AI Tutor
Five friends finish their shifts at a cafΓ© and pool their tips: $12, $18, $15, $45, and $10. If they split the total evenly so everyone walks away with the exact same share, how much does each person get?
This fair-share value is called the arithmetic mean β€” the single balance point of a dataset where all individual values are combined and divided equally among all observations.

The Formula and Sigma Notation

In statistics, raw data refers to unprocessed, individual measurements collected directly from an experiment or sample before being grouped. We represent individual data values with the variable xx and the sample size with nn.
To write the sum of all values concisely, mathematicians use capital sigma (Ξ£\Sigma) notation, where βˆ‘x\sum x represents βˆ‘i=1nxi\sum_{i=1}^n x_i (adding every raw value from i=1i=1 to nn).
The sample arithmetic mean, denoted as xΛ‰\bar{x} (pronounced "x-bar"), is the sum of all individual raw observations divided by the sample size nn: xΛ‰=βˆ‘xn\bar{x} = \frac{\sum x}{n}
Why does this formula work as a balance point? When you calculate deviations from the mean, the sum of differences below the mean exactly cancels the sum of differences above it: βˆ‘(xβˆ’xΛ‰)=0\sum (x - \bar{x}) = 0.
How do we apply this step-by-step to our cafΓ© tips data?

Manual Calculation Step-by-Step

First, sum all raw values to find βˆ‘x\sum x: 12+18+15+45+10=10012 + 18 + 15 + 45 + 10 = 100
Next, count the number of observations (n=5n = 5) and divide: xΛ‰=βˆ‘xn=1005=20\bar{x} = \frac{\sum x}{n} = \frac{100}{5} = 20
A common mistake when calculating manually is rounding intermediate sums or dividing by nβˆ’1n - 1 (a denominator used later for sample variance, not the mean). Always divide by the full count nn.