The following table shows the ages of the patients admitted in a hospital during a year:
| Age (in years) | 5 - 15 | 15 - 25 | 25 - 35 | 35 - 45 | 45 - 55 | 55 - 65 |
|---|---|---|---|---|---|---|
| Number of patients | 6 | 11 | 21 | 23 | 14 | 5 |
Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.
Finding Mode:
The class with maximum frequency is the modal class. Here, maximum frequency = 23, so modal class = 35 - 45. Using the formula: Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
Where: l = 35, f₁ = 23, f₀ = 21, f₂ = 14, h = 10
Mode = 35 + [(23 - 21) / (46 - 21 - 14)] × 10
Mode = 35 + [2 / 11] × 10 = 35 + 1.82 = 36.82 years
Finding Mean:
| Age | fᵢ | xᵢ (class mark) | fᵢ × xᵢ |
|---|---|---|---|
| 5-15 | 6 | 10 | 60 |
| 15-25 | 11 | 20 | 220 |
| 25-35 | 21 | 30 | 630 |
| 35-45 | 23 | 40 | 920 |
| 45-55 | 14 | 50 | 700 |
| 55-65 | 5 | 60 | 300 |
| Total | 80 | 2830 |
Mean = Σ(fᵢ × xᵢ) / Σfᵢ = 2830 / 80 = 35.375 years
Comparison and Interpretation:
Mode (36.82 years) is greater than Mean (35.375 years). The mode indicates that the maximum number of patients admitted are around 36-37 years of age. The mean represents the average age of all patients admitted, which is about 35.4 years. Since mode > mean, the data shows that most patients are in the middle-age group (35-45 years), which is typical for hospital admissions where chronic health issues become more prevalent in this age bracket.
Explanation
This question from Exercise 13.2 tests students on calculating both mode and mean for grouped data. The modal class is identified as 35-45 with highest frequency 23. The mode formula for grouped data requires the lower limit, frequencies of modal class and adjacent classes, and class width. For mean, students must calculate class marks (mid-points) and use the direct method. The comparison shows mode > mean, indicating the distribution is slightly skewed with most patients concentrated in the 35-45 age group.
Solution Steps
Step 1: Identify modal class (35-45) with maximum frequency 23
Step 2: Apply mode formula with l=35, f₁=23, f₀=21, f₂=14, h=10
Step 3: Calculate class marks for each class interval
Step 4: Calculate fᵢ × xᵢ for each class and find totals
Step 5: Apply mean formula Mean = Σ(fᵢ × xᵢ) / Σfᵢ
Step 6: Compare mode and mean values and interpret the result