Final answer:
The mean is the single score that best represents all the scores in a distribution, often used in probability and statistics as a measure of central tendency. In a standard normal distribution, the mean is zero, which allows for the creation of z-scores for comparison across different data sets.
Step-by-step explanation:
The single score that best represents all the scores in a distribution is the definition of the mean. In probability and statistics, especially when discussing the normal distribution, the mean is the central point around which the data are distributed in a bell-shaped curve.
The mean is the average of all the scores in the distribution and is symbolized as μ (mu) in the context of the normal distribution. It represents the expected value and is used as a measure of central tendency, often along with measures such as median and mode.
In a standard normal distribution, the mean is set to 0, and the standard deviation is set to 1. This transforms the original distribution’s scores into z-scores, which indicate how many standard deviations a score is from the mean. Z-scores are incredibly useful for comparing different data sets that may have different means and standard deviations. A score's specific z-score is found using the formula: z = (x - μ) /σ, where x is the score, μ is the mean of the original distribution, and σ (sigma) is the standard deviation of the original distribution.
It is essential for students to understand the concept of the mean as it helps interpret data sets and apply statistical methods like z-scores. By understanding the mean, students can analyze distributions and use this knowledge in various fields such as business, psychology, and scientific research.