For a normal distribution (symmetrical), a value that is two standard deviations below the mean would be closer to which of the following?
Select one:
a. 89th percentile
b. Third Quartile
c. Third percentile
d. Median
To find out where a value two standard deviations below the mean would be located on a normal distribution, you need to consider the properties of the distribution.
A normal distribution is a symmetric distribution where the mean, median, and mode are all equal and located at the center. Additionally, it is characterized by the standard deviation (SD), which measures the average distance of data points from the mean.
Given that the value is two standard deviations below the mean, it means it is relatively far away from the center of the distribution. Since the normal distribution is symmetric, a value two standard deviations below the mean would also be found at the same distance above the mean.
Now, let's consider the given options:
a. 89th percentile: Percentiles represent the relative position of a value in a distribution. For a normal distribution, the 89th percentile falls above the mean, indicating a higher value compared to a value two standard deviations below the mean. So, option a is not correct.
b. Third Quartile: The third quartile (Q3) represents the point at which 75% of the data lies below it in a distribution. In a normal distribution, the third quartile is located above the mean, indicating a higher value compared to a value two standard deviations below the mean. Therefore, option b is not correct.
c. Third percentile: The third percentile represents the point at which only 3% of the data lies below it in a distribution. For a symmetrical normal distribution, the third percentile would be located closer to the mean. Since the value two standard deviations below the mean is relatively far from the mean and is closer to the tails of the distribution, it would not be located at the third percentile. Hence, option c is not correct.
d. Median: The median is the value that divides the distribution into two equal halves. In a symmetrical normal distribution, the median is located at the mean. Therefore, a value two standard deviations below the mean would also be closer to the median. Hence, option d is correct.
Therefore, the correct answer is d. Median.