Standard deviation

The preceding page defined variance as an expected squared deviation from the mean. Suppose the modeled voice onset time variable V has variance

\operatorname{Var}(V)=625\text{ ms}^2.

The value 625 is measured in squared milliseconds. A phonetician cannot compare it directly with a 10 ms difference between two category means because the units differ.

The standard deviation is the nonnegative square root of the variance:

\sigma_X \equiv\sqrt{\operatorname{Var}(X)}.

For the voice onset time variable,

\begin{aligned} \sigma_V &=\sqrt{625\text{ ms}^2}\\ &=25\text{ ms}. \end{aligned}

Taking the square root returns the spread summary to the measurement units of V.

Comparing two category spreads

Suppose a second stop category has variance

900\text{ ms}^2.

Its standard deviation is

\sqrt{900\text{ ms}^2}=30\text{ ms}.

The second variable has greater spread whether we compare variance or standard deviation. The square root preserves the ordering of nonnegative values.

The difference between 25 ms and 30 ms is easier to interpret than the difference between 625 and 900 squared milliseconds because it is on the original voice onset time scale.

Computing and reporting in base R

Code

vot_variance <- c(category_a = 625, category_b = 900)
vot_sd <- sqrt(vot_variance)

vot_sd

The results are 25 and 30. The code assumes that the input variances have already been computed on the millisecond scale.

What standard deviation summarizes

Standard deviation is the square root of the expected squared distance from the mean. It is not the average absolute distance from the mean.

Squaring gives greater weight to large deviations. Two distributions may thus have the same average absolute distance and different standard deviations if one places more probability far from its center.

Standard deviation is also not a complete description of distribution shape. Equal means and standard deviations do not guarantee equal tails, symmetry, or modality.

Standard deviation does not determine coverage

A fixed proportion of observations need not fall within one standard deviation of the mean. A mean and standard deviation alone do not determine that proportion.

Coverage statements require additional information about the distribution’s shape. Until a particular shape has been justified, standard deviation should be interpreted as a root mean squared distance, not as a guaranteed interval containing a fixed percentage of outcomes.

Check your understanding

A duration variable has variance 144\text{ ms}^2. What is its standard deviation?
If a log duration variable has variance .16 in squared log units, what is its standard deviation and in which units should it be reported?
Why do variance and standard deviation order two distributions by spread in the same way?
What additional information would be needed before interpreting one standard deviation as a particular coverage interval?

Standard deviation rescales the second central moment. The next page shows how changing the exponent summarizes other features of deviations from the mean.

--- title: "Standard deviation" --- The [preceding page](variance-and-covariance.qmd) defined variance as an expected squared deviation from the mean. Suppose the modeled voice onset time variable $V$ has variance $$ \operatorname{Var}(V)=625\text{ ms}^2. $$ The value 625 is measured in squared milliseconds. A phonetician cannot compare it directly with a 10 ms difference between two category means because the units differ. The [**standard deviation**](https://bruno.nicenboim.me/bayescogsci/ch-intro.html) is the nonnegative square root of the variance: $$ \sigma_X \equiv\sqrt{\operatorname{Var}(X)}. $$ For the voice onset time variable, $$ \begin{aligned} \sigma_V &=\sqrt{625\text{ ms}^2}\\ &=25\text{ ms}. \end{aligned} $$ Taking the square root returns the spread summary to the measurement units of $V$. ## Comparing two category spreads Suppose a second stop category has variance $$ 900\text{ ms}^2. $$ Its standard deviation is $$ \sqrt{900\text{ ms}^2}=30\text{ ms}. $$ The second variable has greater spread whether we compare variance or standard deviation. The square root preserves the ordering of nonnegative values. The difference between 25 ms and 30 ms is easier to interpret than the difference between 625 and 900 squared milliseconds because it is on the original voice onset time scale. ## Computing and reporting in base R ```{r} #| eval: false vot_variance <- c(category_a = 625, category_b = 900) vot_sd <- sqrt(vot_variance) vot_sd ``` The results are 25 and 30. The code assumes that the input variances have already been computed on the millisecond scale. ## What standard deviation summarizes Standard deviation is the square root of the expected squared distance from the mean. It is not the average absolute distance from the mean. Squaring gives greater weight to large deviations. Two distributions may thus have the same average absolute distance and different standard deviations if one places more probability far from its center. Standard deviation is also not a complete description of distribution shape. Equal means and standard deviations do not guarantee equal tails, symmetry, or modality. ## Standard deviation does not determine coverage A fixed proportion of observations need not fall within one standard deviation of the mean. A mean and standard deviation alone do not determine that proportion. Coverage statements require additional information about the distribution's shape. Until a particular shape has been justified, standard deviation should be interpreted as a root mean squared distance, not as a guaranteed interval containing a fixed percentage of outcomes. ## Check your understanding 1. A duration variable has variance $144\text{ ms}^2$. What is its standard deviation? 2. If a log duration variable has variance $.16$ in squared log units, what is its standard deviation and in which units should it be reported? 3. Why do variance and standard deviation order two distributions by spread in the same way? 4. What additional information would be needed before interpreting one standard deviation as a particular coverage interval? Standard deviation rescales the second central moment. The [next page](central-moments.qmd) shows how changing the exponent summarizes other features of deviations from the mean.