Basic descriptive statistics.
Mean = (x₁ + x₂ + … + xₙ) / n Median = x₍ₙ₊₁₎/₂ [if n is odd] Median = (x₍ₙ/₂₎ + x₍ₙ/₂₊₁₎) / 2 [if n is even] Mode = value(s) with the highest frequency Range = x_max − x_min
Each measure of central tendency uses its own formula. All four apply to a data set of n values.
Mean sums all values and divides by the count.
Median requires sorting the values first, then selecting the middle element (or averaging the two middle elements for an even-sized set).
Mode is found by identifying the value(s) that appear most frequently. A set may have one mode (unimodal), two modes (bimodal), more (multimodal), or no mode if all values are unique.
Range measures spread — the distance from the smallest to the largest value.
**Mean:** Sum = 72 + 85 + 90 + 85 + 78 + 92 + 85 + 68 + 90 + 75 = 820 Mean = 820 / 10 = **82** **Median (sort first):** Sorted: 68, 72, 75, 78, 85, 85, 85, 90, 90, 92 n = 10 (even), so average the 5th and 6th values. Median = (85 + 85) / 2 = **85** **Mode:** 85 appears 3 times — more than any other value. Mode = **85** **Range:** x_max = 92, x_min = 68 Range = 92 − 68 = **24**
Result: Mean = 82 | Median = 85 | Mode = 85 | Range = 24
The mean of 82 tells you the average quiz score across all 10 students. The median of 85 shows the middle score when all scores are ranked — meaning half the class scored 85 or above and half scored 85 or below. The mode of 85 confirms it was the single most common score. The range of 24 reveals a moderate spread between the lowest score (68) and the highest (92), indicating some variation in student performance but no extreme outliers.
The mean is the most widely used measure of center. It is sensitive to outliers — a single very high or very low value can pull the mean away from the bulk of the data. For example, if one student scored 20 on a quiz, the class mean would drop significantly even if everyone else scored around 85.
The median is the middle value of a sorted data set and is resistant to outliers. It is often preferred for skewed distributions. For instance, household income data in economics typically uses the median because a few billionaires would inflate the mean drastically. When n is even, the median is the average of the two central values.
The mode identifies the most frequently occurring value. It is the only measure of central tendency that can apply to non-numerical (categorical) data (e.g., the most popular color or most common surname). A data set can be:
The range is the simplest measure of spread (dispersion). It only uses two values (max and min), so it is easy to calculate but highly sensitive to outliers. For a more robust measure of spread, consider the interquartile range (IQR) or standard deviation.
| Measure | Best Used When | |---|---| | Mean | Data is symmetric with no significant outliers | | Median | Data is skewed or contains outliers | | Mode | Identifying the most common category or value | | Range | Quick snapshot of data spread |
Together, these four statistics give you a solid descriptive summary of any data set.
The mean is highly sensitive to outliers because every value contributes equally to the sum. Adding one extreme value — whether very high or very low — shifts the mean significantly. For example, adding a score of 5 to a set of scores in the 80s will lower the mean noticeably while barely changing the median.
In a perfectly symmetric distribution, mean = median = mode. In a **right-skewed** (positively skewed) distribution, the mean is pulled to the right of the median by high-value outliers (mean > median). In a **left-skewed** (negatively skewed) distribution, the mean is pulled left (mean < median). Comparing mean and median is a quick way to detect skew.
A weighted mean accounts for the fact that some values are more important than others. Multiply each value by its weight, sum the results, then divide by the total weight: Weighted Mean = Σ(wᵢ × xᵢ) / Σwᵢ. For example, if a final exam is worth twice the weight of a quiz, the exam score is multiplied by 2 before averaging.
The IQR is the difference between the 75th percentile (Q3) and the 25th percentile (Q1) of a sorted data set: IQR = Q3 − Q1. Unlike the range, the IQR ignores the extreme values entirely and focuses on the middle 50% of the data, making it far more resistant to outliers as a measure of spread.
In everyday language, 'average' almost always refers to the arithmetic mean. Technically, there are other types of averages (geometric mean, harmonic mean, weighted mean), but when a teacher or textbook says 'average,' they mean the sum of values divided by the count — which is the mean this calculator computes.
Yes. If two values tie for the highest frequency, the data set is **bimodal** and both values are modes. Three or more tied values make it **multimodal**. If every value appears exactly once (or the same number of times), the data set has no mode.
A small number of extremely high earners can pull the mean income far above what a typical person earns. The median is unaffected by these outliers, so it better represents the income of a 'middle' household. The U.S. Census Bureau, for example, reports **median household income** for this reason.
Negative numbers are treated exactly like positive ones. Include them in the sum for the mean, sort them into their correct position (left-most) for the median, and tally their frequency for the mode. The range still equals the maximum value minus the minimum value, which will be a larger positive number if negatives are present.
If every value in the data set is identical (e.g., all 5s), then the mean, median, and mode all equal that value, and the range equals 0.
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