Statistics | Exercise 13.3

Question 7

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

Question diagram 1
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Solution
Understand the Question

To find the median weight from a grouped frequency distribution:

  1. Construct a cumulative frequency (cf\text{cf}) column and find the total frequency N=∑fN = \sum f.
  2. Determine N2\dfrac{N}{2} to identify the median class (the class interval whose cumulative frequency is just greater than or equal to N2\dfrac{N}{2}).
  3. Apply the median formula: Median=l+(N2−cff)×h\text{Median} = l + \left(\dfrac{\dfrac{N}{2} - \text{cf}}{f}\right) \times h where ll is the lower limit of the median class, cf\text{cf} is the cumulative frequency of the preceding class, ff is the frequency of the median class, and hh is the class size.

Step 1 · Construct Cumulative Frequency Table

Create the cumulative frequency distribution table:Diagram 1

Weight (in kg)Number of students (f)Cumulative Frequency (cf)40−452245−503550−5581355−6061960−6562565−7032870−75230\begin{array}{|c|c|c|} \hline \text{Weight (in kg)} & \text{Number of students }(f) & \text{Cumulative Frequency }(\text{cf}) \\ \hline 40 - 45 & 2 & 2 \\ \hline 45 - 50 & 3 & 5 \\ \hline 50 - 55 & 8 & 13 \\ \hline 55 - 60 & 6 & 19 \\ \hline 60 - 65 & 6 & 25 \\ \hline 65 - 70 & 3 & 28 \\ \hline 70 - 75 & 2 & 30 \\ \hline \end{array}

Total number of students:

N=2+3+8+6+6+3+2=30\begin{aligned} N &= 2 + 3 + 8 + 6 + 6 + 3 + 2 \\ &= 30 \end{aligned}

Step 2 · Identify Median Class

Calculate N2\dfrac{N}{2}:

N2=302=15\begin{aligned} \dfrac{N}{2} &= \dfrac{30}{2} \\[0.6em] &= 15 \end{aligned}

The cumulative frequency just greater than 1515 is 1919, which corresponds to the class interval 55−6055 - 60.

Therefore, the median class is 55−6055 - 60.

Step 3 · Calculate Median Weight

From the median class 55−6055 - 60, we have:

  • Lower limit, l=55l = 55
  • Cumulative frequency of preceding class, cf=13\text{cf} = 13
  • Frequency of median class, f=6f = 6
  • Class size, h=45−40=5h = 45 - 40 = 5

Applying the median formula:

Median=l+(N2−cff)×h=55+(15−136)×5=55+(26)×5=55+53=55+1.67=56.67 kg\begin{aligned} \text{Median} &= l + \left(\dfrac{\dfrac{N}{2} - \text{cf}}{f}\right) \times h \\[0.8em] &= 55 + \left(\dfrac{15 - 13}{6}\right) \times 5 \\[0.8em] &= 55 + \left(\dfrac{2}{6}\right) \times 5 \\[0.8em] &= 55 + \dfrac{5}{3} \\[0.8em] &= 55 + 1.67 \\[0.8em] &= 56.67 \text{ kg} \end{aligned}
Answer

56.67 kg56.67\text{ kg}

Common Mistakes
  • Wrong Cumulative Frequency (cf\text{cf}): Taking the cumulative frequency of the median class itself (1919) instead of the class preceding the median class (1313).
  • Confusing ff and cf\text{cf}: Using the cumulative frequency in the denominator instead of the actual frequency f=6f = 6 of the median class.
  • Class Width Calculation: Using incorrect class limits or sizes; ensure h=Upper limit−Lower limit=5h = \text{Upper limit} - \text{Lower limit} = 5.

More questions in Exercise 13.3

Q1

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

Q2

If the median of the distribution given below is 28.5, find the values of xx and yy.

Q3

A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.

Q4

The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table :

Find the median length of the leaves.

(Hint : The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 - 126.5, 126.5 - 135.5, . . . , 171.5 - 180.5.)

Q5

The following table gives the distribution of the life time of 400 neon lamps :

Find the median life time of a lamp.

Q6

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

Q7

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

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