A data set has mean 70 and standard deviation 8. What z-score corresponds to 86
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A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. Z-scores are also known as standard scores. They help us understand how typical or unusual a value is within a dataset.
The z-score formula is z equals the quantity x minus mu, divided by sigma. Here, x represents the individual data value we're examining. Mu is the population mean, which is the average of all data points. Sigma represents the standard deviation, which measures how spread out the data is. By subtracting the mean from our data point and dividing by the standard deviation, we standardize the value.
Now let's look at the specific information given in our problem. We have a dataset with a mean of 70 and a standard deviation of 8. We want to find the z-score that corresponds to the value 86. On this normal distribution curve, you can see the mean at 70, and our data point at 86. The question is asking how many standard deviations 86 is from the mean of 70.
Now we'll calculate the z-score step by step. First, we write down the formula: z equals x minus mu divided by sigma. Next, we substitute our values: z equals 86 minus 70 divided by 8. Then we compute the numerator: 86 minus 70 equals 16. So now we have z equals 16 divided by 8. Finally, we divide 16 by 8 to get 2. Therefore, the z-score corresponding to 86 is 2.
A z-score of 2 has important implications. It means that our value of 86 is exactly 2 standard deviations above the mean. In a normal distribution, this is relatively uncommon - only about 2.3 percent of data points are higher than this value. Approximately 97.7 percent of all data falls below a z-score of 2. This tells us that 86 is quite high compared to the rest of the dataset.
Let's reinforce our understanding with two more examples using the same dataset with mean 70 and standard deviation 8. First, for x equals 62: z equals 62 minus 70 divided by 8, which is negative 8 divided by 8, giving us a z-score of negative 1. This means 62 is one standard deviation below the mean. Second, for x equals 78: z equals 78 minus 70 divided by 8, which is 8 divided by 8, giving us a z-score of 1. This means 78 is one standard deviation above the mean. Notice that negative z-scores indicate values below the mean, while positive z-scores indicate values above the mean.
Let's summarize our key points about z-scores. The z-score formula is z equals x minus mu divided by sigma. Z-scores measure how many standard deviations a data point is from the mean. Negative z-scores indicate values below the mean, while positive z-scores indicate values above the mean. Z-scores are powerful because they standardize data, enabling comparisons across different datasets. In our original problem with mean 70 and standard deviation 8, the z-score for 86 is 2, meaning it's exactly 2 standard deviations above the mean.