Mean

The arithmetic mean equals the total of all observations divided by the number of observations.

The equal-share idea

For data $x_1,x_2,\ldots,x_n$, the mean is

Imagine redistributing the total so every observation receives the same amount. That common amount is the mean. The total is preserved: $n\bar{x}=\sum x_i$.

A basic calculation

Keep the numerator and count visible. Most errors come from an incomplete total or wrong number of observations.

The mean as a balance point

Deviations from the mean sum to zero:

For the example, deviations from $9$ are $-3,-1,-1,1,4$, whose sum is $0$. Values below the mean balance values above it. This property leads to variance, standard deviation, and least-squares methods.

Weighted means

Not every value must count equally. If assessments have weights $w_i$, use

A score of $80$ worth $30\%$ and a score of $90$ worth $70\%$ give $0.30(80)+0.70(90)=87$, not the unweighted average $85$.

Frequency tables

When a value $x$ occurs with frequency $f$, its contribution to the total is $fx$:

This avoids rewriting repeated observations. Check that the frequency total equals the number of data points.

Finding a missing value

If the mean and count are known, recover the total first. Five values with mean $12$ must total $60$. If four known values total $47$, the missing value is $60-47=13$.

This approach is more reliable than creating an equation around a memorized average formula without interpreting the total.

Sensitivity to outliers

The mean uses every value, so an extreme observation can pull it strongly. For household incomes or house prices, a small number of very high values may make the mean larger than what is typical. The median may then describe centre more honestly.

An outlier should not be deleted automatically. Investigate whether it is an error, a legitimate rare case, or a meaningful part of the population.

Context and units

The mean has the same unit as the data. A mean time is measured in seconds or minutes, not squared units. Interpret it with population, measurement method, and spread; two groups can share the same mean but differ greatly in consistency.

Common mistakes

Dividing by the wrong count. Count observations, including repeated values.

Averaging averages without weights. Group sizes may differ.

Assuming the mean must be observed. It can lie between data values.

Calling it “typical” without checking shape or outliers. Compare with median and spread.

Quick self-check

• Did I include every observation exactly once?
• Is the divisor the number of observations or total frequency?
• Are weights represented correctly?
• Could an outlier make median a better summary?

Related topics

• Median
• Mode
• Range
• Standard Deviation