Question

Lower quartile, upper quartile and interquartile range respectively are \(\mathrm{Q}_{1}, \mathrm{Q}_{3}\) and \(\mathrm{Q}\). If the average of \(\mathrm{Q}, \mathrm{Q}_{1}\) and \(\mathrm{Q}_{3}\) is 40 and semi-interquartile range is 6, then find the lower quartile. (1) 24 (2) 36 (3) 48 (4) 60

Short answer

Answer: The value of the lower quartile (Q1) is 48.

Step-by-step solution

Make use of the formula for average of \(Q, Q_1,\) and \(Q_3\)

The given exercise states that the average of \(Q,Q_1\), and \(Q_3\) is \(40\). We can write this as: \((Q + Q_1 + Q_3)/3 = 40\)

Rewrite the interquartile range formula

The semi-interquartile range is given as \(6\). Since semi-interquartile range is half of the interquartile range, we can express this as: \(Q = 2 \cdot 6 = 12\)

Substitute the value of Q

Use the value of \(Q\) calculated in Step 2 in the equation from Step 1: \((12 + Q_1 + Q_3)/3 = 40\) Now, simplify and solve for \(Q_1\): \(12 + Q_1 + Q_3 = 40 \times 3\) \(Q_1 + Q_3 = 40 \times 3 - 12\)

Use the interquartile range formula

We know that the interquartile range is the difference between the upper and lower quartiles, so: \(Q_3 = Q_1 + 12\)

Substitute \(Q_3\) and solve for \(Q_1\)

Substitute the expression for \(Q_3\) from Step 4 back into the equation from Step 3: \(Q_1 + (Q_1 + 12) = 40 \times 3 - 12\) Solve for \(Q_1\): \(2Q_1 + 12 = 108\) \(2Q_1 = 96\) \(Q_1 = 48\) Therefore, the lower quartile \((Q_1)\) is 48, which corresponds to option (3) in the provided exercise options.

Key concepts

Quartiles

Quartiles are a way of dividing a dataset into four equal parts. This helps to understand the distribution of the data, as well as identify any potential outliers.

• The **lower quartile** (also known as the first quartile) is denoted as **\(Q_1\)**, and it marks the 25th percentile of the data. This means 25% of the data falls below \(Q_1\).
• The **upper quartile** (third quartile) is marked as **\(Q_3\)**, representing the 75th percentile, meaning 75% of the data is below \(Q_3\).
• The **median** (or second quartile \(Q_2\)) is the 50th percentile, dividing the data into two equal halves.

Understanding quartiles can help you gain insights into the spread and skewness of the data distribution.
For example, if the quartiles are close to each other, it may indicate that the data points are closely packed, whereas widely spaced quartiles suggest a wider spread.

Interquartile Range

The Interquartile Range (IQR) is the distance between the upper and lower quartiles, denoted as \(Q_3 - Q_1\). It is a measure of statistical dispersion or variability.
The IQR is a useful tool because it is not affected by outliers or extreme values, unlike the range which considers only the maximum and minimum values of the data.Calculating the IQR provides a clear picture of where the majority of data values lie.
To compute the IQR in the provided exercise, the steps are as follows:- Calculate the IQR: \(Q = Q_3 - Q_1\)- In the given solution, the IQR is computed as 12, since the problem specifies a semi-interquartile range of 6, and the full IQR is twice this value.

Semi-Interquartile Range

The Semi-Interquartile Range (SIQR) is simply half of the Interquartile Range. It provides an average spread of the middle 50% of the data around the median.
It is computed as follows:- \( \text{SIQR} = \frac{Q_3 - Q_1}{2} \)The SIQR is particularly valuable for statistically understanding asymmetrical data distributions and minimizing the effects of outliers. It is more stable and reliable than the full range, especially in datasets with a few extreme values.
In the example problem, knowing that the SIQR is 6 allowed for determining that the full interquartile range is 12, which confirms the spread of the quartiles around the median.