The arithmetic mean is the value obtained by adding all observations and dividing by their count. It answers a balancing question: if the total were redistributed equally, how much would each observation receive? The mean uses every value, which makes it informative but also sensitive to unusually large or small observations.
See how one extreme value moves the balance point
The data set in the interactive view keeps four values fixed and lets the fifth vary. The mean moves with every change, while the median stays at the middle ordered value for a much wider interval. This difference matters when salaries, house prices, waiting times, or other skewed data contain extremes.
Compare mean and median as an outlier changes
Move the last observation and watch which measure of center reacts more strongly.
Worked calculation from the total and the count
For the values 6, 8, 9, 12, and 15, first add them to obtain 50. There are five observations, so the mean is 10. The unit is the same as the original data, not a squared or percentage unit unless the observations themselves use one.
A reverse check multiplies the mean by the count: 10 times 5 reconstructs the total 50. Do not divide by the largest value, the number of gaps, or a rounded count. Blank cells and genuinely missing measurements should not silently become zeros because that changes both the total and the count.
A frequency table is a weighted shortcut
When a value occurs repeatedly, multiply it by its frequency, add those products, and divide by the total frequency. This is the same arithmetic mean as writing every observation separately; the table only compresses the data.
In the example, value 2 occurs three times, value 4 occurs twice, and value 7 occurs once. The weighted total is 21 and the total frequency is 6, giving a mean of 3.5. Frequencies must be nonnegative counts. If the numbers are percentages or importance weights rather than occurrence counts, the method is a weighted mean and the denominator is the sum of the weights.
Recovering a missing observation
If the mean and number of observations are known, multiply them to recover the required total. Subtract the sum of the known observations, and the remainder is the missing value.
Here, five values have mean 14, so their total must be 70. The four known values total 55, leaving 15 for the missing observation. Substituting 15 back into the list confirms that the total and mean are correct.
When the mean is representative, and when it is not
The mean is appropriate for quantitative observations whose total has meaning, especially when the distribution is roughly symmetric and extreme values are not dominating it. It cannot be computed meaningfully for unordered labels such as colors. For strongly skewed data, report the median as well; for the most frequent category, use the mode. The comparison in mean, median, and mode shows how to choose.
Never average averages without their group sizes. A class mean based on 10 students and another based on 30 students must be combined using 10 and 30 as weights. Also keep full precision during intermediate work and round the final mean only as far as the measurement quality allows.
