Salary Quartile Calculator

Use the Salary Quartile Calculator to find salary quartiles and analyze income distribution quickly.

Results

Step by Step Solution

Data

Sorted Salaries
Total Salaries (n)

Formula

Q1 = Median of Lower Half

Q2 = Overall Median

Q3 = Median of Upper Half

Interquartile Range (IQR) = Q3 − Q1

Solution

What is a Salary Quartile Calculator?

A salary quartile calculator is a human resources and statistical evaluation tool used to divide a ordered dataset of employee compensation into four equal segments, or quartiles. Each quartile threshold marks a specific boundary representing 25% of the total distribution. In compensation benchmarking and labor analytics, salary quartiles help organizations structure pay bands, analyze pay equity, and evaluate compensation competitiveness. By mapping individual salaries against broader market distribution points, compensation managers can determine whether pay falls into the lower, median, or upper distribution tiers.

Salary Quartile Formulas

To compute the position (\(L_k\)) of the \(k\)-th quartile (\(Q_1, Q_2, Q_3\)) for an ordered dataset of size \(n\), the general position formula is:

$$L_k = \frac{k(n + 1)}{4} \quad \text{for } k \in \{1, 2, 3\}$$

Interpolation and Interquartile Range Formulas

When the rank position \(L_k\) yields a non-integer value with integer part \(i\) and fractional part \(f\), the linear interpolation formula computes the exact quartile value:

$$Q_k = X_i + f \cdot (X_{i+1} - X_i)$$

The spread between the upper and lower pay boundaries, known as the Interquartile Range (\(IQR\)), measures middle-50% salary dispersion:

$$\text{Interquartile Range (IQR)} = Q_3 - Q_1$$

Salary Quartiles for an Odd Dataset Size

A technology startup evaluates annual salaries (in thousands of dollars) for seven software engineers:

$$S = \{\$65, \$72, \$80, \$88, \$95, \$105, \$120\}$$

Step-by-Step Calculation

First, order the dataset in ascending order ($n = 7$). Compute the rank positions for \(Q_1\), \(Q_2\), and \(Q_3\):

$$L_1 = \frac{1(7 + 1)}{4} = 2, \quad L_2 = \frac{2(7 + 1)}{4} = 4, \quad L_3 = \frac{3(7 + 1)}{4} = 6$$

Extract the values at the exact integer position indices in dataset \(S\):

$$Q_1 = S_2 = \$72\text{k}, \quad Q_2 = S_4 = \$88\text{k} \text{ (Median)}, \quad Q_3 = S_6 = \$105\text{k}$$

Calculate the Interquartile Range (\(IQR\)):

$$\text{IQR} = Q_3 - Q_1 = 105 - 72 = \$33\text{k}$$

The 25th percentile (\(Q_1\)) is \$72,000, the median (\(Q_2\)) is \$88,000, the 75th percentile (\(Q_3\)) is \$105,000, and the \(IQR\) is \$33,000.

Salary Quartiles with Linear Interpolation

An HR department evaluates annual compensation for six department managers (in thousands of dollars):

$$S = \{\$60, \$70, \$78, \$85, \$92, \$110\}$$

Step-by-Step Calculation

For $n = 6$, compute the rank positions using the position formula:

$$L_1 = \frac{1(6 + 1)}{4} = 1.75, \quad L_3 = \frac{3(6 + 1)}{4} = 5.25$$

Since \(L_1 = 1.75\), interpolate between the 1st ($S_1 = 60$) and 2nd ($S_2 = 70$) values with fraction $f = 0.75$:

$$Q_1 = S_1 + 0.75 \cdot (S_2 - S_1) = 60 + 0.75 \cdot (70 - 60) = 60 + 7.5 = \$67.5\text{k}$$

For \(L_3 = 5.25\), interpolate between the 5th ($S_5 = 92$) and 6th ($S_6 = 110$) values with fraction $f = 0.25$:

$$Q_3 = S_5 + 0.25 \cdot (S_6 - S_5) = 92 + 0.25 \cdot (110 - 92) = 92 + 4.5 = \$96.5\text{k}$$

The interpolated first quartile (\(Q_1\)) is \$67,500 and the third quartile (\(Q_3\)) is \$96,500.