Question

Consider the IQ model \(N(100,16)\) one last time. a) What IQ represents the 15 th percentile? b) What IQ represents the 98 th percentile? c) What's the IQR of the IQs?

Short answer

15th percentile: 95.84, 98th percentile: 108.2, IQR: 5.4.

Step-by-step solution

Understand the Problem

We have a normal distribution of IQ scores with a mean of 100 and a variance of 16. This means the standard deviation is \(\sigma = \sqrt{16} = 4\). We are tasked with finding IQ scores that correspond to certain percentiles.

Find the 15th Percentile IQ

To find the 15th percentile, we will use the Z-score formula for a normal distribution: \(Z = \frac{X - \mu}{\sigma}\). We need to determine the Z-score for the 15th percentile using a standard normal distribution table or calculator. The Z-score corresponding to the 15th percentile is approximately \(-1.04\). Now apply it to the formula: \(X = \mu + Z \cdot \sigma = 100 + (-1.04) \cdot 4 = 100 - 4.16 = 95.84\). The 15th percentile IQ is approximately 95.84.

Find the 98th Percentile IQ

To find the 98th percentile, we use the Z-score for the 98th percentile which is approximately \(2.05\). Plug this Z-score into our formula: \(X = \mu + Z \cdot \sigma = 100 + 2.05 \cdot 4 = 100 + 8.2 = 108.2\). The 98th percentile IQ is approximately 108.2.

Calculate the Interquartile Range (IQR)

IQR is the difference between the 75th percentile (Q3) and the 25th percentile (Q1). First, find the Z-score for the 25th percentile, which is \(-0.675\), and the Z-score for the 75th percentile, which is \(0.675\). Calculate these IQ values: \(Q1 = 100 + (-0.675) \cdot 4 = 97.3\) and \(Q3 = 100 + 0.675 \cdot 4 = 102.7\). Thus, the IQR is \(Q3 - Q1 = 102.7 - 97.3 = 5.4\).

Key concepts

Percentiles

Percentiles are a way to understand and interpret where a particular value stands in a distribution of data. In the context of a normal distribution like the IQ scores we've been working with, a percentile tells you what percentage of the population scores below a certain value. For example, the 15th percentile indicates that 15% of people have an IQ score below this value.
To find the IQ at the 15th percentile, you'd use the Z-score table to find which Z-score corresponds to 15%. In our example, it is \(-1.04\).
Plug it into the formula, \(X = \mu + Z \cdot \sigma\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation. You calculate: \(100 + (-1.04) \cdot 4 = 95.84\) IQ points for the 15th percentile.
Similarly, for the 98th percentile, meaning 98% of people have an IQ below that score, you find a Z-score of \(2.05\). This calculation yields \(108.2\) IQ points for the 98th percentile.

Standard Deviation

Standard deviation (SD) is a key concept in stats that tells us how much variation there is in a set of data. In a normal distribution, most values cluster around the mean, and the standard deviation quantifies how spread out the numbers are.
A smaller SD means the numbers are closer to the mean. A larger SD indicates more spread. For the IQ model \(N(100,16)\), the standard deviation is \(\sigma = \sqrt{16} = 4\). This means most IQ scores are within 4 points above or below the mean of 100.
In practice, the standard deviation helps us understand how many data points lie within certain distances from the mean. About 68% of the data lies within one SD, \(68\%\) between \(96\) to \(104\) IQ points. \(95\%\) are within two SDs (\(92\) to \(108\) IQ points).
This makes SD a critical component in calculating Z-scores and percentiles.

Interquartile Range (IQR)

The Interquartile Range (IQR) is a measure of statistical dispersion and a better indicator of variability than the range because it is less affected by outliers. IQR is the difference between the first quartile \((Q1)\) and the third quartile \((Q3)\), essentially covering the middle 50% of the data.
For our IQ example, we calculated \(Q1\) at the 25th percentile and \(Q3\) at the 75th percentile. First, find their Z-scores, \(-0.675\) for \(Q1\) and \(0.675\) for \(Q3\). Use those to calculate the IQ values:

• \(Q1 = 100 + (-0.675) \cdot 4 = 97.3\)
• \(Q3 = 100 + 0.675 \cdot 4 = 102.7\)

The IQR is \(Q3 - Q1 = 102.7 - 97.3 = 5.4\), meaning the middle half of IQ scores fall within this 5.4-point range.
Utilizing IQR helps analysts understand the core spread of data and is particularly useful when identifying outliers.