# z-value

(noun)

## Definition of z-value

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 z-score

Source: Boundless Learning - CC BY-SA 3.0

## Examples of z-value in the following topics:

• ### Using the Normal Curve

• The z-value tells us how many standard deviations, or "how many sigmas," the variable is from its respective mean.Areas Under the CurveTo calculate the probability that a variable is within a range, we have to find the area under the curve.
• The intersection of the 6th row and 2nd column is 0.6950, which tells us that there is a 69.50% percent chance that a variable is less than 0.51 sigmas (or standard deviations) above the mean.A common mistake is to look up a z-value 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 z-value.
• The table only gives the probabilities to the left of the z-value.
• Since the total area under the curve is 1, all we need to do is subtract the value found in the table from 1.
• What if we wished instead to find out the probability that a value falls below a z-value of -0.51, or 0.51 standard deviations below the mean?
• The normal curve is used to find the probability that a value falls within a certain standard deviation away from the mean.
• ### Estimating a Population Proportion

• 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 Z-value of 1.960 (found using a critical value table: ), and we would find p to be estimated as 0.52$\pm$0.04896.
• ### Change of Scale

• This conversion process is called standardizing or normalizing.Standard scores are also called z-values, z-scores, normal scores, and standardized variables.
• The use of "Z" is because the normal distribution is also known as the "Z distribution".
• They are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with μ = 0 and σ = 1).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.A key point is that calculating z requires the population mean and the population standard deviation, not the sample mean or sample deviation.
• In cases where it is impossible to measure every member of a population, a random sample may be used.The Z value measures the sigma distance of actual data from the average and provides an assessment of how off-target a process is operating.