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Standard Deviation
Measuring Spread
Related Labs
Variance and standard deviation are measures of how spread out data values are around the mean. They help describe whether a dataset is tightly clustered or widely scattered. This matters in science, economics, testing, and engineering because two datasets can have the same average but very different variability. Understanding spread gives a fuller picture than the mean alone.
Variance measures the average squared distance from the mean, while standard deviation is the square root of variance. Because standard deviation is in the same units as the original data, it is often easier to interpret. In a normal distribution, standard deviation also helps describe how much of the data lies near the mean. These ideas are used to compare consistency, detect unusual values, and model uncertainty.
Key Facts
- Population variance: σ^2 = Σ(x - μ)^2 / N
- Sample variance: s^2 = Σ(x - x̄)^2 / (n - 1)
- Population standard deviation: σ = √[Σ(x - μ)^2 / N]
- Sample standard deviation: s = √[Σ(x - x̄)^2 / (n - 1)]
- A larger standard deviation means data are more spread out from the mean.
- For a normal distribution, about 68% of values lie within 1 standard deviation, 95% within 2, and 99.7% within 3.
Vocabulary
- Mean
- The mean is the average value found by adding all data values and dividing by the number of values.
- Variance
- Variance is the average squared distance of data values from the mean.
- Standard deviation
- Standard deviation is the square root of the variance and measures spread in the original units of the data.
- Population
- A population is the complete set of values or individuals being studied.
- Sample
- A sample is a smaller group taken from a population and used to estimate population characteristics.
Common Mistakes to Avoid
- Using n instead of n - 1 for sample variance, which gives a biased estimate of population spread when working from sample data.
- Forgetting to square the deviations before averaging, which is wrong because positive and negative deviations would cancel out.
- Confusing variance with standard deviation, which is wrong because variance is in squared units while standard deviation is in the original units.
- Assuming a larger mean always means a larger standard deviation, which is wrong because center and spread describe different features of a dataset.
Practice Questions
- 1 Find the population variance and population standard deviation of the data set 2, 4, 4, 4, 5, 5, 7, 9.
- 2 A sample of quiz scores is 6, 8, 10, 12. Find the sample mean, sample variance, and sample standard deviation.
- 3 Two classes have the same average test score of 75. Class A has a standard deviation of 4 and Class B has a standard deviation of 12. Explain which class has scores that are more consistent and why.