Understanding median, mode and range
Each statistic describes a different aspect of the data. The table below summarizes what each one tells you.
| Statistic | What it describes | Sensitivity to outliers |
|---|---|---|
| Median | The middle of the ordered data; 50% of values lie at or below it | Low — depends only on the order of values |
| Mode | The most frequent value(s); the peak(s) of the distribution | Low — unaffected by extreme values |
| Range | Total spread from smallest to largest value | High — determined entirely by the two extremes |
- When the mean is noticeably larger than the median, the data are typically right-skewed (a tail of large values, as with incomes); a mean below the median suggests left skew.
- A data set can have no mode (all values distinct), one mode, or several modes; this calculator lists all values tied for the highest frequency.
- The median of a sample estimates the population median; like all sample statistics it is subject to sampling variation.
What are the median, mode and range?
The median is the value that splits an ordered data set in half. To find it, sort the values from smallest to largest; with an odd number of values the median is the single middle value, and with an even number it is the arithmetic mean of the two middle values. Because the median depends only on the order of the data, not the magnitudes of extreme values, it is robust to outliers.
The mode is the value (or values) that occur most frequently in the data. A data set can have one mode, several modes (for example, bimodal data with two peaks), or no mode at all when every value occurs exactly once. The mode is the only measure of central tendency that also applies to categorical data.
The range is the difference between the largest and smallest value. It is the simplest measure of spread, but because it uses only the two extreme observations it is highly sensitive to outliers; the standard deviation or interquartile range describe spread more robustly.
How to use this median calculator
- Enter your numbers separated by commas, for example: 7, 12, 3, 9, 21, 14. The calculator sorts them for you.
- Read the median. With an even count of values it is the average of the two middle values of the sorted list.
- Read the mode: the most frequent value or values. A dash is shown when no value repeats, meaning the data have no mode.
- Read the range, the maximum minus the minimum — a quick indication of how spread out the data are.
How the median, mode and range are calculated
Median: sort the n values. If n is odd, the median is the value at position (n + 1) / 2. If n is even, the median is the mean of the values at positions n/2 and n/2 + 1.
Worked example: the values 7, 12, 3, 9, 21, 14. Sorted: 3, 7, 9, 12, 14, 21. There are n = 6 values (even), so the median is the mean of the 3rd and 4th values: (9 + 12) / 2 = 10.5. No value repeats, so there is no mode. The range is 21 - 3 = 18.
Second example with a mode: 2, 4, 4, 7, 9. Sorted, the middle (3rd) value is 4, so the median is 4. The value 4 occurs twice while every other value occurs once, so the mode is 4. The range is 9 - 2 = 7.
Common mistakes
- Forgetting to sort the data before locating the middle value — the median is defined on the ordered list.
- Taking one of the two middle values instead of their average when the count is even: for 3, 7, 9, 12, 14, 21 the median is 10.5, not 9 or 12.
- Assuming every data set has a mode — when no value repeats, the mode does not exist.
- Using the range as a robust measure of spread; it depends only on the two most extreme values and ignores everything between them.
Câu hỏi thường gặp
How do I find the median of a data set?
Sort the values from smallest to largest. If the count is odd, the median is the middle value. If the count is even, the median is the average of the two middle values. For example, for 7, 12, 3, 9, 21, 14 the sorted list is 3, 7, 9, 12, 14, 21 and the median is (9 + 12) / 2 = 10.5.
When is the median better than the mean?
When the data are skewed or contain outliers. The median depends only on the ordering of the values, so a single extreme observation barely moves it, whereas the mean can shift substantially. Incomes and house prices are typically summarized by the median for this reason.
Can a data set have more than one mode?
Yes. If two or more values are tied for the highest frequency, the data set is multimodal and all tied values are modes. For example, in 1, 2, 2, 5, 7, 7, 9 both 2 and 7 occur twice, so the data are bimodal with modes 2 and 7. If no value repeats, the data set has no mode.
What does the range tell me?
The range is the maximum minus the minimum — the total span of the data. It is quick to compute and easy to interpret, but it uses only the two most extreme values, so one outlier can inflate it dramatically. For a fuller picture of spread, use the standard deviation or the interquartile range.
Is the median always a real data value?
Not always. With an odd number of values the median is one of the observed values. With an even number it is the average of the two middle values, which may not appear in the data — for example, the median 10.5 of the list 3, 7, 9, 12, 14, 21 is not one of the six observations.
Tài liệu tham khảo
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.1: Measures of Location. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (median, mode and resistant measures).
- Weisstein, Eric W. "Statistical Median." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.