Mean, Median, Mode & Range Calculator

Descriptive Statistics Guide

What should I know about Mean, Median, Mode?

Mean: sum of all values / number of values. Sensitive to outliers. Best for symmetrical, normally distributed data. Median: middle value when sorted. Odd n: middle value. Even n: average of two middle values. Resistant to outliers. Best for skewed data (house prices, incomes). Mode: most frequently occurring value. Can have multiple modes (bimodal) or none (all values unique). Best for categorical data. Range: maximum − minimum. Interquartile range (IQR): Q3 − Q1 (middle 50% of data). More robust to outliers than the range alone, since it ignores the most extreme 25% of values at each end of the dataset.

What's the key thing to understand about Standard Deviation?

Variance σ² = Σ(x − x̄)² / n. Population variance (divide by n). Sample variance (divide by n−1) — unbiased estimate. Standard deviation σ = √variance. Interpretation: approximately 68% of data within 1σ of mean. 95% within 2σ. 99.7% within 3σ (if normally distributed). Example: data 2, 4, 4, 4, 5, 5, 7, 9. Mean = 5. Deviations: −3, −1, −1, −1, 0, 0, 2, 4. Squared: 9,1,1,1,0,0,4,16. Sum: 32. Variance = 32/8 = 4. SD = 2. Coefficient of variation: σ/x̄ — useful for comparing the spread of datasets with different units or scales, since it expresses spread as a proportion of the mean rather than in the original measurement units.

When to Use Which Average?

Mean: test scores (normally distributed, no outliers). Average speed on a journey. Temperature readings. Median: house prices (skewed by luxury properties). Salary data (skewed by very high earners). Any data with significant outliers. Mode: shoe size sales (which size to stock most). Most popular choice from a survey. Favourite colour, most common grade. Comparing measures: if mean > median: right-skewed distribution (few high values pulling mean up). If mean < median: left-skewed. If mean is approximately equal to median: the distribution is roughly symmetric, with no strong skew in either direction.

What should I know about GCSE Statistics Exam Tips?

Interpreting spread: standard deviation is more informative than range because it uses all data points. IQR ignores outliers (uses middle 50%). Box plots: display min, Q1, median, Q3, max. Outliers: typically defined as values more than 1.5 × IQR below Q1 or above Q3. Comparing two datasets: use the mean (or median) to compare average performance. Use standard deviation (or IQR) to compare consistency. Example: class A mean 65%, SD 5. Class B mean 65%, SD 15. Same average but B has much more spread — a common exam question testing whether students understand that identical means can still represent very different underlying distributions.

What should I know about Skewness and Choosing the Right Average?

In a perfectly symmetric distribution, mean, median, and mode all coincide. In a skewed distribution, they separate in a predictable order. Right-skewed (long tail toward high values, like household income or house prices): mode < median < mean, because a few very high values pull the mean upward while the median stays near the bulk of the data. Left-skewed (long tail toward low values, like exam scores where most people do well but a few score very low): mean < median < mode. This is exactly why median house price and median income are reported in the news rather than the mean — a handful of very expensive houses or very high earners would drag the mean well above what's typical for most people, while the median reports the genuinely middle value unaffected by those extremes.

What do I need to know about the Weighted Mean?

A weighted mean accounts for some values mattering more than others, calculated as Σ(value×weight) / Σ(weight), rather than simply summing values and dividing by count. For a module grade made of a 20% coursework score of 70 and an 80% exam score of 85: weighted mean = (70×0.2 + 85×0.8)/(0.2+0.8) = (14+68)/1 = 82. This differs from the simple average of 70 and 85 (which would be 77.5) because the exam carries four times the weight of the coursework. Weighted means are used whenever different data points carry different importance or frequency — grade calculations, portfolio returns weighted by investment size, or a grouped frequency table where each value's 'weight' is how many times it occurred — and forgetting to weight when weights genuinely differ is a common error that understates the influence of the heavier-weighted components.

What's the key thing to understand about How Outliers Affect Mean vs Median Differently?

The mean is calculated from every value in a dataset, which makes it highly sensitive to outliers — a single extreme value can shift the mean substantially, especially in a small dataset. The median, by contrast, only depends on which value sits in the middle position once the data is sorted, so an outlier can only pull the median toward itself gradually (and only if there are enough outliers to actually shift what counts as "the middle"), never in one dramatic jump. A classic illustration: the salaries $30k, $32k, $35k, $38k, $40k have a mean of $35k and a median of $35k — nearly identical — but if the top value becomes $500k instead of $40k, the mean rockets to $127k while the median stays exactly $35k, since the middle position in the sorted list hasn't changed. This is precisely why income and house-price statistics are typically reported using the median rather than the mean.

How to Find the Mean, Median, and Mode from Grouped (Frequency) Data

When data is already summarised into a frequency table rather than listed individually, you estimate the mean using class midpoints. Multiply each midpoint by its frequency, sum those products, and divide by the total frequency. The median class is the one containing the middle-ranked value, and the modal class is simply the one with the highest frequency.

Total frequency = 4 + 9 + 12 + 5 = 30. Sum of f × x = 258.0 + 670.5 + 1014.0 + 472.5 = 2415.0. Estimated mean = 2415.0 ÷ 30 = 80.5. The 15th and 16th values (the middle of 30 ordered values) both fall in the 80-89 class, so that is the median class, and since it also has the highest frequency (12), it is the modal class too — a common situation in roughly bell-shaped grouped data. This grouped-data technique is exactly what GCSE and A-Level statistics papers test when they give you a frequency table instead of raw scores, and it's the same logic spreadsheets use internally for histogram-based summaries.

Worked Comparison: How One Outlier Changes Mean, Median, and Mode

Six employees report these monthly commute distances in kilometres: 12, 15, 14, 16, 13, 95 (the last person works from a different office). This dataset shows exactly why relying on a single "average" without checking the others can mislead. Mean = (12 + 15 + 14 + 16 + 13 + 95) ÷ 6 = 165 ÷ 6 = 27.5 km — pulled far above every individual commuter's actual experience by the one outlier. Median: sort the values → 12, 13, 14, 15, 16, 95. With 6 values, the median is the average of the 3rd and 4th: (14 + 15) ÷ 2 = 14.5 km — a far more representative "typical" commute. Mode: every value is unique, so there is no mode for this dataset. If you dropped the 95 km outlier entirely, the remaining five values (12, 13, 14, 15, 16) give a mean of 14.0 km, almost identical to the median of the full dataset (14.5 km). That confirms the median is the more robust measure here — it barely moved when the outlier was included, while the mean shifted by nearly 100%. This is the practical reason median income, median house price, and median commute time are reported in the news instead of the mean: a handful of extreme values can distort an average without changing what's "typical" for most people.

Quick Reference: Which Average Fits Which Data Type

Statisticians don't pick an average arbitrarily — the right measure depends on the type of data and its shape:

This is exactly why government statistics agencies report median household income and median house prices rather than the mean — a handful of very high earners or luxury properties would otherwise inflate the "average" well above what a typical household actually earns or pays, while the mode would simply be meaningless for continuous income data.