The most familiar statistical measure is the arithmetic mean, or average. A second important statistical measure is the standard deviation, which is a measure of how far the data are from the mean. For example, the mean score on the Wechsler IQ test is 100 and the standard deviation is 15. This means that people within one standard deviation of the mean have IQ scores that are 15 points higher or lower than the mean.

 

One year, the mean mathematics score on the ACT test was 20.9 with a standard deviation of 5.3. Write an absolute value equation to find the maximum and minimum scores within one standard deviation of the mean. Let t represent the test score.



To express the situation using an absolute value equation, we need to consider how far a test score \( t \) can deviate from the mean score within one standard deviation. 

The mean mathematics score on the ACT test is \( 20.9 \) and the standard deviation is \( 5.3 \). This means that the test scores within one standard deviation range from \( 20.9 - 5.3 \) to \( 20.9 + 5.3 \), or from \( 15.6 \) to \( 26.2 \).

The absolute value equation represents the distance between any score \( t \) and the mean \( 20.9 \) being less than or equal to the standard deviation \( 5.3 \). The equation is:

\[
|t - 20.9| = 5.3
\]

This equation says that the distance from the test score \( t \) to the mean \( 20.9 \) is 5.3, corresponding to the one standard deviation range. Solving this equation gives you the maximum and minimum scores:

\[
t - 20.9 = 5.3 \quad \text{or} \quad t - 20.9 = -5.3
\]
\[
t = 20.9 + 5.3 = 26.2 \quad \text{(maximum score)}
\]
\[
t = 20.9 - 5.3 = 15.6 \quad \text{(minimum score)}
\]

Thus, the maximum score within one standard deviation of the mean is \( 26.2 \), and the minimum score is \( 15.6 \).

Would you like further clarification or more details?

Here are 5 related questions to explore:
1. What is the significance of standard deviation in a data set?
2. How do you interpret scores that are more than one standard deviation away from the mean?
3. What are other measures of central tendency besides the arithmetic mean?
4. How is the standard deviation calculated from a data set?
5. What is the z-score, and how does it relate to standard deviation?

**Tip**: In statistics, about 68% of the data in a normal distribution lies within one standard deviation from the mean.

One year, the mean mathematics score on the ACT test was 20.9 with a standard deviation of 5.3. Write an absolute value equation to find the maximum and minimum scores within one standard deviation of the mean. Let t represent the test score.

Learn how to calculate the maximum and minimum ACT test scores within one standard deviation of the mean score of 20.9, with a standard deviation of 5.3. This problem explores key concepts in statistics, such as mean, standard deviation, and absolute value equations. Suitable for students in Grades 9-12.

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