standard deviation

(noun)

a measure of how spread out data values are around the mean, defined as the square root of the variance

Related Terms

(noun)

shows how much variation or dispersion exists from the average (mean), or expected value

Related Terms

Examples of standard deviation in the following topics:

  • Standard Deviation: Definition and Calculation

    • If the standard deviation were zero, then all men would be exactly 70 inches tall.
    • This quantity is known as the standard deviation.
    • The sample standard deviation, $s$, is a statistic known as an estimator.
    • As mentioned above, most often the standard deviation is estimated using the corrected sample standard deviation (using $N-1$).
    • Dark blue is one standard deviation on either side of the mean.

  • Which Standard Deviation (SE)?

    • Although they are often used interchangeably, the standard deviation and the standard error are slightly different.
    • The standard error is the standard deviation of the sampling distribution of a statistic.
    • However, the mean and standard deviation are descriptive statistics, whereas the mean and standard error describes bounds on a random sampling process.
    • Standard deviation will be unaffected by sample size.
    • This is an example of two sample populations with the same mean and different standard deviations.

  • Areas Under Normal Distributions

    • State the proportion of a normal distribution within 1 standard deviation of the mean
    • Figure 1 shows a normal distribution with a mean of 50 and a standard deviation of 10.
    • Normal distribution with a mean of 100 and standard deviation of 20. 68% of the area is within one standard deviation (20) of the mean (100)
    • Normal distribution with a mean of 50 and standard deviation of 10. 68% of the area is within one standard deviation (10) of the mean (50).
    • A normal distribution with a mean of 75 and a standard deviation of 10. 95% of the area is within 1.96 standard deviations of the mean

  • The Standard Normal Distribution

    • The standard normal distribution is a normal distribution of standardized values called z-scores.
    • A z-score is measured in units of the standard deviation.
    • For example, if the mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard deviations above (or to the right of) the mean.
    • The mean for the standard normal distribution is 0 and the standard deviation is 1.
    • The value x comes from a normal distribution with mean µ and standard deviation σ.

  • Pooled standard deviation estimate (special topic)

    • The pooled standard deviation of two groups is a way to use data from both samples to better estimate the standard deviation and standard error.
    • If s1 and s2 are the standard deviations of groups 1 and 2 and there are good reasons to believe that the population standard deviations are equal, then we can obtain an improved estimate of the group variances by pooling their data:
    • The benefits of pooling the standard deviation are realized through obtaining a better estimate of the standard deviation for each group and using a larger degrees of freedom parameter for the t distribution.
    • Caution: Pooling standard deviations should be done only after careful research
    • A pooled standard deviation is only appropriate when background research indicates the population standard deviations are nearly equal.

  • Variance and standard deviation

    • Here, we introduce two measures of variability: the variance and the standard deviation.
    • The standard deviation is defined as the square root of the variance:
    • The standard deviation is the square root of the variance.
    • The σ2 population variance and for the standard deviation.
    • In the num char data, 41 of the 50 emails (82%) are within 1 standard deviation of the mean, and 47 of the 50 emails (94%) are within 2 standard deviations.

  • Normal distribution model

    • Specifically, the normal distribution model can be adjusted using two parameters: mean and standard deviation.
    • Figure 3.2 shows the normal distribution with mean 0 and standard deviation 1 in the left panel and the normal distributions with mean 19 and standard deviation 4 in the right panel.
    • If a normal distribution has mean µ and standard deviation σ, we may write the distribution as N(µ,σ).
    • Write down the short-hand for a normal distribution with (a) mean 5 and standard deviation 3, (b) mean -100 and standard deviation 10, and (c) mean 2 and standard deviation 9.
    • The normal distribution with mean 0 and standard deviation 1 is called the standard normal distribution.

  • Measures of the Spread of the Data

    • The standard deviation is always positive or 0.
    • At market A, the standard deviation for the waiting time is 2 minutes; at market B the standard deviation for the waiting time is 4 minutes.
    • The deviations are used to calculate the standard deviation.
    • You can think of the standard deviation as a special average of the deviations.
    • It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean.

  • Estimating the Accuracy of an Average

    • The standard error of the mean is the standard deviation of the sample mean's estimate of a population mean.
    • In general terms, the standard error is the standard deviation of the sampling distribution of a statistic.
    • Generally, the SEM is the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size:
    • Where $\sigma$ is the standard deviation of the population.
    • Note that the standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and deviations because the standard error of the mean is a biased estimator of the population standard error.

  • Standard Error

    • The standard error is the standard deviation of the sampling distribution of a statistic.
    • Quite simply, the standard error is the standard deviation of the sampling distribution of a statistic.
    • $s$ is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
    • The standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and deviations.
    • The standard error and standard deviation are often considered interchangeable.