A Z-score is a statistical measure that tells us how many standard deviations a data point is away from the mean. It's calculated using the formula Z equals X minus mu divided by sigma, where X is the individual data point, mu is the population mean, and sigma is the population standard deviation. This standardization allows us to compare data points from different distributions.
Let's break down the Z-score calculation into simple steps. First, find the difference between your data point and the mean. Then, divide this difference by the standard deviation. For example, if we have a data point of 85, a mean of 75, and a standard deviation of 10, we calculate Z equals 85 minus 75, divided by 10, which equals 10 divided by 10, giving us a Z-score of 1.0. This means our data point is exactly one standard deviation above the mean.
Understanding what Z-scores mean is crucial for data analysis. A positive Z-score indicates that the data point is above the mean, while a negative Z-score means it's below the mean. When the Z-score equals zero, the data point is exactly at the mean. The magnitude of the Z-score tells us how far the data point is from the mean - the larger the absolute value, the farther it is from the center of the distribution.
Let's work through a practical example. A student scores 92 on a test where the class average is 78 with a standard deviation of 8. To find the Z-score, we use our formula: Z equals X minus mu divided by sigma. Substituting our values: Z equals 92 minus 78, divided by 8, which gives us 14 divided by 8, resulting in a Z-score of 1.75. This means the student scored 1.75 standard deviations above the class average, indicating an excellent performance.
To summarize what we've learned about Z-scores: They measure how many standard deviations a data point is from the mean using the formula Z equals X minus mu divided by sigma. Positive Z-scores indicate values above average while negative ones show below average performance. Z-scores are essential for comparing data from different distributions and are a fundamental tool in statistics and data analysis.