Question 10 of 21intermediate🔧 ApplyNumerical4 marks

The following table shows the ages of the patients admitted in a hospital during a year:

Age (in years)5 - 1515 - 2525 - 3535 - 4545 - 5555 - 65
Number of patients6112123145

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Correct Answer

Mode = 36.82 years, Mean = 35.38 years. The mode is greater than the mean, indicating that the maximum number of patients admitted are around 36-37 years of age, while the average age is about 35 years.

Exercise: EXERCISE 13.2 | Q: 1 | (Page 16)
For More Understanding

Explanation

This question from Exercise 13.2 tests students' understanding of calculating mode and mean for grouped data. The modal class is identified as the class with maximum frequency (35-45 with 23 patients). For mode, we use the formula: Mode = L + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h. For mean, we use the step-deviation method with assumed mean. The comparison helps students interpret that mode represents the most common age group of patients, while mean gives the average age.

Solution Steps

  1. Step 1: Identify the modal class. The maximum frequency is 23, corresponding to class 35-45.

  2. Step 2: Calculate Mode using formula: L = 35, f₁ = 23, f₀ = 21, f₂ = 14, h = 10. Mode = 35 + [(23-21)/(2×232 \times 23-21-14)] × 10 = 35 + [2/11] × 10 = 35 + 1.82 = 36.82 years

  3. Step 3: Calculate Mean using step-deviation method. Take assumed mean a = 40, h = 10.

  4. Step 4: Find class marks (xᵢ): 10, 20, 30, 40, 50, 60. Calculate uᵢ = (xᵢ - a)/h: -3, -2, -1, 0, 1, 2

  5. Step 5: Calculate fᵢuᵢ: -18, -22, -21, 0, 14, 10. Sum = -37. Σfᵢ = 80

  6. Step 6: Mean = a + (Σfᵢuᵢ/Σfᵢ) × h = 40 + (-37/80) × 10 = 40 - 4.625 = 35.38 years

  7. Step 7: Compare: Mode (36.82 years) > Mean (35.38 years). This shows maximum patients are around 37 years old, while average age is 35 years.

PreviousNext