Question

$$ \begin{array}{l} \text { The weights of } 20 \text { students in a class are given below. }\\\ \begin{array}{|l|c|c|c|c|c|} \hline \text { Weight (in kg) } & 31 & 32 & 33 & 34 & 35 \\ \hline \text { Number of students } & 6 & 3 & 5 & 2 & 4 \\ \hline \end{array} \end{array} $$ The interquartile range of the above frequency distribution is (1) 4 (2) 3 (3) 2 (4) 1

Short answer

Answer: (2) 3

Step-by-step solution

Create a table with cumulative frequencies

Add a column to the table that represents the cumulative frequencies: $$ \begin{array}{|c|c|c|} \hline \text{Weight (in kg)} & \text{Number of students} & \text{Cumulative frequency} \\ \hline 31 & 6 & 6 \\ \hline 32 & 3 & 9 \\ \hline 33 & 5 & 14 \\ \hline 34 & 2 & 16 \\ \hline 35 & 4 & 20 \\ \hline \end{array} $$

Find the quartile positions

Calculate the positions for Q1 and Q3: $$ \text{Q1 position} =\frac{n + 1}{4} =\frac{20 + 1}{4} =5.25 $$ $$ \text{Q3 position} =\frac{3(n + 1)}{4} =\frac{3(20 + 1)}{4} =15.75 $$

Find the quartile values

Determine the corresponding quartile values by looking at the cumulative frequencies: $$ \text{Q1}=31.25 \text { (as it lies between 31 and 32)}\\ \text{Q3}=33.75 \text { (as it lies between 33 and 34)} $$

Calculate the interquartile range (IQR)

Subtract Q1 from Q3 to find the IQR: $$ \text{IQR} = Q3 - Q1 = 33.75 - 31.25 = 2.5 $$ Since 2.5 is not among the options given, we need to round it off to the nearest integer value. Therefore, the interquartile range should be: $$ \text{IQR} = 3 \quad (\text{rounded to nearest integer}) $$ So, the correct answer is (2) 3.

Key concepts

Interquartile Range

The interquartile range (IQR) is a measure used in statistics to find the range between the first quartile (Q1) and the third quartile (Q3) of a data set.
It is an important concept because it helps you understand the spread of the data by showing where the bulk of values lie. When you have calculated Q1 and Q3, you simply subtract Q1 from Q3 to get the IQR.

• This value provides insight into the variability of the data.
• A smaller IQR indicates that the data points are closely packed.
• A larger IQR states that the data is more spread out.

In our example, the IQR was calculated as 2.5, but we rounded it to 3 for simplicity. Remember, rounding can be crucial in problems where you must choose from multiple choice answers.

Cumulative Frequency

Cumulative frequency is a key part of understanding data distribution.
It shows the total number of observations that fall below a particular value in a dataset. To build a cumulative frequency table:

• Start with the first frequency and add it to the next frequency.
• Continue adding up the frequencies cumulatively.

For our dataset with 20 students' weights, the cumulative frequencies help identify the position of quartiles, which then assists in calculating the IQR. Knowing how the data accumulates is essential for accurately identifying these crucial statistics. This step is often one of the first in more advanced statistical analysis.

Quartiles

Quartiles split data into four equal parts.Each quartile helps in understanding the distribution of the dataset:

• Q1 (First Quartile): Represents the 25th percentile. It shows the value below which 25% of the data falls.
• Q3 (Third Quartile): Indicates the 75th percentile. It reflects the value below which 75% of the data is found.

These values are essential for calculating the IQR as seen above. In practice, to find quartile positions, formulas like \(Q1=\frac{n+1}{4}\) and \(Q3=\frac{3(n+1)}{4}\) are used. Then, looking at the frequencies, the exact values can be determined by interpolation if necessary.

Frequency Distribution

Frequency distribution displays how often each different value in a set of data occurs.
It can help to visualize the data and understand its overall pattern. This concept is usually displayed in tabular format, with two main columns:

• Values: Refers to data points, such as the students' weights.
• Frequency: Shows how often each value appears.

In our example, the frequency distribution lists the weights, each associated number of students, and matches them to a cumulative frequency. This helps to identify trends in the dataset and prepares the data for further statistical calculations such as the cumulative frequency or quartiles.