Mean and median tell you where the center is. But two datasets can have the same mean and be completely different. You also need to know how spread out the data is — that's where range, variance, standard deviation, and IQR come in.
The Complete Statistics Toolkit
| Statistic | What It Measures | Formula | |-----------|-----------------|---------| | Mean | Center (average) | Σx / n | | Median | Center (middle) | Middle value when sorted | | Mode | Center (most common) | Most frequent value | | Range | Spread (total) | Max − Min | | Variance (σ²) | Spread (average squared deviation) | Σ(x−μ)² / n | | Std Dev (σ) | Spread (in original units) | √Variance | | Q1 | 25th percentile | First quartile | | Q3 | 75th percentile | Third quartile | | IQR | Spread (middle 50%) | Q3 − Q1 |
Full Worked Example
Dataset: 12, 15, 18, 22, 25, 28, 30, 35
Central Tendency
- Mean = (12+15+18+22+25+28+30+35)/8 = 185/8 = 23.125
- Median = (22+25)/2 = 23.5 (even count, average of middle two)
- Mode = No mode (all unique)
Spread
- Range = 35 − 12 = 23
- Min = 12, Max = 35
Variance & Standard Deviation
| x | x − μ | (x − μ)² | |---|-------|----------| | 12 | −11.125 | 123.77 | | 15 | −8.125 | 66.02 | | 18 | −5.125 | 26.27 | | 22 | −1.125 | 1.27 | | 25 | 1.875 | 3.52 | | 28 | 4.875 | 23.77 | | 30 | 6.875 | 47.27 | | 35 | 11.875 | 141.02 |
- Population Variance (σ²) = 432.875/8 = 54.11
- Population Std Dev (σ) = √54.11 = 7.36
- Sample Variance (s²) = 432.875/7 = 61.84
- Sample Std Dev (s) = √61.84 = 7.86
Quartiles
- Q1 = 15 (25th percentile)
- Q3 = 30 (75th percentile)
- IQR = 30 − 15 = 15
Population vs. Sample — Which to Use?
| Type | Divide By | When to Use | |------|----------|-------------| | Population (σ) | n | Data IS the entire population | | Sample (s) | n − 1 | Data is a SAMPLE of a larger population |
Example: If you have ALL 50 students' scores in a class → use population (divide by 50). If you surveyed 50 students out of 500 → use sample (divide by 49).
The n−1 is Bessel's correction — it compensates for the fact that samples tend to underestimate population variance.
The 68-95-99.7 Empirical Rule
For normally distributed data:
| Within | Range | % of Data | |--------|-------|-----------| | 1 std dev of mean | μ ± σ | 68% | | 2 std dev of mean | μ ± 2σ | 95% | | 3 std dev of mean | μ ± 3σ | 99.7% |
For our example (μ = 23.125, σ = 7.36):
- 68% of data between 15.77 and 30.49
- 95% between 8.41 and 37.85
The Trench Truth: If someone reports only the mean, they're hiding half the story. "Average salary ₹6L" tells you nothing. Is everyone near ₹6L (low std dev) or do half earn ₹2L and half ₹10L (high std dev)? Always ask for the standard deviation or at least the range.
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Related: Standard Deviation Calculator · Derivative Calculator · Square Root Calculator
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