How to find median mode and quartiles?

The basics of central tendency and spread

Understanding these key statistical measures gives you the tools to summarize data accurately. When analyzing numerical data, the mean does not always tell the whole story. A single extreme number can pull the mean far away from typical values, while the median remains stable. The mode tells you what happens most often, and quartiles show how the data spreads out across four equal sections.

Step-by-step method for median, mode, and quartiles

First, sort your list in ascending order. Count your total values, n. The mode is the number that appears with the highest frequency; a dataset can have one mode, more than one mode, or no mode at all. For grouped data, look for the modal class, which is the interval with the highest frequency.

Next, find the median (Q2). If n is odd, the median is the exact middle value at position (n + 1) / 2. If n is even, average the two middle values at positions n / 2 and (n / 2) + 1.

Finally, split the sorted list into a lower half and an upper half, excluding the median if n is odd. The lower quartile (Q1) is the median of the lower half (the 25th percentile). The upper quartile (Q3) is the median of the upper half (the 75th percentile). Subtracting Q1 from Q3 yields the interquartile range (IQR), which measures the spread of the middle 50 percent.

A concrete example and what to watch for

Take the test scores: 65, 70, 70, 80, 85, 90, 100. The mode is 70. There are 7 values, so the median is the 4th value, 80. The lower half consists of 65, 70, 70, making Q1 equal to 70. The upper half consists of 85, 90, 100, making Q3 equal to 90. The interquartile range is 90 minus 70, which is 20.

Watch out for outliers when choosing summary statistics. If an eighth student scored 10 on this test, the mean would plummet from 80 to 71.25. However, the median would only adjust slightly from 80 to 75. Use the median and IQR when your dataset contains extreme values or skew.