A successful basketball player has a height of 6 feet 9 inches, or 206 cm. Based on statistics from a data set, his height converts to the z score of 4.45. How many standard deviations is his height above the mean? The player’s height is ? standard deviation(s) above the mean.
The Correct Answer and Explanation is:
The z-score is a measure of how many standard deviations a particular value is from the mean. The formula to calculate the z-score is:z=X−μσz = \frac{X – \mu}{\sigma}z=σX−μ
Where:
- XXX is the data point (the player’s height in this case),
- μ\muμ is the mean height of the dataset,
- σ\sigmaσ is the standard deviation.
In this case, the z-score for the player’s height is given as 4.45, which means the player’s height is 4.45 standard deviations above the mean. To clarify:
- A z-score of 4.45 means the player’s height is 4.45 times the size of the standard deviation above the average height of players in the dataset.
- The z-score formula indicates that for each standard deviation, the player’s height deviates from the mean in a certain proportion.
Since the z-score is positive (4.45), we know the player’s height is above the average height of the dataset.
Explanation:
A z-score represents the number of standard deviations a value is from the mean. A z-score of 4.45 indicates that the basketball player’s height is far above the average for the population or dataset in question.
- In general, a z-score around 0 means the value is close to the average.
- A z-score greater than 2 (or smaller than -2) typically indicates that the value is quite unusual or rare relative to the dataset.
For example, if the average height of players in this dataset were 180 cm, and the standard deviation were 5 cm, a z-score of 4.45 means the player’s height is 4.45×5=22.254.45 \times 5 = 22.254.45×5=22.25 cm above the mean. Thus, the player’s height is much taller than most players in the dataset.
Thus, the player’s height is 4.45 standard deviations above the mean.
