As a subject matter expert in statistics, I can provide a comprehensive explanation of how to determine the number of standard deviations from the mean that corresponds to the 75th percentile.
The standard deviation is a measure of the amount of variation or dispersion in a set of values. A z-score is a way to standardize a value from a data set by calculating how many standard deviations away from the mean that value lies. The percentile is a value that indicates the relative standing of an element within a data set. The 75th percentile, for example, is the value below which 75% of the data falls.
To find out how many standard deviations from the mean is the 75th percentile, we can use the concept of the z-score. The z-score is calculated using the following formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where \( X \) is the value of interest, \( \mu \) is the mean of the data set, and \( \sigma \) is the standard deviation.
The z-score that corresponds to the 75th percentile can be found by looking at the standard normal distribution table, also known as the z-table, or by using statistical software that can calculate it for you. The standard normal distribution is a bell-shaped curve that represents all values falling between negative and positive infinity with a mean (μ) of 0 and a standard deviation (σ) of 1.
According to the standard normal distribution, the z-score for the 75th percentile is approximately 0.674. This means that to be at the 75th percentile, a value must be 0.674 standard deviations above the mean. The formula to find the value (X) that corresponds to a given z-score is:
\[ X = \mu + z \cdot \sigma \]
Substituting the z-score for the 75th percentile:
\[ X = \mu + (0.674) \cdot \sigma \]
This equation tells us that to find the value at the 75th percentile, you take the mean and add to it the product of the z-score (0.674) and the standard deviation (σ) of the data set.
It's important to note that the actual value of X will depend on the specific data set you are working with. The mean (μ) and standard deviation (σ) can vary widely from one data set to another, so the actual value that represents the 75th percentile will also vary.
In summary, the 75th percentile is 0.674 standard deviations above the mean in a normal distribution. To find the exact value that represents the 75th percentile for a specific data set, you would need to know the mean and standard deviation of that data set and apply the formula \( X = \mu + (0.674) \cdot \sigma \).

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The value of z is 0.674. Thus, one must be .674 standard deviations above the mean to be in the 75th percentile. a little algebra demonstrates that X = --+ z --.read more >>