Mathwords logoReference LibraryMathwords

Standard Deviation

Standard deviation is a number that tells you how spread out the values in a data set are from the mean. A small standard deviation means values are close together; a large one means they are more spread out.

The standard deviation is the square root of the average squared distance of each data value from the mean. For a population of N values, it is calculated using the formula below. A standard deviation of zero means all values are identical. The further values tend to stray from the mean, the larger the standard deviation.

Key Formula

σ=1N∑i=1N(xi−μ)2\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}
Where:
  • σσ = the population standard deviation
  • NN = the number of values in the data set
  • xix_i = each individual data value
  • μμ = the mean of the data set

Worked Example

Problem: Find the standard deviation of the data set: 2, 4, 4, 6, 9.
Step 1: Find the mean of the data set.
μ=2+4+4+6+95=255=5\mu = \frac{2 + 4 + 4 + 6 + 9}{5} = \frac{25}{5} = 5
Step 2: Subtract the mean from each value and square the result.
(2−5)2=9,(4−5)2=1,(4−5)2=1,(6−5)2=1,(9−5)2=16(2-5)^2 = 9,\quad (4-5)^2 = 1,\quad (4-5)^2 = 1,\quad (6-5)^2 = 1,\quad (9-5)^2 = 16
Step 3: Find the mean of those squared differences (this is the variance).
Variance=9+1+1+1+165=285=5.6\text{Variance} = \frac{9 + 1 + 1 + 1 + 16}{5} = \frac{28}{5} = 5.6
Step 4: Take the square root of the variance to get the standard deviation.
σ=5.6≈2.37\sigma = \sqrt{5.6} \approx 2.37
Answer: The standard deviation is approximately 2.37, meaning the values typically differ from the mean by about 2.37 units.

Visualization

Why It Matters

Standard deviation appears throughout statistics and science because it gives a single number summarising how consistent or variable a data set is. In AP Stats, it underpins confidence intervals, hypothesis tests, and the normal distribution. In everyday contexts, it is used to compare things like test score consistency across classes or variation in manufacturing quality.

Common Mistakes

Mistake: Forgetting to square root at the end, leaving the answer as the variance.
Correction: The variance and standard deviation are different measures. Standard deviation is always the square root of the variance, so the final step is essential.
Mistake: Confusing population standard deviation (÷N) with sample standard deviation (÷(n−1)).
Correction: When your data is a sample taken from a larger population, divide by (n−1) instead of n. This gives the sample standard deviation, written s rather than σ.

Related Terms

  • Mean — The mean is the center point standard deviation measures spread around
  • Variance — Standard deviation is the square root of the variance
  • Interquartile Range — Another measure of spread, based on quartiles instead of the mean