The mean is the most commonly used measure of central tendency in quantitative research, and for good reason: when the conditions are right, it uses every value in your dataset efficiently and forms the basis for a wide range of powerful statistical tests. It has three main types.
- Arithmetic mean
- Weighed mean
- Geometric mean
The Arithmetic Mean
The arithmetic mean is what most people mean when they say ‘the average.’ Add up all values in a dataset and divide by the number of observations.
Formula: Mean (x̄) = (Σx) ÷ n
Where Σx is the sum of all values and n is the number of observations.
Example: Seven students take a statistics test and score: 48, 55, 61, 64, 68, 73, 80.
Sum = 449. n = 7. Mean = 449 ÷ 7 = 64.1 (rounded to 1 decimal place).
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Why Arithmetic Mean Is Preferred For Normally Distributed Data
When data follows a normal (bell-shaped) distribution, the mean, median, and mode are all approximately equal. In this situation, the mean is the most statistically efficient estimator of the population centre, it uses every data point and minimises the average squared distance from the centre, which makes it ideal as the foundation for t-tests, ANOVA, regression, and other parametric methods.
The Weighted Mean
A standard arithmetic mean treats every observation as equally important. A weighted mean assigns different weights to different values before averaging, reflecting their relative importance or frequency.
Formula: Weighted Mean = Σ(w × x) ÷ Σw
Where w is the weight for each value x.
| Scenario | Why Weighted Mean Is Needed | Example |
|---|---|---|
| Module grades | Modules carry different credit weights | Dissertation (60 credits) weighted more than a seminar (20 credits) |
| National income averages | Regional populations differ in size | London’s mean income given higher weight than the Highlands’ |
| Survey responses | Response groups have unequal sizes | Oversampled minority groups re-weighted to population proportions |
| School league tables | Schools have different pupil numbers | Larger schools influence national averages more |
Example: A student’s final grade is 40% coursework, 60% exam. Coursework = 71, Exam = 58. Weighted mean = (71×0.4) + (58×0.6) = 28.4 + 34.8 = 63.2. An unweighted mean of (71+58)÷2 = 64.5 would overstate the final grade because it does not reflect the assessment weighting.
The Geometric Mean
Less commonly taught at undergraduate level but worth knowing: the geometric mean is appropriate when data represents rates, ratios, or percentages, especially growth rates over time.
Formula: Geometric Mean = (x₁ × x₂ × … × xₙ)^(1/n)
It is particularly useful in finance (average investment returns over multiple periods) and biology (population growth rates). When values can span several orders of magnitude, the geometric mean is more representative than the arithmetic mean.
The Biggest Weakness Of Using Mean
The arithmetic mean is influenced by every data point, which is its strength in normal distributions and its weakness when data is skewed or contains outliers.
Common Mistake: If you include a single very high or very low value in your dataset, the mean can move dramatically while the median remains stable. Before reporting the mean as your primary measure, always check your data’s distribution. If your histogram is skewed, or if skewness is above ±1 in SPSS output, consider whether the median is a more honest summary.
| Data Type | Mean Appropriate? | Better Alternative If Not |
|---|---|---|
| Normally distributed interval/ratio | Yes | – |
| Skewed ratio data (income, house prices) | Caution – report alongside median | Median + IQR |
| Ordinal (Likert items, degree grades) | Debated – see note below | Median + non-parametric tests |
| Nominal (gender, blood type) | Never | Mode + frequency table |
For ordinal data specifically, the debate about whether to use the mean is ongoing.
Mean And The Measures Of Variability
The mean rarely appears alone in a results section. It is almost always paired with the standard deviation, which tells your reader how spread out the data is around the mean. If the mean is 64 and the standard deviation is 2, data points cluster tightly. If SD is 20, they are widely scattered.
Reporting The Mean In APA Style
APA 7th edition format:
- In text: M = 64.1, SD = 10.3
- In a table: Mean (M) and standard deviation (SD) in column headers
- When comparing groups: report M and SD for each group before stating the result of your t-test or ANOVA
Student Tip: Always report the mean and standard deviation together. A mean without a standard deviation is like a map without a scale, you know where the centre is, but you have no idea how spread out the territory is.
Frequently Asked Questions
They both are not as much the same as much as being two sides of the same coin. Average is a sum of all the values in a dataset divided by the total number of values. Mean is also the same, theoretically, but it can involve the sum of more than one dataset. It’s a statistical way of describing an average of a sample or dataset.
Median is the middle value of a dataset. It has to be calculated for an even number of values in a dataset, but not for odd-numbered datasets. However, the central tendency in mean always has to be calculated; it’s not a middle value but a central tendency from the OVERALL values in a dataset.
The sample mean (x̄) is calculated from the observations in your study. The population mean (μ) is the true average across the entire population you are studying, which you almost never have direct access to. The sample mean is used to estimate the population mean.
Use the geometric mean when your data represents rates of change, proportions, or multiplicative processes, for example, annual growth rates of GDP, investment returns over multiple years, or biological population growth. For most student research in social sciences, the arithmetic mean is appropriate.
Technically, a single Likert item is ordinal, and the mean is not strictly appropriate. However, for composite Likert scales (multiple items summed into a total score), many researchers treat the total as approximately interval and report the mean. This is common in psychology, education, and health research. Whatever you choose, state your rationale in your methodology chapter.
Exactly what you would expect, the centre of your data is below zero. This is perfectly interpretable for variables that can take negative values, such as temperature in Celsius, a financial loss expressed in pounds, or a change score (post-test minus pre-test score) where some participants declined.
