Mean Median Mode Range Calculator
Calculate mean, median, mode, range, standard deviation, variance, min, max, sum, and count from any dataset.
Mean Median Mode Range Calculator
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Formula: Mean = Σx / n | Median = middle value | Mode = most frequent
Worked example — Mean: 85.6 | Median: 86.5 | Mode: 85 | Range: 19
Formula
Mean = Σx / n | Median = middle value | Mode = most frequent
Mean is the sum of all values divided by count. Median is the middle value in sorted data (average of two middle values for even counts). Mode is the value that appears most frequently. Range is the difference between maximum and minimum values.
Worked Examples
Example 1: Student Test Scores
Problem:Find the mean, median, mode, and range of these test scores: 85, 92, 78, 90, 85, 88, 76, 95, 85, 82
Solution:Sorted: 76, 78, 82, 85, 85, 85, 88, 90, 92, 95 Mean = 856 / 10 = 85.6 Median = (85 + 88) / 2 = 86.5 Mode = 85 (appears 3 times) Range = 95 - 76 = 19
Result:Mean: 85.6 | Median: 86.5 | Mode: 85 | Range: 19
Example 2: Daily Temperatures
Problem:Analyze these temperatures (°F): 72, 68, 75, 71, 69, 73, 70
Solution:Sorted: 68, 69, 70, 71, 72, 73, 75 Mean = 498 / 7 = 71.14 Median = 71 (middle value) Mode = No mode (all unique) Range = 75 - 68 = 7 Population Std Dev = 2.23
Result:Mean: 71.14 | Median: 71 | Mode: None | Range: 7 | Population SD: 2.23
Frequently Asked Questions
What is the difference between mean, median, and mode?
The mean (average) is the sum of all values divided by the count. The median is the middle value when data is sorted — it splits the data into two equal halves. The mode is the most frequently occurring value. For symmetric distributions, mean ≈ median ≈ mode. For skewed distributions they differ: in right-skewed data, mean > median > mode; in left-skewed data, mean < median < mode. The median is more robust to outliers than the mean.
When should I use the median instead of the mean?
Use the median when your data has outliers or is skewed. For example, income data: if 9 people earn $50K and 1 earns $5M, the mean is $545K (misleading), but the median is $50K (representative). The median is also preferred for ordinal data, house prices, response times, and any dataset where extreme values could distort the average. Use the mean when data is roughly symmetric and you need to include all values in the measure.
Can a dataset have more than one mode?
Yes. A dataset is unimodal if it has one mode (e.g., 1,2,2,3 — mode is 2), bimodal if it has two modes (e.g., 1,2,2,3,3,4 — modes are 2 and 3), and multimodal if it has more than two. If all values occur with equal frequency, there is no mode. Mode is the only measure of central tendency that works for categorical (non-numeric) data, such as favorite colors or brands.
What does range tell us about data?
Range = Maximum - Minimum. It measures the total spread of the data. A larger range indicates more variability. However, range is very sensitive to outliers since it only uses the two most extreme values. For example, the data set {10, 12, 11, 13, 100} has a range of 90, driven entirely by the outlier 100. Standard deviation and interquartile range (IQR) are more robust measures of spread.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides by N (total count) and is used when you have data for the entire population. Sample standard deviation (s) divides by N-1 (Bessel's correction) and is used when your data is a sample from a larger population. Dividing by N-1 corrects for the bias that a sample tends to underestimate population variance. In practice, if you're analyzing a subset of data to make inferences about a larger group, use sample standard deviation.
How does variance relate to standard deviation?
Variance is the mean of the squared deviations from the mean, and standard deviation is its square root. Squaring keeps every deviation positive and penalises large departures more heavily, but it also squares the units, so a dataset in dollars produces a variance in dollars squared. Taking the root returns the spread to the original units, which is why standard deviation is what gets reported. Variance remains the working quantity in the mathematics because variances of independent quantities add, while standard deviations do not.
What is a weighted mean, and when do you need one?
A weighted mean is the sum of each value times its weight, divided by the sum of the weights, so values carrying more importance or representing more observations pull the result harder. You need it whenever the items being averaged do not count equally: course grades where the final exam is worth more than a quiz, or combining group averages of different sizes. Averaging two class averages of 70 and 90 gives 80 only if both classes have the same number of students; with 40 and 10 students the correct figure is 74.
Why does the median of an even-sized dataset average the two middle values?
With an even count there is no single middle position after sorting, so the convention takes the mean of the values at positions n/2 and n/2 plus 1. For ten sorted scores that is the average of the fifth and sixth values. A consequence is that the median need not be a value that actually appears in the data, which is why ten scores can have a median of 86.5 even though every score is a whole number. Any point between those two middle values splits the data evenly, and the midpoint is chosen for consistency.
What is the interquartile range, and why is it preferred over the range?
The interquartile range is the third quartile minus the first quartile, so it describes the spread of the middle half of the data and ignores the tails entirely. The range uses only the single smallest and largest values, which makes it collapse or explode on the strength of one observation. A common rule of thumb flags points more than 1.5 times the interquartile range beyond either quartile as candidate outliers. Report the range when you need the full extent, the interquartile range when you need a description that survives a stray extreme value.
What is the difference between the arithmetic mean and the geometric mean?
The arithmetic mean adds the values and divides by the count; the geometric mean multiplies them and takes the nth root. Use the geometric mean for quantities that compound or are ratios, such as annual growth factors or index changes, because it reproduces the correct overall multiplier while the arithmetic mean overstates it. Growth factors of 2.0 and 0.5 average arithmetically to 1.25 but geometrically to exactly 1.0, which is the honest answer since doubling then halving returns you to the start. The geometric mean requires all values to be positive and never exceeds the arithmetic mean.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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