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.
 raw score (noun) an original observation that has not been transformed to a zscore
 zscore (noun) The standardized value of observation x from a distribution that has mean and standard deviation .

 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.