Mean, Median & Mode Calculator

About Mean, Median & Mode Calculator

What Are Mean, Median, and Mode?

Mean, median, and mode are the three primary measures of central tendency, each with a different definition of "typical". The mean weights every value equally, the median ignores values except for their rank, and the mode only counts frequency. These three measures of central tendency each answer the question "what is a typical value?" in a different way:

Measure	Definition	Best Used When
Mean	Sum of all values divided by the count	Data is symmetric with no extreme outliers
Median	The middle value when data is sorted	Data is skewed or has outliers
Mode	The most frequently occurring value	Data is categorical or you want the "most common"

How to Calculate Each Measure

Mean:

Mean = (sum of all values) / (number of values)

Dataset: 4, 7, 2, 9, 7, 3, 8

• Sum = 4 + 7 + 2 + 9 + 7 + 3 + 8 = 40
• Count = 7
• Mean = 40 / 7 = 5.714

Median:

1. Sort the data: 2, 3, 4, 7, 7, 8, 9
2. Count = 7 (odd), so the median is the middle value
3. Middle position = (7 + 1) / 2 = 4th value
4. Median = 7

For even-count datasets: the median is the average of the two middle values.

• Dataset: 3, 5, 7, 9 (4 values)
• Two middle values: 5 and 7
• Median = (5 + 7) / 2 = 6

Mode:

• Dataset: 2, 3, 4, 7, 7, 8, 9
• 7 appears twice, all others appear once
• Mode = 7

When Mean and Median Disagree

The relationship between mean and median tells you about the shape of the data:

Relationship	Shape	Example
Mean ≈ Median	Symmetric	Heights of adults: mean and median both around 170 cm
Mean > Median	Right-skewed (positive skew)	Income: a few high earners pull the mean up
Mean < Median	Left-skewed (negative skew)	Retirement age: most retire at 65, a few retire very early

Classic example: In a room of 10 people earning £30K each, the mean and median income are both £30K. If one person gets a raise to £1 million, the mean jumps to £127K but the median stays at £30K. This is why median income is more informative than mean income for understanding "typical" earnings.

Real-world skew: This is not just a textbook trick. UK household income is visibly right-skewed: ONS figures for the financial year ending 2025 put median disposable household income at £719 per week before housing costs, up 5% in real terms on the previous year. The mean sits well above the median because a small number of very high earners pull the average up. The US Census Bureau reported 2024 median household income at $83,730, and the mean is consistently several thousand dollars higher for the same reason. When journalists and governments report "typical" income they almost always use the median - reporting mean income would overstate what an ordinary household takes home.

Types of Mode

Type	Meaning	Example Dataset
No mode	All values appear equally often	1, 2, 3, 4, 5
Unimodal	One mode	1, 2, 2, 3, 4
Bimodal	Two modes	1, 1, 2, 3, 3
Multimodal	Three or more modes	1, 1, 2, 2, 3, 3

Additional Summary Statistics

Beyond the three central tendency measures, the calculator also shows:

Statistic	Definition	For dataset 2, 3, 4, 7, 7, 8, 9
Range	Maximum - Minimum	9 - 2 = 7
Sum	Total of all values	40
Count	Number of values	7
Minimum	Smallest value	2
Maximum	Largest value	9

Which Average to Report?

Context	Best Measure	Why
Test scores (symmetric)	Mean	Uses all data, most precise when distribution is normal
House prices	Median	Extreme values (mansions) skew the mean
Salary data	Median	A few high earners distort the mean
Shoe sizes sold	Mode	You want the most popular size to stock
Survey ratings (1-5)	Mode or Median	Ordinal data - mean of 3.7 stars is harder to interpret
Scientific measurements	Mean	Random errors cancel out; mean is the best estimate

Weighted Mean

When some values count more than others, use a weighted mean:

Weighted mean = Σ(value × weight) / Σ(weights)

Example: A course grade with homework (30%), midterm (30%), and final (40%):

