The distribution of scores on a test is mound-shaped and symmetric with a mean score of 78. If 68% of the scores fall between 72 and 84, which of the following is most likely to be the standard deviation of the distribution? 3, 1, 4, 6?
3
To determine the most likely standard deviation of the distribution, we can use the concept of the empirical rule, also known as the 68-95-99.7 rule, which applies to mound-shaped and symmetric distributions.
According to the empirical rule, in a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% falls within two standard deviations.
- Approximately 99.7% falls within three standard deviations.
Since we are given that 68% of the scores fall between 72 and 84, we can conclude that this interval represents one standard deviation on each side of the mean. Therefore, the standard deviation would be half of the range between 72 and 84.
The range between 72 and 84 is 84 - 72 = 12.
Half of this range is 12 / 2 = 6.
Therefore, the most likely standard deviation of the distribution is 6.
Among the given options (3, 1, 4, 6), the value that matches this answer is 6.
The given information implies that 68% of the scores fall within one standard deviation of the mean score. In a normal distribution, this corresponds to the interval between the mean minus one standard deviation and the mean plus one standard deviation.
To find the interval, we can subtract and add the mean from the given values:
Mean - 1 standard deviation = 78 - x
Mean + 1 standard deviation = 78 + x
Given that 68% of the scores fall between 72 and 84, we can set up the following equations:
78 - x = 72
78 + x = 84
Simplifying these equations, we have:
-x = -6 (subtract 78 from both sides)
x = 6 (multiply both sides by -1)
Therefore, the most likely standard deviation is 6.