Examples of zscore in the following topics:

 If the observation is one standard deviation above the mean, its Z score is 1.
 If it is 1.5 standard deviations below the mean, then its Z score is 1.5.
 Using µSAT = 1500, σSAT = 300, and xAnn = 1800, we ﬁnd Ann's 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.

 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.
 The conversion of a raw score, x, to a zscore can be performed using the following equation:
 Zscores for this standard normal distribution can be seen in between percentiles and tscores.
 Define zscores and demonstrate how they are converted from raw scores.

 To find the kth percentile when the zscore is known: k = µ + ( z ) σ

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

 If X is a normally distributed random variable and X ∼ N ( µ, σ ) , then the zscore is:
 The zscore tells you how many standard deviations that the value x is above (to the right of) or below (to the left of) the mean, µ.
 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 allows us to compare data that are scaled differently.

 If the observations are normally distributed, then their Z scores will approximately correspond to their percentiles and thus to the zi in Table 3.16.

 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.

 Cumulative SAT scores are approximated well by a normal model, N(µ = 1500,σ = 300).
 The simplest way to ﬁnd the shaded area under the curve makes use of the Z score of the cut oﬀ value.
 With µ = 1500, σ = 300, and the cutoﬀ value x = 1630, the Z score is computed as
 However, the percentile describes those who had a Z score lower than 0.43.
 The probability Shannon scores at least 1630 on the SAT is 0.3336.

 Standard scores are also called zvalues, zscores, normal scores, and standardized variables.
 The zscore is only defined if one knows the population parameters.
 The absolute value of z represents the distance between the raw score and the population mean in units of the standard deviation.
 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.

 This will be useful in a wide range of practical settings, especially when trying to make a quick estimate without a calculator or Z table.