Mean, median, mode and range together
Mean, median, mode and range together
“Average” usually means the arithmetic mean, but there are three common measures of central tendency and they answer different questions.
Mean — add every value and divide by how many there are. It uses all the data, which makes it informative and also sensitive to extremes.
Mean = Σx ÷ n
Median — sort the values and take the middle one. With an even count, average the two middle values. It ignores how extreme the outliers are, only where they sit relative to the centre.
Mode — the value appearing most often. A data set can have one mode, several, or none.
Consider the salaries in a small firm: $38k, $41k, $43k, $45k, and $480k. The mean is $129,400. Not one person earns close to that. The median is $43,000, which describes a typical employee accurately.
This is why income, house prices, and wealth are almost always reported as medians. Distributions with a long tail on one side pull the mean toward the tail, away from where most of the data actually sits.
The mean remains the right choice when values are roughly symmetric and every observation should count — test scores in a class, daily temperatures, measurement repeats. The practical habit is to calculate both: when they are close, the distribution is fairly even; when they diverge sharply, outliers or skew are present and the median is the safer summary.
Range is simply the largest value minus the smallest. It is the crudest measure of spread and still useful as a first look.
Two data sets can share a mean of 50 while one runs from 48 to 52 and the other from 0 to 100. The averages are identical; the datasets are not remotely comparable. Range flags that immediately, though it only reflects the two extreme values and says nothing about the distribution in between — which is what standard deviation measures.
None is more correct in general. Each is a different summary, and quoting one without saying which was used is how statistics get misread.
A plain mean treats every value equally. Often they should not be. If a course grade is 30% coursework and 70% exam, averaging the two marks directly gives the wrong answer.
Weighted mean = Σ(value × weight) ÷ Σweight
With 80 on coursework and 60 on the exam: (80 × 0.3) + (60 × 0.7) = 24 + 42 = 66, not the 70 a simple average would give. The same logic applies to GPA calculations, portfolio returns, and any figure where components differ in importance.
The mean adds all values and divides by the count. The median is the middle value when sorted. The mean is pulled by extreme values; the median is not, which is why income and house prices are usually reported as medians.
Use the mean for symmetric data where every value should count. Use the median when outliers or skew are present. Use the mode for categorical data where averaging makes no sense.
Yes. A set with two values tied for most frequent is bimodal, and more than two is multimodal. If every value appears once, the set has no mode.
It is the difference between the largest and smallest values, giving a rough sense of spread. It only reflects the two extremes, so standard deviation is more informative about how the data is distributed.