deviation

(noun)

For interval variables and ratio variables, a measure of difference between the observed value and the mean.

Related Terms

Examples of 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.
    • The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate.
    • 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

  • Variance and standard deviation

    • We call the distance of an observation from its mean its deviation.
    • Notice that squaring the deviations does two things.
    • The standard deviation is the square root of the variance.
    • The σ2 population variance and for the standard deviation.
    • Usually 70% of the data will be within one stan- dard deviation of the mean and about 95% will be within two standard deviations.

  • 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.

  • The Standard Normal Distribution

    • 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.

  • Summary of Formulas

    • ( x − $\bar{x}$ ) or ( x − µ ) = Deviations from the mean (how far a value is from the mean)
    • ( x − $\bar{x}$)2 or ( x − µ )2 = Deviations squared
    • f ( x − $\bar{x}$ )2 or f ( x − µ )2 = The deviations squared and multiplied by their frequencies
    • value = mean + (#ofSTDEVs)(standard deviation), where #ofSTDEVs = the number of standard deviations

  • Measures of the Spread of the Data

    • The standard deviation is always positive or 0.
    • 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.
    • By graphing your data, you can get a better "feel" for the deviations and the standard deviation.

  • 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.

  • Median and Mean

    • State whether it is the mean or median that minimizes the mean squared deviation
    • The mean is the point on which a distribution would balance, the median is the value that minimizes the sum of absolute deviations, and the mean is the value that minimizes the sum of the squared deviations.
    • You can see that the sum of absolute deviations from the median (20) is smaller than the sum of absolute deviations from the mean (22.8).
    • On the other hand, the sum of squared deviations from the median (174) is larger than the sum of squared deviations from the mean (134.8).
    • Absolute and squared deviations from the median of 4 and the mean of 6.8