A Z-score, also known as a standard score, is a measurement of how many standard deviations a data point is above or below the mean in a normal distribution. It is calculated by subtracting the mean from the data point and then dividing by the standard deviation.
Z-scores are used to calculate probabilities in a normal distribution. A Z-score tells us the likelihood of a data point falling within a certain number of standard deviations from the mean. This can then be converted to a probability using a Z-score table or a statistical software.
A Z-score of 0 indicates that the data point is equal to the mean of the distribution. In other words, it is at the center of the distribution and has a probability of 50% of occurring. This is because the mean of a normal distribution is always at the 50th percentile.
Yes, a Z-score can be negative if the data point is below the mean, and it can be greater than 3 if it is more than three standard deviations above the mean. However, these are considered extreme values and have very low probabilities of occurring in a normal distribution.
Z-scores are used to calculate p-values in hypothesis testing. A p-value is a measure of the probability of obtaining the observed data or more extreme data if the null hypothesis is true. It is compared to a predetermined significance level to determine if the results are statistically significant.