Question 21 of 21intermediate🔧 ApplyNumerical3 marks

The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

Weight (in kg)40 - 4545 - 5050 - 5555 - 6060 - 6565 - 7070 - 75
Number of students2386632
Correct Answer

56.67 kg

Exercise: EXERCISE 13.3 | Q: 7 | (Page 30)
For More Understanding

Explanation

To find the median weight, we first construct the cumulative frequency table. The median class is identified using n/2, which is 15. The cumulative frequency just exceeding 15 is 19, corresponding to the class 55-60. We then apply the median formula provided in the textbook context: Median = l + [(n/2 - cf) / f] * h, where l is the lower limit, cf is the cumulative frequency of the preceding class, f is the frequency of the median class, and h is the class size.

Solution Steps

  1. Step 1: Construct the cumulative frequency table.

    Weight (in kg)Frequency (ff)Cumulative Frequency (cf\text{cf})
    40 - 4522
    45 - 5035
    50 - 55813
    55 - 60619
    60 - 65625
    65 - 70328
    70 - 75230
    Here, n=30n = 30.

  2. Step 2: Identify the median class. Calculate n2=302=15\frac{n}{2} = \frac{30}{2} = 15. The cumulative frequency just greater than 15 is 19, so the median class is 55 - 60.

  3. Step 3: Apply the median formula. Lower limit (ll) = 55 Cumulative frequency of preceding class (cf\text{cf}) = 13 Frequency of median class (ff) = 6 Class size (hh) = 5

  4. Step 4: Calculate the median.

    Median=l+(n2−cff)×h\text{Median} = l + \left( \frac{\frac{n}{2} - \text{cf}}{f} \right) \times h Median=55+(15−136)×5\text{Median} = 55 + \left( \frac{15 - 13}{6} \right) \times 5 Median=55+26×5\text{Median} = 55 + \frac{2}{6} \times 5 Median=55+53\text{Median} = 55 + \frac{5}{3} Median=55+1.67\text{Median} = 55 + 1.67 Median=56.67\text{Median} = 56.67

    Thus, the median weight is 56.67 kg.

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