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The concept of a z-score is fundamental when working with normal distributions. A z-score, often referred to as a standard score, is a way of describing a data point's position relative to the mean of a group of points. The formula to calculate a z-score is \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the data point, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation. By converting scores into z-scores, we can determine how many standard deviations away a particular score is from the mean.

• If a z-score is 0, the data point's score is exactly average or equal to the mean.
• Positive z-scores indicate the score is above the mean, while negative z-scores show it's below the mean.
• This normalization is crucial for comparing scores from different distributions or calculating probabilities.

Understanding z-scores allows you to make sense of how extreme or typical a certain score is within a distribution.

The z-table, also known as the standard normal distribution table, is used to find the probability of a statistic (z-score) falling below or above a certain point in a standard normal distribution.Each z-score corresponds to a unique probability in the z-table. By convention:

• The table shows probabilities for z-scores less than or equal to a given value.
• For probabilities greater than a z-score, subtract the table value from 1.
• Rows and columns intersect to provide the cumulative probability from the left of the standard normal curve up to the z-score.

Using the z-table is a step-by-step process:

• Convert the raw score to a z-score using \( z = \frac{x - \mu}{\sigma} \).
• Locate the corresponding cumulative probability for the z-score in the table.
• For two z-scores, calculate areas for both and subtract to find inter-quartile probability.

This process is the bedrock for calculating probabilities in normally distributed data.

Calculating probability involves using z-scores and the z-table to understand the likelihood of a particular score range within a normal distribution. Here’s a simple breakdown: - **Calculate the z-score:** Start by converting your raw data point into a z-score, which we learned earlier. This tells you how far and in which direction a data point deviates from the mean in terms of standard deviations. - **Look up the probability in the z-table:** Once you have a z-score, find the corresponding probability in the z-table. The z-table gives you cumulative probabilities, which is the probability that a random variable is less than or equal to a given z-score. - **Interpreting the probability:**

• If you're interested in the probability that the score is greater than a specific z-score, subtract the z-table value from 1.
• To find the probability between two z-scores, subtract the probability of the lower z-score from the higher one.
• The subtraction accounts for the area under the curve, which represents the probability.

Understanding these calculations is crucial for quantifying uncertainty and making predictions based on normal distribution.