Examples of zscore in the following topics:

 Thus, a positive $z$score represents an observation above the mean, while a negative $z$score represents an observation below the mean.
 $z$scores are also called standard scores, $z$values, normal scores or standardized variables.
 The conversion of a raw score, $x$, to a $z$score can be performed using the following equation:
 $z$scores for this standard normal distribution can be seen in between percentiles and $t$scores.
 Define $z$scores and demonstrate how they are converted from raw scores

 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:
 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.

 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.
 A zscore is measured in units of the standard deviation.
 x = µ + ( z ) σ = 5 + ( 3 )( 2 ) = 11 (6.1)
 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).
 The zscore for y = 4 is z = 2.
 The zscores are 1 and +1 for 6 and 6, respectively.

 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 zi in Table 3.16.
 The zi 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.

 However, the percentile describes those who had a Z score lower than 0.43.
 TIP: always draw a picture ﬁrst, and ﬁnd the Z score second
 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.
 3.17: Remember: draw a picture ﬁrst, then ﬁnd the Z score.

 Standard scores are also called $z$values, $z$scores, normal scores, and standardized variables.
 The $z$score 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, $Z$scores, $T$scores, and standard nine.

 SAT scores closely follow the normal model with mean µ = 1500 and standard deviation σ = 300. ( a) About what percent of test takers score 900 to 2100?
 (b) What percent score between 1500 and 2100?
 To ﬁnd the area between Z = −1 and Z = 1, use the normal probability table to determine the areas below Z = −1 and above Z = 1.
 Repeat this for Z = −2 to Z = 2 and also for Z = −3 to Z = 3.
 (b) Since the normal model is symmetric, then half of the test takers from part (a) (95% / 2 = 47.5% of all test takers) will score 900 to 1500 while 47.5% score between 1500 and 2100.