Median Percentile Quartile: What They Mean

This calculator works out the median, quartiles (Q1, Q3), and percentile rank for a set of numbers you enter. Type or paste your data into the list, choose the percentile you want, and you get the value that separates that percentage of the data from the rest. It also shows the interquartile range (IQR), which is the middle 50% spread of your data.

Formula

For a sorted dataset of n values, the median (Q2) is the middle value when n is odd, or the average of the two middle values when n is even. The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half. The interquartile range is IQR = Q3 − Q1. For a given percentile p (0 to 100), the value is found by locating the position k = (p/100) × (n + 1) in the sorted list; if k is not an integer, interpolate between the two nearest values.

Worked Example

Using the default data: 4, 8, 15, 16, 23, 42 (n = 6). First sort: 4, 8, 15, 16, 23, 42. Median: since n is even, average the 3rd and 4th values: (15 + 16) / 2 = 15.5. Lower half: 4, 8, 15 → Q1 = 8. Upper half: 16, 23, 42 → Q3 = 23. IQR = 23 − 8 = 15. For the 50th percentile (p = 50), position k = 0.50 × (6 + 1) = 3.5, so interpolate between 15 and 16: 15 + 0.5 × (16 − 15) = 15.5. The calculator shows the same results: median 15.5, Q1 8, Q3 23, IQR 15, and the 50th percentile value 15.5.

When to Use Median, Quartiles, and Percentiles

These measures are used when you want to understand the distribution of a dataset without being skewed by extreme values. The median is the central value; quartiles divide the data into four equal parts; percentiles divide it into 100 parts. They are common in education (test scores), finance (income distributions), and health (growth charts). For example, the 90th percentile of a test score means 90% of scores are below that value.

How to Read the Results

The median tells you the middle of your data. Q1 and Q3 give you the range where the central 50% of values lie. The IQR is the distance between Q1 and Q3; a larger IQR means more spread in the middle half of the data. The percentile value you request (e.g., the 90th) is the score below which that percentage of the data falls. If you enter 50 as the percentile, you get the median itself.

Common Mistakes and Misconceptions

• Mixing up quartile methods: Different software may compute Q1 and Q3 slightly differently (e.g., including or excluding the median in the halves). The calculator here uses the common method of splitting the data into two halves and taking the median of each half, which is the textbook approach.
• Forgetting to sort: Percentiles and quartiles are defined on sorted data. The calculator sorts automatically, but if you compute manually, always sort first.
• Interpreting percentile as a score: A percentile is not the number of correct answers; it is a relative rank. Being in the 80th percentile means you scored higher than 80% of the group, not that you got 80% correct.
• Using mean instead of median: If your data has outliers, the median is often a better measure of center than the mean. For example, incomes are usually reported with the median because a few very high incomes pull the mean up.

Limits of the Formula

These measures assume your data is at least ordinal (can be ranked). They do not tell you about the shape of the distribution beyond the central spread. For example, two datasets can have the same median and IQR but very different distributions (one symmetric, one skewed). Also, percentiles are less reliable with small sample sizes; with fewer than 10 values, extreme percentiles (like the 1st or 99th) can be imprecise because there is not enough data to interpolate accurately.

Frequently Asked Questions

What is the difference between a quartile and a percentile?

A quartile is a specific type of percentile. Quartiles divide data into four equal parts: the 25th, 50th, and 75th percentiles are called Q1, Q2 (median), and Q3. Percentiles are more general; the 90th percentile, for example, is not a quartile.

How do I find the percentile rank of a specific value?

To find the percentile rank of a value x, count how many values are less than x, divide by the total number of values, and multiply by 100. For example, if 8 out of 20 values are below your score, your percentile rank is (8/20) × 100 = 40th percentile.

Why do different calculators give different quartiles?

There are several accepted methods for calculating quartiles. The most common differences arise in whether the median is included when splitting the data for Q1 and Q3, and how interpolation is done. The method used here (median of lower and upper halves) is the one taught in most basic statistics courses, but some software uses linear interpolation or other methods, leading to slightly different results.

What does the interquartile range tell me?

The IQR measures the spread of the middle 50% of your data. It is a robust measure of variability because it is not affected by extreme values. A small IQR means the central half of your data is tightly clustered; a large IQR means it is spread out.

Can I use these measures for non-numeric data?

No, you need numeric or at least ordinal data (categories that can be ranked, like letter grades). For nominal categories (like colors), medians and percentiles do not make sense.

What is the 50th percentile called?

The 50th percentile is the median. It is also the second quartile (Q2). Half of the data falls below this value, and half falls above it.

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in list | Data values | 4, 8, 15, 16, 23, 42 | 0
in p | Percentile to find | % | 50 | 0
out q | Median (Q2) | value | median(list) | 2
out q1 | First quartile (Q1) | value | q1(list) | 2
out q3 | Third quartile (Q3) | value | q3(list) | 2
out iqr | Interquartile range (IQR) | value | q3(list) - q1(list) | 2
out pval | Percentile value | value | percentile(list, p) | 2