We can convert any normal distribution into the standard normal distribution in order to find probability and apply the properties of the standard normal. In order to do this, we use the z-value. Show Z-value, Z-score, or Z The Z-value (or sometimes referred to as Z-score or simply Z) represents the number of standard deviations an observation is from the mean for a set of data. To find the z-score for a particular observation we apply the following formula: \(Z = \dfrac{(observed\ value\ - mean)}{SD}\) Let's take a look at the idea of a z-score within context. For a recent final exam in STAT 500, the mean was 68.55 with a standard deviation of 15.45.
Is it always good to have a positive Z score? It depends on the question. For exams, you would want a positive Z-score (indicates you scored higher than the mean). However, if one was analyzing days of missed work then a negative Z-score would be more appealing as it would indicate the person missed less than the mean number of days. Characteristics of Z-scores
From Z-score to ProbabilityFor any normal random variable, if you find the Z-score for a value (i.e standardize the value), the random variable is transformed into a standard normal and you can find probabilities using the standard normal table. For instance, assume U.S. adult heights and weights are both normally distributed. Clearly, they would have different means and standard deviations. However, if you knew these means and standard deviations, you could find your z-score for your weight and height. You can now use the Standard Normal Table to find the probability, say, of a randomly selected U.S. adult weighing less than you or taller than you.
How do you find the probability of a standard normal random variable z?The probability that a standard normal random variables lies between two values is also easy to find. The P(a < Z < b) = P(Z < b) - P(Z < a). For example, suppose we want to know the probability that a z-score will be greater than -1.40 and less than -1.20.
What is the probability of Z value?Examine the table and note that a "Z" score of 0.0 lists a probability of 0.50 or 50%, and a "Z" score of 1, meaning one standard deviation above the mean, lists a probability of 0.8413 or 84%.
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