Which standard score indicates the number of standard deviations from the mean?

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Standardizing in mathematical statistics
External links
1.6.2 Z-Scores
1.6.3 General Z-Score Properties
1.6.4 Gaussian Z-Score Properties
What is the number of standard deviations from the mean?
What does the z
What is the score of standard deviation?

Compares the various grading methods in a normal distribution. Includes: Standard deviations, cummulative precentages, percentile equivalents, Z-scores, T-scores, standard nine, percent in stanine

Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

In statistics, the standard score, also called the z-score or normal score, is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing.

The standard score indicates how many standard deviations an observation is above or below the mean. It allows comparison of observations from different normal distributions, which is done frequently in research.

The standard score is not the same as the z-factor used in the analysis of high-throughput screening data, but is sometimes confused with it.

Formula

The standard score is:	<math> z = \frac{x - \mu}{\sigma}</math>
where:	x is a raw score to be standardized σ is the standard deviation of the population μ is the mean of the population.

The quantity 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. It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest. But knowing the true standard deviation of a population is often unrealistic except in cases such as standardized testing, where the entire population is measured. In cases where it is impossible to measure every member of a population, the standard deviation may be estimated using a random sample. For example, a population of people who smoke cigarettes is not fully measured.

When a population is normally distributed, the percentile rank may be determined from the standard score and statistical tables.

Standardizing in mathematical statistics

In mathematical statistics, a random variable X is standardized using the theoretical (population) mean and standard deviation:

<math>Z = {X - \mu \over \sigma}</math>

where μ = E(X) is the mean and σ = the standard deviation of the probability distribution of X.

If the random variable under consideration is the sample mean:

then the standardized version is

<math>Z={X-\bar{X}\over\sigma/\sqrt{n}}.</math>

References

Abdi, H. (2007). Z-scores. In N.J. Salkind (Ed.), Encyclopedia of Measurement and Statistics. Thousand Oaks, CA: Sage.

External links

Template:Spoken Wikipedia

Z-Score to percentile conversion table With a given Z-Score, calculate the value's percentile rank.

The z-score is particularly important because it tells you not only something about the value itself, but also where the value lies in the distribution. Typically, for example, if the value is 3 standard deviations above the mean you know it's three times the average distance above the mean and represents one of the higher scores in the sample. On the other hand, if the value is one standard deviation below the mean then you typically know it is on the low end of the midrange of the values from the sample. But, there is much more that is important about z-scores. In fact, the z-score opens the door to doing statistical inference for quantitative variables.

Standard Scores

Practice
Exercise 1:

The z-score tells you the distance the value is above or below the mean in:

No Response
Raw score units
Mean units
Standard deviation units
Interquartile range units

Lesson 1: Summary Measures of Data 1.6 - 2

Biostatistics for the Clinician

1.6.2 Z-Scores

For every value from a sample, a corresonding z-score can be computed. The z-score is simply the signed distance the sample value is from the mean in standard deviations. This statement defining a z-score is represented concisely in the simple formula for computing z-scores you see below (see Figure). In the formula x represents the sample value, the greek letter mu represents the mean, and the greek letter sigma represents the standard deviation.

Z-Score Formula

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 Scores
Practice 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

1.6.3 General Z-Score Properties

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 Scores
Practice Exercise 3:	The mean of the z-scores is equal to: No Response 0 1 100 68

1.6.4 Gaussian Z-Score Properties

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).

Standard Normal, Gaussian, or Bell Curve

Standard Scores
Practice Exercise 4:	The percentage of values that lie within one standard deviation of the mean in a Gaussian distribution is approximately: No Response 2.1% 50% 68% 75% 95%

What is the number of standard deviations from the mean?

The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution: Around 68% of values are within 1 standard deviation of the mean. Around 95% of values are within 2 standard deviations of the mean.

Is z

The Z-score, by contrast, is the number of standard deviations a given data point lies from the mean. For data points that are below the mean, the Z-score is negative. In most large data sets, 99% of values have a Z-score between -3 and 3, meaning they lie within three standard deviations above and below the mean.

What does the z

Z-score indicates how much a given value differs from the standard deviation. The Z-score, or standard score, is the number of standard deviations a given data point lies above or below mean. Standard deviation is essentially a reflection of the amount of variability within a given data set.

What is the score of standard deviation?

A lower standard deviation score means that the measurement is closer to the average or mean, while a high standard deviation score means that the value is further from the average or mean. A negative SDS indicates that the value is below the average or mean and a positive value means it is above the average or mean.