What each statistic means
| Statistic | Formula or rule | What it tells you |
|---|---|---|
| Mean (x̄) | Σx ÷ n | The balance point of the data; pulled toward outliers. |
| Median | Middle value of the sorted list (average of the two middle values if n is even) | The typical value; resistant to outliers. |
| Mode | Most frequent value(s) | The most common outcome; may not exist or may be several values. |
| Range | max − min | Total spread, using only the two extremes. |
| Sample standard deviation (Sx) | √[ Σ(x − x̄)² ÷ (n − 1) ] | Typical distance from the mean when the data is a sample. |
| Population standard deviation (σx) | √[ Σ(x − x̄)² ÷ n ] | Same idea when the data is the whole population. |
| Variance (s², σ²) | Standard deviation squared | Spread in squared units; used in further calculations. |
| Q1, Q3 | Medians of the lower and upper halves | The 25th and 75th percentile boundaries. |
| IQR | Q3 − Q1 | Spread of the middle 50%; 1.5 × IQR beyond Q1 or Q3 is the usual outlier fence. |
| Standard error | Sx ÷ √n | How much the sample mean would vary from sample to sample. |
Worked example
Data: 12, 15, 15, 18, 20, 22, 22, 22, 25, 30 (n = 10)
- Sum = 201, so mean = 201 ÷ 10 = 20.1
- Sorted middle values are 20 and 22, so median = 21
- 22 appears three times, so mode = 22
- Lower half 12, 15, 15, 18, 20 → Q1 = 15; upper half 22, 22, 22, 25, 30 → Q3 = 22; IQR = 7
- Σ(x − x̄)² = 254.9; Sx = √(254.9 ÷ 9) ≈ 5.322; σx = √(254.9 ÷ 10) ≈ 5.049
Load the example in the calculator above to see every value and check your own arithmetic against it.
Sample vs population: which one to report
Textbooks and DeltaMath problems usually say which. “A random sample of 10 students” → sample (Sx). “The heights of all 10 players on the team” → population (σx). When a problem just says “find the standard deviation” without context, most Algebra 2 and introductory Statistics courses expect the sample version, but check the Show Example problem for the convention your assignment uses.
Common mistakes on statistics assignments
- Forgetting to sort before finding the median or quartiles.
- Reporting σx when the problem describes a sample, or vice versa.
- Rounding the mean before using it to compute deviations; keep full precision until the last step.
- Using a spreadsheet quartile function that interpolates, then wondering why Q1 does not match.
- Writing “no mode” when two values tie; list both.