Examples of zscore in the following topics:

 Define zscores and demonstrate how they are converted from raw scores.
 A positive zscore represents an observation above the mean, while a negative zscore represents an observation below the mean.
 Zscores provide an assessment of how offtarget a process is operating.
 Thus, a positive zscore represents an observation above the mean, while a negative zscore represents an observation below the mean.
 Zscores are also called standard scores, zvalues, normal scores or standardized variables.
 zscore (noun) The standardized value of observation x from a distribution that has mean μ and standard deviation σ.
 raw score (noun) an original observation that has not been transformed to a zscore

 If it is 1.5 standard deviations below the mean, then its Z score is 1.5.
 We compute the Z score for an observation x that follows a distribution with mean µ and standard deviation σ using: Z=(xµ)/σ
Use Tom's ACT score, 24, along with the ACT mean and standard deviation to compute his Z score.
 Observations above the mean always have positive Z scores while those below the mean have negative Z scores.
 SAT score of 1500), then the Z score is 0.
 ( a) Find the Z score of x.

 X ∼ N ( µ,σ )
µ = the mean σ = the standard deviation
Z ∼ N ( 0,1 )
z = a standardized value (zscore)
mean = 0 standard deviation = 1
To find the kth percentile when the zscore is known: k = µ + ( z ) σ
$z = \frac{x \mathrm{\mu}}{\sigma }$
Formula 6.5: Finding the area to the left
The area to the left: P ( X x )
Formula 6.6: Finding the area to the right
The area to the right: P ( X x ) = 1 − P ( X x )

 The standard normal distribution is a normal distribution of standardized values called zscores.
 A zscore is measured in units of the standard deviation.
 The calculation is:
x = µ + ( z ) σ = 5 + ( 3 )( 2 ) = 11 (6.1)
The zscore is 3.
 The transformation z = (x − µ)/σ produces the distribution Z ∼ N ( 0,1 ) .

 Values of x that are larger than the mean have positive zscores and values of x that are smaller than the mean have negative zscores.
 If x equals the mean, then x has a zscore of 0.
 The zscore when x = 10 pounds is z = 2.5 (verify).
 (6.6)
The zscore for y = 4 is z = 2.
 The zscores are 1 and +1 for 6 and 6, respectively.

 Identify the Z score corresponding to each percentile.
 Create a scatterplot of the observations (vertical) against the Z scores (horizontal).
 If the observations are normally distributed, then their Z scores will approximately correspond to their percentiles and thus to the z_{i} in Table 3.16.
 To create the plot based on this table, plot each pair of points, (z_{i},x_{i}).
 The z_{i} in Table 3.16 are not the Z scores of the observations but only correspond to the percentiles of the observations.

 Example 3.7 Ann from Example 3.2 earned a score of 1800 on her SAT with a corresponding Z = 1.
 A normal probability table, which lists Z scores and corresponding percentiles, can be used to identify a percentile based on the Z score (and vice versa).
 We use this table to identify the percentile corresponding to any particular Z score.
 We can also ﬁnd the Z score associated with a percentile.
 We determine the Z score for the 80th percentile by combining the row and column Z values: 0.84.

 For the following data, plot the theoretically expected z score as a function of the actual z score (a QQ plot).
3.

 However, the percentile describes those who had a Z score lower than 0.43.
 Finally, the height x is found using the Z score formula with the known mean µ, standard deviation σ, and Z score Z = 0.92.
 Finally, the height x is found using the Z score formula with the known mean µ, standard deviation σ, and Z score Z = 0.92.
 (b) Z_{Jim} = (76−7)/3.3 = 1.82 → 0.9656.
3.17: Remember: draw a picture ﬁrst, then ﬁnd the Z score.
 Knowing Z_{95} = 1.65, µ = 1500, and σ = 300, we setup the Z score formula: 1.65 = (x_{95}−1500) / 300.

 Standard scores are also called zvalues, zscores, normal scores, and standardized variables.
 The zscore is only defined if one knows the population parameters.
 The standard score of a raw score x is:
$z=\frac { x\mu }{ \sigma }$,
where μ is the mean of the population, and is the standard deviation of the population.
 z is negative when the raw score is below the mean, positive when above.
 Includes: standard deviations, cumulative percentages, percentile equivalents, Zscores, Tscores, and standard nine.
 standard score (noun) The number of standard deviations an observation or datum is above the mean.