A successful basketball player has a height of 6 feet 5 inches, or
196 cm. Based on statistics from a data set, his height converts to the z score of 3.02. How many standard deviations is his height above the mean?
(Round to two decimal places as needed.)
The Correct Answer and Explanation is :
The z-score of 3.02 directly tells us that the basketball player’s height is 3.02 standard deviations above the mean height. Thus, the direct answer to your question is 3.02 standard deviations.
Explanation:
A z-score is a numerical measurement that describes a value’s relationship to the mean of a group of values, measured in terms of standard deviations from the mean. The formula to calculate a z-score is:
[ Z = \frac{(X – \mu)}{\sigma} ]
Where:
- ( Z ) is the z-score.
- ( X ) is the value being measured (in this case, the basketball player’s height).
- ( \mu ) is the mean of the values.
- ( \sigma ) is the standard deviation of the values.
A z-score of 3.02 implies that the basketball player’s height is 3.02 standard deviations above the mean height of the population from which this data set is drawn. Here’s what this tells us in terms of statistics:
- Standard Deviations and Distribution: Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, whereas a high standard deviation means that the values are spread out over a wider range.
- Interpreting the Z-Score:
- A z-score of 0 indicates that the data point’s score is identical to the mean score.
- A z-score of 1.0 indicates a value that is one standard deviation from the mean.
- This player’s z-score of 3.02 indicates that his height is significantly above the average, lying far outside the typical range found in the general population.
- Implications in a Real-World Context: In sports, especially in basketball, being taller can be a significant advantage, contributing to a player’s success. The fact that this player is over three standard deviations taller than the average person could be one of the factors contributing to his success in basketball.
- Statistical Rarity: A z-score of 3.02 suggests a rarity in height, as it is far from the mean. In a normally distributed curve, about 99.7% of values lie within three standard deviations of the mean. This player is thus taller than the vast majority of the population, highlighting how unusual his height is.
In summary, the z-score of 3.02 means the player’s height is quite exceptional in the context of the general population, and this exceptional stature likely provides advantages in the sport of basketball.