Understanding the Relationship Between Standard Scores and Percentiles to Determine IQ Percentile Rank

Use the normal distribution of IQ​ scores, which has a mean of 125 and a standard deviation of 16​, and the following table with the standard scores and percentiles for a normal distribution to find the indicated quantity.

To find the indicated quantity, we need to understand the relationship between standard scores and percentiles in a normal distribution

To find the indicated quantity, we need to understand the relationship between standard scores and percentiles in a normal distribution. The standard score, also known as a z-score, is used to measure how many standard deviations an individual observation is from the mean. The percentile represents the percentage of scores in a distribution that are below a specific value.

First, we need to convert the given IQ score into a standard score (z-score) using the formula:

z = (X – μ) / σ

Where:
– X is the given IQ score
– μ is the mean of the distribution (125)
– σ is the standard deviation of the distribution (16)

Once you calculate the z-score, you can use the table provided with standard scores and percentiles to find the corresponding percentile. The table typically includes the z-score values and the corresponding percentile or area under the normal curve.

For example, if you want to find the percentile of an IQ score of 140, you would calculate the z-score first:

z = (140 – 125) / 16

After calculating the z-score, you can use the table to find the corresponding percentile or area under the curve associated with that z-score. For instance, if you find that the z-score of 2.00 corresponds to a percentile of 97.72%, it means that the IQ score of 140 is at the 97.72th percentile.

So, to find the indicated quantity, always start by calculating the z-score using the formula mentioned above, and then use the table to find the corresponding percentile.

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