IQ scores are approximately normally distributed with a mean of 100 and a standard deviation of 15. What IQ score is about 1.5 standard deviations above the mean?
A. 77.5
B. 120
C. 115
D. 122.5
The Correct Answer and Explanation is:
To determine which IQ score is 1.5 standard deviations above the mean, we need to calculate the value based on the given mean and standard deviation.
Step 1: Understanding the Given Information
The IQ scores are normally distributed with:
- A mean (( \mu )) of 100.
- A standard deviation (( \sigma )) of 15.
In a normal distribution, scores that deviate from the mean by certain standard deviations can be calculated using the formula:
[
\text{Score} = \mu + (z \times \sigma)
]
where ( z ) is the number of standard deviations from the mean.
Step 2: Plugging in the Values
Here, ( z = 1.5 ) (1.5 standard deviations above the mean), ( \mu = 100 ), and ( \sigma = 15 ).
- Calculate the contribution of the standard deviation:
[
1.5 \times 15 = 22.5
] - Add this value to the mean:
[
100 + 22.5 = 122.5
]
So, an IQ score that is 1.5 standard deviations above the mean is 122.5.
Step 3: Selecting the Correct Answer
From the answer choices:
- Option D. 122.5 matches the calculated result.
Explanation of Normal Distribution and Standard Deviations
A normal distribution is a symmetric, bell-shaped curve where most values cluster around the mean, with values becoming less common as they move away from the mean. In IQ scoring, the mean is set at 100, and the spread (standard deviation) is 15. Being “1.5 standard deviations above the mean” means we move 1.5 times the standard deviation (15 points) above 100. This measurement helps us understand relative positioning in the distribution.
In practice, approximately 68% of IQ scores fall within 1 standard deviation of the mean (between 85 and 115), while 95% fall within 2 standard deviations (between 70 and 130). Thus, a score of 122.5 is above average but still within a common range for IQ scores.
Final Answer
The correct answer is D. 122.5.