Examples of zvalue in the following topics:

 A common mistake is to look up a zvalue in the table and simply report the corresponding entry, regardless of whether the problem asks for the area to the left or to the right of the zvalue.
 The table only gives the probabilities to the left of the zvalue.
 There is another note of caution to take into consideration when using the table: The table provided only gives values for positive zvalues, which correspond to values above the mean.
 What if we wished instead to find out the probability that a value falls below a zvalue of 0.51, or 0.51 standard deviations below the mean?
 This table can be used to find the cumulative probability up to the standardized normal value z.



 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).
 Generally, we round Z to two decimals, identify the proper row in the normal probability table up through the ﬁrst decimal, and then determine the column representing the second decimal value.
 We can also ﬁnd the Z score associated with a percentile.
 For example, to identify Z for the 80th percentile, we look for the value closest to 0.8000 in the middle portion of the table: 0.7995.
 We determine the Z score for the 80th percentile by combining the row and column Z values: 0.84.

 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 use of "z" is because the normal distribution is also known as the "z distribution.
 The absolute value of z represents the distance between the raw score and the population mean in units of the standard deviation.
 Define zscores 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:
 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.
 One observation x1 is said to be more unusual than another observation x2 if the absolute value of its Z score is larger than the absolute value of the other observation's Z score: Z1 > Z2.


 For each significance level, the Ztest has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's ttest which has separate critical values for each sample size.
 If T is a statistic that is approximately normally distributed under the null hypothesis, the next step in performing a Ztest is to estimate the expected value θ of T under the null hypothesis, and then obtain an estimate s of the standard deviation of T.
 We then calculate the standard score Z = (T − θ) / s, from which onetailed and twotailed pvalues can be calculated as Φ(−Z) (for uppertailed tests), Φ(Z) (for lowertailed tests) and 2Φ(−Z) (for twotailed tests) where Φ is the standard normal cumulative distribution function.
 To calculate the standardized statistic Z = (X − μ0) / s , we need to either know or have an approximate value for σ2, from which we can calculate s2 = σ2 / n.
 For larger sample sizes, the ttest procedure gives almost identical pvalues as the Ztest procedure.

 On the table of values, find the row that corresponds to 1.5 and the column that corresponds to 0.00.
 However, this is the probability that the value is less than 1.17 sigmas above the mean.
 The difficulty arrises from the fact that our table of values does not allow us to directly calculate P(Z≤1.16).
 This table gives the cumulative probability up to the standardized normal value z.
 Interpret a zscore table to calculate the probability that a variable is within range in a normal distribution

 The computation of a confidence interval on the population value of Pearson's correlation (ρ) is complicated by the fact that the sampling distribution of r is not normally distributed.
 The conversion of r to z' can be done using a calculator.
 This calculator shows that the z' associated with an r of 0.654 is 0.78.
 The Z for a 95% confidence interval (Z.95) is 1.96, as can be found using the normal distribution calculator (setting the shaded area to .95 and clicking on the "Between" button).
 The r associated with a z' of 1.13 is 0.81 and the r associated with a z' of 0.43 is 0.40.