Estimated Percentile/Quartile (Grouped)

Statement

The estimated percentile or quartile for grouped data is a method used to find approximate values that divide a dataset into equal parts. Since grouped data is organised into intervals, we cannot identify exact values of quartiles or percentiles. Instead, we interpolate within the relevant class to estimate their values using the formula:

\[ Q_k \; \text{or} \; P_p = L + \left(\frac{kN - \text{CF}}{f}\right)\times w \]

Where:

• \(L\) = lower boundary of the required class
• \(N\) = total frequency
• \(\text{CF}\) = cumulative frequency before the class
• \(f\) = frequency of the class containing the target position
• \(w\) = class width
• \(kN\) = the position of the target value (e.g. \(0.25N\) for Q1, \(0.5N\) for median, \(0.75N\) for Q3, \(0.9N\) for 90th percentile).

Why it works

• Percentiles and quartiles divide data into equal-sized sections. Quartiles split data into four parts; percentiles split into 100 parts.
• Since grouped data hides individual values, we estimate positions proportionally inside the class interval.
• The formula interpolates between class boundaries depending on where the target lies.

Recipe (Steps)

1. Find \(N\), the total frequency.
2. Compute the required position:
   • \(Q_1 = N/4\)
   • \(Q_2 = N/2\) (the median)
   • \(Q_3 = 3N/4\)
   • \(P_p = pN/100\) for the \(p\)-th percentile.
3. Locate the class containing this position using cumulative frequencies.
4. Identify \(L, \text{CF}, f, w\).
5. Substitute into the formula.

Spotting it

Use this formula when the question asks for an estimated quartile, percentile, or decile from a grouped frequency table. Look for class intervals and cumulative frequency data.

Common pairings

• Estimated median formula (special case of quartiles).
• Cumulative frequency diagrams for graphical estimation.
• Interquartile range: \(Q_3 - Q_1\).

Mini examples

1. Find \(Q_1\): Data of 40 students with grouped ages. The first quartile lies in the 10–20 class. Answer: 13.5.
2. Find \(P_{90}\): Exam scores grouped in intervals. The 90th percentile falls in the 70–80 interval. Answer: 77.8.

Pitfalls

• Forgetting to multiply by \(N\) when finding percentiles.
• Confusing cumulative frequency before the class with the frequency of the class itself.
• Using midpoints instead of class boundaries.
• Rounding too early—keep decimals until the final step.

Exam Strategy

• Write the formula at the start of your working.
• Show clearly which class contains the quartile/percentile.
• Check that your answer lies within the class interval—it should never fall outside.

Worked Micro Example

A table shows marks:

Total \(N=30\). For \(Q_1\), position = \(N/4 = 7.5\). The 7.5th value lies in the 10–20 class.

Here, \(L=10\), \(w=10\), \(\text{CF}=4\), \(f=6\).

So, \(Q_1 = 10 + \frac{7.5-4}{6}\times 10 = 16.0\).

Summary

The estimated percentile/quartile formula extends the median method. By calculating positions such as \(N/4\), \(N/2\), \(3N/4\), or \(pN/100\), we can estimate quartiles and percentiles from grouped data. This is essential for measures of spread like the interquartile range, and for comparing different data sets in exams and real-world statistics.