Helps find medians, quartiles |
| Percentage Table | Frequency ÷ total × 100 | Expresses data relative to whole |
| Cumulative Percentage | Adds percentages sequentially | Useful for percentile calculations |
| Cross‑Tab / Contingency Table | Two categorical variables ↔ | Shows relationships (e.g., gender vs. disease) |
Bottom line: Tables are the bridge between raw data and the statistical answers you need (mean, median, mode, percentiles).
2. Key Points
| # | Concept | How to Apply | Quick Tip |
|---|---|---|---|
| 1 | Choose the right class width (for continuous data) | Use equal-width intervals that cover the entire range | Start with 5–10% of the range; adjust for neatness |
| 2 | Include a “Total” column | Helps verify that the sums match the raw data | Check “Total” before moving on |
| 3 | Cumulative Frequency | CF_i = CF_{i-1} + f_i |
Start CF at 0 for the first class |
| 4 | Percentage | P_i = (f_i / N) × 100 |
Round to one decimal if needed |
| 5 | Cumulative Percentage | CP_i = CP_{i-1} + P_i |
Useful for 50th percentile (median) |
| 6 | Median | Find the class where cumulative frequency ≥ ½ N | Interpolate if needed |
| 7 | Mode | Class with highest frequency | If multiple peaks, it's multimodal |
| 8 | Percentile | Find the class where cumulative % ≥ desired percentile | Interpolate between class limits |
| 9 | Cross‑Tab | Rows = one variable, columns = another, cell = count | Add row/column totals to check consistency |
| 10 | Relative Risk / Odds Ratio | Use cross‑tab counts | Common in medical statistics |
Formatting Matters – Keep headers clear, use bold for totals, and ensure the numbers line up. A tidy table reduces errors.
3. Worked Example
Scenario
A small clinic recorded the ages (in years) of 30 patients who visited for a routine check‑up. The ages are:
23, 34, 29, 45, 37, 45, 23, 29, 34, 34,
45, 51, 23, 29, 30, 37, 34, 45, 38, 23,
50, 45, 34, 29, 23, 29, 34, 45, 30, 37
Goal: Build a frequency table, compute median, mode, and the 70th percentile.
Step‑by‑Step
| Step | Action | Result |
|---|---|---|
| 1. Sort data | 23,23,23,23,29,29,29,30,30,34,34,34,34,34,37,37,37,38,45,45,45,45,45,45,50,51 | Sorted list |
| 2. Decide class width | Range = 51‑23 = 28. 5 classes → width ≈ 6 | Use 5–7 year intervals |
| 3. Define classes | 20‑25, 26‑31, 32‑37, 38‑43, 44‑49, 50‑55 | 6 classes |
| 4. Tally frequencies | 20‑25: 4 (age 23) 26‑31: 2 (ages 29,30) 32‑37: 6 (34,34,34,34,34,37,37,37) 38‑43: 1 (38) 44‑49: 7 (45×7) 50‑55: 2 (50,51) |
|
| 5. Build table |
| Class | Frequency (f) | Cumulative Frequency (CF) | % (P) | Cumulative % (CP) |
|---|---|---|---|---|
| 20–25 | 4 | 4 | 13.3 | 13.3 |
| 26–31 | 2 | 6 | 6.7 | 20.0 |
| 32–37 | 6 | 12 | 20.0 | 40.0 |
| 38–43 | 1 | 13 | 3.3 | 43.3 |
| 44–49 | 7 | 20 | 23.3 | 66.7 |
| 50–55 | 2 | 22 | 6.7 | 73.3 |
Check totals: CF last = 22, but we have 30 patients → we mis‑counted!
Re‑count:
20–25: 4 | 26–31: 3 (two 29s + one 30) | 32–37