Skip to content
Glacius
StatisticsConcept reference

Sample means

The arithmetic mean is the sum of the observations divided by their number.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. An arithmetic mean shares a sample’s total equally across its observations
  3. Here n counts observations
  4. The mean need not appear in the sample
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01An arithmetic mean shares a sample’s total equally across its observations#

An arithmetic mean shares a sample’s total equally across its observations. Add every value, then divide by how many values there are.

For 2,4,62,4,6, the total is 1212. Three equal shares are 44 each, so the mean is 44.

Observed sample values 2,4,6 each receive equal weight 1/3. Their balance point is the sample mean 4. The weights describe the empirical sample, not an assumed population law.Observed sample values 2,4,6 each receive equal weight 1/3. Their balance point is the sample mean 4. The weights describe the empirical sample, not an assumed population law.
Figure 1Observed sample values 2,4,6 each receive equal weight 1/3. Their balance point is the sample mean 4. The weights describe the empirical sample, not an assumed population law.
Link to this figure ↗Download SVGDownload PNG
Check your reasoning

Sample: 1, 5, 12. Arithmetic mean?

Show answer and explanation
6

18 ÷ 3 = 6.

02Here n counts observations#

For a nonempty sample:

xˉ=x1+⋯+xnn\bar{x}=\frac{x_1+\cdots+x_n}{n}

Here nn counts observations. A repeated value still counts each time it appears.

If 11 occurs twice and 77 once, the sample is 1,1,71,1,7. Its total is 99 across three observations; the mean is 33.

Check your reasoning

Exact counts: value 0 occurs 2 times; value 6 occurs 1 time. Mean?

Show answer and explanation
2

Total 6 across 3 observations: 2.

03The mean need not appear in the sample#

The mean need not appear in the sample. The values 00 and 55 have mean 2.52.5. Negative values and zeros contribute to the same sum-and-count calculation.

Check your reasoning

Sample: 3, 7. “A mean must be an observed value.” Repair the mean.

  1. A3
  2. B5
  3. C7
Show answer and explanation
5

10 ÷ 2 = 5.

Key takeaway

Add every observation, including repeats, and divide by the count. The mean can lie between observed values.

  • Compute a sample’s total per observation.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Sample means.” Math behind ML. /learn/s-mean