The first quartile (Q1) is a fundamental measure used in statistics to describe the distribution and spread of a data set. Quartiles are values that divide ordered data into four equal segments, with each segment containing 25% of the total observations. Q1 represents the 25th percentile, establishing the point where one-quarter of the data values fall below it. This measure is a component of the five-number summary, which also includes the minimum, median (Q2), third quartile (Q3), and maximum value.
Preparing the Data Set
Before calculation, the raw data must be organized in a systematic way. The entire collection of numbers must be arranged in ascending order, moving from the smallest value to the largest value. This ordering is a necessary step because quartiles rely on the relative position of a number within the sequence, not just its magnitude.
Once the data is ordered, the total number of observations must be counted, which is denoted by the variable $N$. This count represents the size of the data set and is necessary for determining the precise mathematical location of the first quartile. For example, if a data set contains 15 individual numbers, the value of $N$ is 15.
Determining the Position of Q1
The initial step in locating the first quartile is to calculate its position within the ordered data set using a standard formula. This procedure identifies the rank, or index number, of the Q1 value, rather than the value itself. The position of the first quartile is calculated using the expression $(N+1) / 4$, where $N$ is the total number of data points.
Interpreting the result depends on whether the calculated value is a whole number or a fraction. If the position calculation results in a whole number, such as 5, then the value of Q1 is simply the data point at that exact position in the ordered list. For instance, the 5th number in the sequence would be the first quartile.
If the calculation results in a fractional position, such as a number ending in $.25$, $.50$, or $.75$, interpolation is required to determine the final Q1 value. A result ending in $.50$, for example, means the first quartile lies exactly halfway between two consecutive data points. To find the Q1 value in this case, one must average the two data points surrounding the fractional position.
For a more complex fractional result, such as a position of 4.25, the value of Q1 is found by moving a proportional distance between the two surrounding data points. Specifically, a position of 4.25 means the Q1 value is located 25% of the way from the 4th data point to the 5th data point.
Illustrative Calculation Example
Consider a data set of nine numbers: 7, 3, 8, 1, 9, 5, 6, 2, 4. The first step is to arrange these $N=9$ observations in ascending order, yielding the sequence: 1, 2, 3, 4, 5, 6, 7, 8, 9.
Applying the position formula, $(N+1)/4$, the calculation becomes $(9+1)/4$, which simplifies to $10/4$, resulting in a positional index of 2.5. Since the result is a fraction, Q1 does not correspond to a single data point and must be determined through interpolation between the 2nd and 3rd observations in the ordered list. The 2nd data point is 2, and the 3rd data point is 3.
A position of 2.5 indicates that the first quartile is located exactly halfway between the 2nd and 3rd values. The final Q1 value is found by averaging these two surrounding data points: $(2 + 3) / 2$. This averaging calculation yields $5 / 2$, which results in a final first quartile value of 2.5.