Examples of zvalue in the following topics:

 The zvalue tells us how many standard deviations, or "how many sigmas," the variable is from its respective mean.
 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?
 zvalue (noun) the standardized value of an observation found by subtracting the mean from the observed value, and then dividing that value by the standard deviation; also called zscore
 standard deviation (noun) a measure of how spread out data values are around the mean, defined as the square root of the variance

 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.

 Standard scores are also called zvalues, zscores, normal scores, and standardized variables.
 The use of "Z" is because the normal distribution is also known as the "Z distribution".
 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.
 The Z value measures the sigma distance of actual data from the average and provides an assessment of how offtarget a process is operating.

 A positive zscore represents an observation above the mean, while a negative zscore represents an observation below the mean.
 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.
 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 .
 Student's tstatistic (noun) a ratio of the departure of an estimated parameter from its notional value and its standard error

 Learning Objectives
State how the shape of the sampling distribution of r deviates from normality
Transform r to z'
Compute the standard error of z'
Calculate the probability of obtaining an r above a specified value
Assume that the correlation between quantitative and verbal SAT scores in a given population is 0.60.
 Naturally different samples of 12 students would yield different values of r.
 The greater the value of ρ, the more pronounced the skew.
 The first step is to convert both 0.60 and 0.75 to their z' values, which are 0.693 and 0.973, respectively.
 The standard error of z' for N = 12 is 0.333.

 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 )

 To estimate a population proportion to be within a specific confidence interval, we use the formula: $\hat{p}\pm z^{*}\sqrt{\frac{\hat{p}(1\hat{p})}{n}}$.
 This is computed slightly differently, using the formula $\hat{p}\pm z^{*}\sqrt{\frac{\hat{p}(1\hat{p})}{n}}$, where z* is the upper critical value of the standard normal distribution.
 In the above example, if we wished to calculate p with a confidence of 95%, we would use a Zvalue of 1.960 (found using a critical value table: ), and we would find p to be estimated as 0.52$\pm$0.04896.
 Ttable used for finding z* for a certain level of confidence.
 standard error (noun) A measure of how spread out data values are around the mean, defined as the square root of the variance.

 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 ) .
 The value x comes from a normal distribution with mean µ and standard deviation σ.

 Let the ordered values be denoted by
$\displaystyle{z_{(1)} z_{(2)} z_{(3)} ...
 z_{(n1)} z_{(n)}}$
These n ordered values will play the role of the sample quantiles.
 Therefore, 10% of the distribution is below a z value of 1.28.
 Consider the first ordered value, z_{(1)}.
 What might we expect the value of Φ_{(z(1))} to be?

 If X is a normally distributed random variable and X ∼ N ( µ, σ ) , then the zscore is:
$z = \frac{x \mathrm{\mu}}{\sigma } $ (6.2)
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.
 The values 6 and 6 are within 1 standard deviation of the mean 50.
 The values 12 and 12 are within 2 standard deviations of the mean 50.
 The values 18 and 18 are within 3 standard deviations of the mean 50.