Statistics Calculator – Mean, Median, Mode
Calculate mean, median, mode, range, and standard deviation for any dataset.
About This Calculator
Calculate mean, median, mode, range, and standard deviation for any dataset. Use the calculator above for instant results.
Understanding Center and Spread in a Data Set
Descriptive statistics summarize two different features of data: where the values are centered and how widely they are spread. The mean uses every observation, the median identifies the middle of the ordered list, and the mode identifies the most frequent value or values. None of these measures alone describes the entire distribution.
The mean is sensitive to extreme values because each observation contributes directly to the sum. The median is more resistant to outliers, which is why it often describes the center of strongly skewed data better. The range is maximum minus minimum and gives a quick measure of total spread, but it depends only on two observations.
Standard deviation measures typical distance from the mean by squaring deviations, averaging those squares, and taking a square root. This page uses the population formula, dividing by n. If the entered values are a sample used to estimate variability in a larger population, many statistics courses instead use the sample standard deviation, which divides the squared-deviation sum by n−1.
Always sort values before locating a median. With an odd count, the median is the single middle value; with an even count, it is the average of the two middle values. A data set can be multimodal if several values share the highest frequency.
Shape matters when interpreting these summaries. In a symmetric distribution without strong outliers, mean and median are often close. A long right tail typically pulls the mean above the median, while a long left tail can pull it below. Comparing center measures can therefore reveal information about skewness even before drawing a graph.
Worked Examples
- Mean: 4.86 | Median: 4 | Mode: 4,7
Answer: Mean=4.86
- Mean: 30 | Median: 30 | Range: 40
Answer: Mean=30
- Mean: 1.8 | Mode: 1 | Std Dev: 1.6
Answer: Mean=1.8
Data: 2, 3, 3, 5, 7.
- Mean = 20/5 = 4.
- Median = 3, mode = 3, range = 5.
- Population variance = 3.2, so population SD ≈ 1.79.
Answer: mean 4, median 3, mode 3, SD ≈ 1.79
Data: 1, 2, 2, 3, 20.
- Mean = 28/5 = 5.6, while median = 2.
Answer: the mean is pulled upward by 20
Who Uses This Calculator?
Quick summary statistics.
Stats homework and tests.
Descriptive statistics.
Sales and performance data.
Common Mistakes to Avoid
❌ Mean vs median
Mean is skewed by outliers; median is not. Use median for income data.
❌ Multiple modes
A dataset can have multiple modes or no mode at all.
❌ Using a sample formula when a population formula is required
Population variance divides by n; sample variance commonly divides by n−1. State which data interpretation the problem requires.
❌ Finding the median before ordering the data
The median is based on position in the sorted list. The middle entry of an unsorted list has no special statistical meaning.
Frequently Asked Questions
Mean vs median?
Mean: sum/count. Median: middle value. Use median for skewed data.
Standard deviation?
Measures spread from mean. Low = clustered, High = spread out.
68-95-99.7 rule?
Normal distribution: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.
When is median more informative than mean?
Median is often more representative for skewed data or data with large outliers because it depends on order rather than the magnitude of every observation. Income and home-price distributions are common examples.
Can a data set have more than one mode?
Yes. If two or more distinct values tie for the greatest frequency, the set is multimodal. Some classroom conventions say a set has no mode when every value occurs equally often.
Why are deviations squared in variance?
Raw deviations from the mean sum to zero, so positive and negative differences would cancel. Squaring makes every contribution nonnegative and gives larger deviations greater influence.
Does standard deviation have the same units as the data?
Yes. Variance has squared units, but taking the square root returns standard deviation to the original data units, making its scale easier to interpret.
Formula Explorer connections
Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.