A data set has mean 74 and standard deviation 8. What z-score corresponds to 90
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Z-scores are standardized values that tell us how many standard deviations a data point is from the mean. In this example, we have a dataset with a mean of 74 and a standard deviation of 8. We want to find the z-score for the value 90, which is represented on this number line.
The formula for calculating a z-score is z equals (x minus mu) divided by sigma. Here, x represents the individual value we're examining, mu is the population mean, and sigma is the standard deviation. This formula standardizes any raw score by expressing it in terms of standard deviations from the mean.
For our specific problem, we've been given these values: the population mean mu equals 74, the standard deviation sigma equals 8, and our value of interest x equals 90. We'll substitute these values into the z-score formula to find our answer.
Let's substitute our values into the formula. We have z equals (90 minus 74) divided by 8. First, we calculate the numerator: 90 minus 74 equals 16. Then we divide by the denominator: 16 divided by 8 equals 2. Therefore, the z-score is 2.
A z-score of 2 tells us that the value 90 is exactly 2 standard deviations above the mean. On a normal distribution curve, this places the value in the upper portion of the distribution. Since about 95% of values fall within 2 standard deviations of the mean, a z-score of 2 represents a relatively high value in our dataset.
To reinforce our understanding, let's calculate z-scores for a few more values in this dataset with mean 74 and standard deviation 8. For x equals 66, we get z equals (66 minus 74) divided by 8, which equals negative 1. For x equals 82, we get z equals (82 minus 74) divided by 8, which equals 1. And for x equals 58, we get z equals (58 minus 74) divided by 8, which equals negative 2. These examples show how the z-score formula consistently transforms raw scores into standardized values.