四分位数

Quartiles are measures of relative standing that divide a dataset into four equal groups, each containing 25 percent of the data. They separate the data into quarters based on ordered values. The second quartile is the median, which separates the lower 50 percent from the upper 50 percent. The first and third quartiles divide the lower and upper halves respectively, creating a framework for understanding data distribution.

Q1 (first quartile) is the middle value of the lower half of ordered data, with 25 percent of values at or below it. Q2 (second quartile) is the median, separating the dataset in half. Q3 (third quartile) is the middle value of the upper half, with 75 percent of values below it. Together, these three quartiles provide key reference points for understanding data spread and location.

The interquartile range (IQR) is calculated by subtracting the first quartile from the third quartile: IQR = Q3 – Q1. This value represents the spread of the middle 50 percent of the data. The IQR indicates how concentrated or dispersed the central portion of your dataset is, providing insight into data variability without being affected by extreme values.

The IQR helps identify potential outliers by establishing boundaries around the middle 50 percent of data. Any value falling below 1.5 times the IQR below Q1, or above 1.5 times the IQR above Q3, is considered a potential outlier. This method isolates extreme values that deviate significantly from the central data distribution, making it useful for detecting what are outliers in datasets.

Quartiles may or may not be part of the original dataset. When calculating quartiles using formulas, the result might fall between two existing data values rather than matching an actual observation. For example, a calculated first quartile might be 43, representing a value between the 2nd and 3rd data points. Whether quartiles are actual data values depends on the dataset size and distribution.

If 75 percent of data is below Q3 (third quartile), it means that three-quarters of all ordered values fall at or below that point. Conversely, only 25 percent of the data exceeds Q3. This relationship holds true by definition: Q3 marks the boundary where exactly three-fourths of the dataset lies below it, making it a key measure of relative standing in data analysis.

To find quartiles, first arrange data from lowest to highest. Locate the median (Q2) to split the data in half. Q1 is the middle value of the lower half, and Q3 is the middle value of the upper half. Use the formula approach for larger datasets: the first quartile position yields the 2.5th value, meaning Q1 falls between the 2nd and 3rd ordered observations, interpolating as needed.