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Generate Range & Standard Deviation Worksheets

Understand measures of dispersion, such as range and standard deviation, to analyze data spread.

How Spread Out: Range & Standard Deviation

While measures of center describe the typical value, measures of spread describe how far data values are from that center. Range gives the simplest picture — the gap between the highest and lowest values — while standard deviation quantifies the average distance each data point sits from the mean, providing a much richer understanding of variability.

Components of Range & Standard Deviation

This section covers the key measures of spread:

  • Range: The difference between the maximum and minimum values in a dataset: Range = max - min.
  • Variance: The average of the squared differences from the mean, measuring overall dispersion.
  • Standard Deviation: The square root of the variance, returning the measure to the original units of the data.
  • Interpreting Spread: A small standard deviation means data is clustered near the mean; a large one means data is widely scattered.

Examples of Range & Standard Deviation

Range Examples

  • Test scores: 65, 72, 88, 91, 95. Range = 95 - 65 = 30 points.
  • Daily temperatures: 58°F, 62°F, 65°F, 70°F, 72°F. Range = 72 - 58 = 14°F.
  • Two classes both scored an average of 80, but Class A has a range of 10 (tight cluster) while Class B has a range of 40 (wide spread).

Variance Examples

  • Data: 4, 8, 6, 2, 10. Mean = 6. Squared differences: (4-6)² + (8-6)² + (6-6)² + (2-6)² + (10-6)² = 4 + 4 + 0 + 16 + 16 = 40. Variance = 40/5 = 8.
  • Data: 5, 5, 5, 5. Mean = 5. All differences are 0, so variance = 0 (no spread at all).
  • Data: 1, 10, 1, 10. Mean = 5.5. Squared differences sum to 81, variance = 81/4 = 20.25 (high variability).

Standard Deviation Examples

  • From the first variance example (variance = 8): standard deviation = √8 ≈ 2.83.
  • Test scores: 80, 82, 78, 84, 76. Mean = 80. Squared differences: 0 + 4 + 4 + 16 + 16 = 40. Variance = 8. SD = √8 ≈ 2.83 points.
  • Heights in cm: 160, 165, 170, 175, 180. Mean = 170. Squared differences: 100 + 25 + 0 + 25 + 100 = 250. Variance = 50. SD = √50 ≈ 7.07 cm.

Interpreting Spread Examples

  • Machine A produces bolts with a mean of 10 mm and SD of 0.1 mm. Machine B has the same mean but SD of 0.5 mm. Machine A is more consistent.
  • If exam scores have mean 75 and SD of 10, roughly 68% of students scored between 65 and 85 (within one standard deviation).
  • Two basketball players both average 20 points per game. One has SD of 2 (consistent), the other has SD of 8 (highly variable performance).