What is the standard deviation of the Z distribution?

One very special member of the normal distribution family is called the standard normal distribution, or Z-distribution. In statistics, the Z-distribution is used to help find probabilities and percentiles for regular normal distributions (X). It serves as the standard by which all other normal distributions are measured.

The Z-distribution is a normal distribution with mean zero and standard deviation 1; its graph is shown here. Almost all (about 99.7%) of its values lie between –3 and +3 according to the Empirical Rule. Values on the Z-distribution are called z-values, z-scores, or standard scores. A z-value represents the number of standard deviations that a particular value lies above or below the mean. For example, z = 1 on the Z-distribution represents a value that is 1 standard deviation above the mean. Similarly, z = –1 represents a value that is one standard deviation below the mean (indicated by the minus sign on the z-value). And a z-value of 0 is — you guessed it — right on the mean. All z-values are universally understood.

Three normal distributions, with means and standard deviations of a) 90 and 30; b) 120 and 30; and c) 90 and 10, respectively.

The above figure shows some examples of normal distributions. To compare and contrast the distributions shown here, you first see they are all symmetric with the signature bell shape. Examples (a) and (b) have the same standard deviation, but their means are different; the mean in Example (b) is located 30 units to the right of the mean in Example (a) because its mean is 120 compared to 90. Examples (a) and (c) have the same mean (90), but Example (a) has more variability than Example (c) due to its higher standard deviation (30 compared to 10). Because of the increased variability, most of the values in Example (a) lie between 0 and 180 (approximately), while the most of the values in Example (c) lie only between 60 and 120.

Finally, Examples (b) and (c) have different means and different standard deviations entirely; Example (b) has a higher mean which shifts the graph to the right, and Example (c) has a smaller standard deviation; its data values are the most concentrated around the mean.

Note that the mean and standard deviation are important in order to properly interpret values located on a particular normal distribution. For example, you can compare where the value 120 falls on each of the normal distributions in the above figure. In Example (a), the value 120 is one standard deviation above the mean (because the standard deviation is 30, you get 90 + 1[30] = 120). So on this first distribution, the value 120 is the upper value for the range where the middle 68% of the data are located, according to the Empirical Rule.

In Example (b), the value 120 lies directly on the mean, where the values are most concentrated. In Example (c), the value 120 is way out on the rightmost fringe, 3 standard deviations above the mean (because the standard deviation this time is 10, you get 90 + 3[10] = 120). In Example (c), values beyond 120 are very unlikely to occur because they are beyond the range where the middle 99.7% of the values should be, according to the Empirical Rule.

Now, based on the above figure and the discussion regarding where the value 120 lies on each normal distribution, you can calculate z-values. In Example (a), the value 120 is located one standard deviation above the mean, so its z-value is 1. In Example (b), the value 120 is equal to the mean, so its z-value is 0. Example (c) shows that 120 is 3 standard deviations above the mean, so its z-value is 3.

Given z-scores, now you can take a whole bunch of data like life expectancies and instantly find values for people that express where they rank with respect to others. In other words, the z-score formula gives you a way of normalizing or collapsing the data to a common standard based on how many standard deviations values lie from the mean. To put it another way. Subtracting the value of the mean from each one of the values and dividing each of these differences by its standard deviation parametizes the original distribution so that it has a mean of 0 all the time and a standard deviation of 1. So, given the shape of the distribution, you can build one table for it. In other words, no matter what your data looks like, no matter what the mean value is, you can reduce it to one standard table by reformulating your data using the z-score formula. You then can take all kinds of experiments and build tables for them because you can normalize it or reduce it by doing things like forming a z-value.

Another way to illustrate this is to present the following problem, "Suppose you have two people. One has an IQ of 130 on the WAIS IQ test which has a mean of 100 and a standard deviation of approximately 10. The other has an IQ of 145 on the Stanford Binet IQ test which also has a mean of 100, but has a standard deviation of approximately 15. According to the IQ tests, who is the smartest?" Given no knowledge of statistics the answer is far from obvious. On the other hand, with z-scores you can quickly calculate that each person has an IQ 3 standard deviations above the mean. In other words, you can quickly use z-scores to find that both have approximately the same intelligence.

Standard ScoresPractice
Exercise 2:Z-scores provide a common standard for comparison of different measures.

No Response
True
False

Lesson 1: Summary Measures of Data 1.6 - 3

Biostatistics for the Clinician

Because every sample value has a correponding z-score it is possible then to graph the distribution of z-scores for every sample. The z-score distributions share a number of common properties that it is worthwhile to know. These are summarized below.Properties of Z-Scores

The mean of the z-scores is always 0.
The standard deviation of the z-scores is always 1.
The graph of the z-score distribution always has the same shape as the original distribution of sample values.
The sum of the squared z-scores is always equal to the number of z-score values.
Z-scores above 0 represent sample values above the mean, while z-scores below 0 represent sample values below the mean.

Standard ScoresPractice
Exercise 3:The mean of the z-scores is equal to:

No Response
0
1
100
68

Given the z-score properties above, it is obvious that if the sample values have a Gaussian (normal) distribution then the z-scores will also have a Gaussian distribution. The distribution of z-scores having a Gaussian distribution has a special name because of its fundamental importance in statistics. It is called the standard normal distribution. All Gaussian or normal distributions can be transformed using the z-score formula to the standard normal distribution.

Statisticians know a great deal about the standard normal distribution. Consequently, they also know a great deal about the entire family of Gaussian distributions. All of the previous properties of z-score distributions hold for the standard normal distribution. But, in addition, probability values for all sample values are known and tabled. So, for example, it is known then that for any normal distribution, approximately 68% of values lie within one standard deviation of the mean. Approximately 95% of values lie with 2 standard deviations of the mean. Approximately 2.1% of values lie below 2 standard deviations below the mean. Approximately 2.1% of values lie above 2 standard deviations above the mean. In general, all probabilities associated with the normal distribution have already been computed and are tabled (see Figure below).

What is the standard deviation of the Z distribution quizlet?

True, The mean of the standard normal (z) distribution is 0 and the standard deviation is 1.

Why is the standard deviation of Z scores always 1?

We might have to do a little math to convert our data from one unit of measurement to another, but the thing we are measuring remains unchanged. When we convert our data into z scores, the mean will always end up being zero (it is, after all, zero steps away from itself) and the standard deviation will always be one.