A population distribution is standardized to produce a new …

A population distribution is standardized to produce a new distribution. What is the standardized distribution called? What information is provided by the sign (+/-) of a -score? What information is provided by the numerical value of the -score?

The standardized distribution that results from standardizing a population distribution is commonly referred to as the z-distribution. This distribution is characterized by a mean of zero and a standard deviation of one. The z-distribution is widely used in inferential statistics and serves as a basis for various statistical tests and confidence interval estimations.

The sign (+/-) of a z-score provides information about the relative position of an observation within the distribution compared to the mean. A positive z-score indicates that the observation is above the mean, while a negative z-score indicates that the observation is below the mean. The sign of the z-score enables us to determine whether an observation is above or below the population mean, thereby providing information about its relative position within the distribution.

The numerical value of the z-score provides information about the distance between an observation and the mean of the distribution. It represents the number of standard deviations an observation is away from the mean. If a z-score has a value of 1, it means that the observation is exactly one standard deviation away from the mean. A z-score of 2 indicates that the observation is two standard deviations away, and so on.

By using the numerical value of the z-score, we can determine the percentile rank or probability associated with a specific observation in the population distribution. The z-score is often used to calculate probabilities by referring to a standard normal distribution known as the standard normal table or z-table. This table provides the area under the curve of the standard normal distribution for different z-score values. By referencing the z-table, one can find the proportion or probability associated with a specific z-score, which allows for various statistical analyses and hypothesis testing.

The z-score also enables us to compare observations from different distributions with varying means and standard deviations. By standardizing the data using z-scores, we can remove the differences in scale and easily compare observations from different populations or variables. This standardization facilitates the comparison and interpretation of data, particularly in fields such as psychology, economics, and social sciences.

Moreover, the z-score allows for the identification of outliers or extreme observations in a distribution. Since the z-score measures the distance between an observation and the mean in terms of standard deviations, observations with high absolute z-scores indicate values that deviate significantly from the mean. These extreme values can be considered potential outliers or influential data points that may have a substantial impact on the analysis or interpretation of the data.

In conclusion, the standardized distribution resulting from standardizing a population distribution is known as the z-distribution. The sign (+/-) of a z-score provides information about the relative position of an observation in relation to the mean, while the numerical value indicates the distance of the observation from the mean in terms of standard deviations. The z-score is used to calculate probabilities, compare observations from different distributions, identify outliers, and facilitate statistical analyses.