• Homework: 85, Midterm: 78, Final: 92
• Weighted mean = (85 × 0.3) + (78 × 0.3) + (92 × 0.4) = 25.5 + 23.4 + 36.8 = 85.7
• The unweighted mean would be (85 + 78 + 92) / 3 = 85.0 - slightly different

How Outliers Distort the Mean

A single extreme value can move the mean by a large amount while leaving the median almost untouched, which is the core reason statisticians calculate both. Consider this salary dataset for a 10-person team:

• Original: 28, 30, 31, 32, 34, 35, 36, 38, 40, 42 (all in £000s)
• Mean = 346 / 10 = £34.6K, Median = (34 + 35) / 2 = £34.5K - almost identical
• Replace the top salary with a founder on £500K: 28, 30, 31, 32, 34, 35, 36, 38, 40, 500
• Mean = 804 / 10 = £80.4K, Median = £34.5K - unchanged

The founder's salary alone shifts the mean by £45.8K, a 132% jump, while the median does not move. This is why "average salary at company X" is almost always quoted as a median on UK job sites and in ONS releases. When data is heavily skewed - income, house prices, insurance claims, website session durations - the median is the more honest summary.

What Does "No Mode" Actually Mean?

A dataset has no mode when every value appears the same number of times, which usually signals that the data is either very small or uniformly distributed. In the set {1, 2, 3, 4, 5}, every value appears once so there is no single most-frequent value. Some textbooks instead say every value is a mode in this situation, but the more common convention in UK GCSE and A-level specifications (AQA, OCR, Edexcel) is to report "no mode". This calculator follows that convention. In practice the mode is only really informative on larger datasets with genuine repetition - for continuous measurements like heights or weights, the raw mode is rarely meaningful and analysts usually bin the data into intervals first and report the modal class instead.

Common Mistakes When Averaging

• Using the mean on skewed data. House prices, salaries, and reaction times are all right-skewed. Report the median instead, or report both and explain the gap.
• Averaging percentages directly. Taking the mean of 50% and 25% only gives 37.5% if both percentages are based on the same denominator. If they are not, you need a weighted mean.
• Averaging ratios or rates. The mean of two speeds covering equal distances is the harmonic mean, not the arithmetic mean. 60 mph out and 30 mph back gives an average speed of 40 mph, not 45.
• Forgetting zeros. If 100 customers visit but only 10 buy, the "average purchase value" depends on whether you include the 90 zero-purchase sessions. Both answers can be valid - just be explicit about which you mean.
• Rounding too early. Rounding each value before summing can introduce error. Keep full precision for intermediate calculations and round only the final answer.
• Confusing sample mean with population mean. The symbol x̄ (x-bar) denotes the sample mean; μ (mu) denotes the population mean. They are computed the same way but are conceptually different, which matters for standard deviation calculations.

Which Average to Use in School and Exams?

GCSE and A-level maths exams expect students to pick the right measure for the context and justify the choice. A quick rule of thumb: if the data has extreme values or a long tail, report the median. If every value matters equally and the distribution looks symmetric, report the mean. If the data is categorical (favourite colour, shoe brand, test grade A/B/C/D), the mode is the only measure that makes sense. For the percentage calculator side of data analysis - converting counts into percentages for reports - keep separate intermediate values so you can cross-check your work.

For standard deviation and variance calculations that build on the mean, the standard deviation calculator takes the analysis further. For a simpler tool focused just on averages, the average calculator handles weighted and unweighted means.

All calculations run in your browser. No data is sent to any server, so the dataset stays on your device.

Sources

• ONS - Average household income, UK: financial year ending 2025
• US Census Bureau - Income in the United States: 2024
• GOV.UK - Households Below Average Income: FYE 1995 to FYE 2025
• Investopedia - Central Tendency: Mean, Median, and Mode
• Wikipedia - Central Tendency
• Khan Academy - Summarising Quantitative Data