Praxis Core Academic Skills for Educators: Mathematics (5733)Statistics and ProbabilityHard
A researcher collects data on the heights of 100 individuals and finds that the mean height is 170 cm with a standard deviation of 5 cm. If the data is normally distributed, approximately what percentage of individuals have a height between 165 cm and 175 cm?
- A34%
- B99.7%
- C95%
- D68%
Show answer & explanationAnswer & explanation
Correct answer: D. 68%
For a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. The mean is 170 cm and the standard deviation is 5 cm. Therefore, one standard deviation below the mean is 170 - 5 = 165 cm, and one standard deviation above the mean is 170 + 5 = 175 cm. So, 68% of individuals have heights between 165 cm and 175 cm.
Why the other options are wrong
- A. This is approximately the percentage within one standard deviation on one side of the mean (e.g., 170 to 175 cm).
- B. This is the approximate percentage within ±3 standard deviations of the mean.
- C. This is the approximate percentage within ±2 standard deviations of the mean.
Empirical Rule (68-95-99.7 Rule)
For a normal distribution, approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.
- Applies specifically to data that follows a normal (bell-shaped) distribution.
- Standard deviation (σ) measures the spread of data around the mean (μ).
- Used to estimate probabilities or proportions within certain ranges without complex calculations.
Memory trick: Normal bell curve, 68-95-99.7 for the steps.