Question

. Exercise 26 proposes modeling IQ scores with $N(100,16).$ What IQ would you consider to be unusually high? Explain.

Short answer

An IQ score higher than 108 is considered unusually high.

Step-by-step solution

Understand the Normal Distribution

The problem states that IQ scores follow a normal distribution, denoted as $N(100,16)$. This implies that the mean (average) IQ score is 100, and the variance is 16. The standard deviation, which is the square root of the variance, is $\sqrt{16}=4$. Consequently, IQ scores are distributed with a mean of 100 and a standard deviation of 4.

Define Unusually High IQ Scores

In a normal distribution, an observation is typically considered unusual if it lies more than 2 standard deviations away from the mean. For IQ scores, this means any score $X$ such that $X>\mu +2\sigma$ is considered unusually high, where $\mu =100$ and $\sigma =4$.

Calculate the Threshold for Unusually High IQ

Calculate $\mu +2\sigma$ to determine the threshold for unusually high IQ scores:$$100+2\times 4=100+8=108$$Therefore, an IQ score higher than 108 is considered unusually high.

Key concepts

IQ Scores

IQ scores are a standardized way of measuring human intelligence. They are typically scored on a scale that has a mean (average) of 100. This means that most people score around this number. A person's IQ score is relative to the average, allowing for a comparison of intellectual abilities among individuals.

IQ tests are designed to have a specific distribution pattern. This pattern follows a bell curve, known as a normal distribution. The design ensures that a certain percentage of the population scores below, at, or above the mean. This statistical arrangement helps to gauge where an individual stands in comparison to the broad population.

Here's what you need to remember about IQ scores:

• A mean IQ score of 100 is considered average intelligence.
• IQ tests are constructed to have a specific distribution pattern.
• Score variability is a natural part of human diversity in intelligence.

The design and scoring of IQ tests allow for a relatively fair comparison across different individuals.

Standard Deviation

Standard deviation is a crucial concept when understanding data variability. It gives insight into how spread out the data values are around the mean. In the context of IQ scores, the standard deviation indicates the typical distance of scores from the average score of 100.

If the IQ scores follow a distribution that is described as $N(100,16)$, the variance is 16. The standard deviation, which is the square root of the variance, is calculated as follows:
$$\sigma =\sqrt{16}=4$$

This tells us that most people's IQ scores will fall within 4 points above or below the mean of 100. This means that:

• An IQ of 96 or 104 is very common.
• The smaller the standard deviation, the less spread out the scores are.
• A larger standard deviation would imply more variability in scores.

Understanding the standard deviation helps us to make informed conclusions about what typical or unusual scores are.

Statistical Thresholds

Statistical thresholds help identify what is considered typical or unusual within a dataset. They set boundaries beyond which data points are seen as significant departures from the average. In terms of IQ scores, the statistical threshold determines what is unusually high or low.

In a normal distribution, a common rule of thumb is to look at scores that lie more than 2 standard deviations away from the mean. For the IQ distribution $N(100,16)$:

• The mean $\mu$ is 100.
• The standard deviation $\sigma$ is 4.

To find the threshold for an unusually high IQ score, calculate:
$$100+2\times 4=100+8=108$$

Any IQ score higher than 108 is considered unusually high. This rule helps in categorizing intelligence levels and identifying individuals with particularly different intelligence from the average population.

Understanding statistical thresholds is crucial for interpreting data results effectively and applying them to real-world scenarios.