How to Find the Percentile Rank of an IQ Score Using Standard Scores and Percentiles

Use the normal distribution of IQ​ scores, which has a mean of 85 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 use the standard scores and percentiles for a normal distribution in conjunction with the given mean and standard deviation of the IQ scores

To find the indicated quantity, we need to use the standard scores and percentiles for a normal distribution in conjunction with the given mean and standard deviation of the IQ scores.

The standard score, also known as the z-score, represents the number of standard deviations an individual score is from the mean. It is calculated using the formula:

z = (x – μ) / σ,

where:
– z is the standard score,
– x is the individual score,
– μ is the population mean,
– σ is the population standard deviation.

The table with standard scores and percentiles can be used to determine the percentage of scores below a certain standard score (z) or the percentile associated with a specific z-score.

Now, let’s demonstrate how to use this information to find the indicated quantity.

Example: Find the percentile rank of an IQ score of 105.

1. Calculate the standard score (z-score):
z = (x – μ) / σ,
z = (105 – 85) / 16,
z = 20 / 16,
z ≈ 1.25.

2. Using the standard scores and percentiles table, locate the closest value to 1.25 under the “z” column. From the table, we can see that the closest value is 1.3, with a corresponding percentile of 90.32%.

3. Since the given standard score is between 1.2 and 1.3, we can interpolate to find a more accurate percentile rank. To do this, we need to determine the proportion between the two closest percentiles:
Proportion = (1.25 – 1.2) / (1.3 – 1.2) = 0.25 / 0.1 = 2.5.

4. Apply the proportion to the difference between the percentiles:
Difference = 90.32% – 89.52% = 0.8%.

5. Finally, calculate the interpolated percentile:
Interpolated Percentile = 89.52% + (0.8% * 2.5) = 89.52% + 2% = 91.52%.

Therefore, an IQ score of 105 corresponds to approximately the 91.52 percentile. This means that approximately 91.52% of individuals have an IQ score below 105.

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