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Mean, median, mode, and range are the four fundamental measures of central tendency and spread in descriptive statistics. Whether you are a GCSE student in the UK calculating the average of a frequency table, a US middle school student working on data sets, or a data analyst summarising a distribution, these four measures give you a quick and clear picture of your data. Our free calculator computes all four instantly, along with interquartile range (IQR), quartiles, outlier detection, and a complete frequency table.
The mean is calculated by adding all the values together and dividing by the count:
Mean = Sum of all values / Number of values
xΜ = (xβ + xβ + ... + xβ) / n
Example: The exam scores 55, 62, 70, 78, 85 have mean = (55+62+70+78+85)/5 = 350/5 = 70.
The mean is sensitive to outliers. A single very high or very low value can pull the mean significantly. This is why income data is often reported using median rather than mean β a small number of very high earners pull the mean salary well above what most people earn.
UK: The UK Office for National Statistics (ONS) reports that the median annual salary for full-time employees was approximately Β£34,900 in 2024, while the mean was notably higher at around Β£39,000 β illustrating the effect of high earners on the mean.
US: The US Census Bureau similarly shows median household income being consistently lower than mean household income due to the right-skewed distribution of incomes β a classic example of mean vs median in practice.
The median is the middle value when data is sorted in order. It divides the dataset in half β 50% of values are above and 50% below.
Example (odd): Sorted data: 3, 7, 9, 12, 15. Median = 9 (position 3 of 5).
Example (even): Sorted data: 3, 7, 9, 12. Median = (7+9)/2 = 8.
The median is resistant to outliers β adding a value of 1,000,000 to the dataset above would barely change the median but would dramatically increase the mean.
The mode is the value (or values) that appear most frequently in a dataset. A dataset can have:
Example: In the dataset {2, 3, 3, 5, 7, 7, 7, 9}, the mode is 7 (appears 3 times).
Mode is the only measure of central tendency applicable to categorical (non-numeric) data. "The most common hair colour in the class is brown" β that is a mode.
Range = Maximum value β Minimum value. It is the simplest measure of spread:
Example: {4, 8, 15, 16, 23, 42} β Range = 42 β 4 = 38.
Range is highly sensitive to outliers. Remove the 42 and the range drops to 19. The range gives no information about how the middle values are distributed.
The IQR measures the spread of the middle 50% of data, making it much more robust to outliers than the range:
IQR = Q3 β Q1
Quartiles divide sorted data into four equal parts:
The box plot (box-and-whisker diagram) displays Q1, Q2, Q3, and the extent of the whiskers β a visual summary of the five-number summary (minimum, Q1, median, Q3, maximum).
A commonly used rule for identifying outliers:
This is the rule used by Tukey's box plots and is the standard in most statistics courses and software.
| Measure | Best Used When | Sensitive to Outliers? |
|---|---|---|
| Mean | Data is roughly symmetric with no extreme outliers; most statistical calculations | Yes β highly |
| Median | Data is skewed; there are outliers; reporting income, house prices, salaries | No β resistant |
| Mode | Categorical data; finding the most common response; bimodal distributions | No |
| Range | Quick overview of spread; very small datasets | Yes β extremely |
| IQR | Robust spread measure; box plots; outlier detection | No β resistant |
In a perfectly symmetric distribution, mean = median = mode. In skewed distributions:
The direction in which the tail points is the direction of the skew. A right-skewed distribution has a long right tail.
In England, Wales, and Northern Ireland, mean, median, mode, and range are core GCSE Mathematics topics, appearing in Foundation and Higher tier papers. Students are expected to calculate these measures from raw data and from frequency tables. GCSE Statistics goes further to include quartiles, IQR, box plots, and comparing distributions. All major exam boards (AQA, Edexcel, OCR) test these topics frequently.
In the US, measures of central tendency are introduced in Grade 6 Math under Common Core State Standards (6.SP standards). AP Statistics covers all descriptive statistics including quartiles, IQR, outlier rules, and comparison of distributions in Unit 1. The AP exam regularly asks students to compare two distributions using shape, centre (mean/median), and spread (SD/IQR).
UK: According to the UK Land Registry, the mean house price in England in 2024 was significantly higher than the median, pulled up by expensive London properties. The Office for National Statistics recommends the median as a more representative measure of typical house prices.
US: The US Census Bureau's American Housing Survey similarly reports both median and mean home values. The median is the standard reporting figure for housing market health.
Mean = sum of all values Γ· count (the arithmetic average). Median = the middle value when data is sorted. Mode = the most frequently occurring value. All three are measures of the "typical" value in a dataset, but they respond differently to outliers and skewed data.
Use median when data is skewed or contains outliers. Income, salaries, house prices, and healthcare costs are typically reported using median rather than mean because a small number of very high values would make the mean unrepresentative of the typical person's experience.
Yes. If two values appear the same number of times (and more than any other value), the dataset is bimodal and has two modes. If three or more values share the highest frequency, the dataset is multimodal. If all values appear exactly once, there is no mode.
IQR (Interquartile Range) = Q3 β Q1. It measures the spread of the middle 50% of data. Unlike range, it is not affected by outliers, making it a robust measure of spread. It is used in box plots and to define the Tukey fence for outlier detection (1.5 Γ IQR below Q1 or above Q3).
Sort the data in ascending order. If n is even, the median is the average of the two middle values: (value at position n/2 + value at position n/2 + 1) / 2. For {3, 7, 9, 12}: median = (7+9)/2 = 8.
An outlier is a value that is unusually far from the rest of the data. The Tukey rule defines outliers as values below Q1 β 1.5ΓIQR or above Q3 + 1.5ΓIQR. Outliers may be genuine extreme values or data entry errors. Their effect should be considered before deciding whether to include them in analysis.
No. In a symmetric distribution they are equal. In a right-skewed distribution, mean > median. In a left-skewed distribution, mean < median. This is why income data (right-skewed) has mean salary always higher than median salary.
Yes. Mean, median, mode, and range are core GCSE Mathematics topics at both Foundation and Higher tier. GCSE Statistics also covers quartiles, IQR, and box plots. All major UK exam boards (AQA, Edexcel, OCR) test these topics in every exam series.
Results are for educational and informational purposes. Statistical calculations depend on correct data entry. Different software may use slightly different quartile calculation methods.