For a hypothetical normal distribution of test scores, approximately 95% fall between 38 and 62, 2.5% are below 38, and 2.5% are above 62. Given this information, (a) the mode=_____ and (b) the standard deviation=_____.
Mean = (38 + 62)/2 = 100/2 = 50
50 - 38 = 12 = 2 standard deviations below the mean.
62 - 50 = 12 = 2 standard deviations above the mean.
Therefore, 1 standard deviation = 6
I hope this helps.
Yes, I understand how you worked out the mean. I understand about the 12
above and below the mean, but how did you arrive at 12 = 2 standard deviation points.
To find the mode and standard deviation based on the given information about a hypothetical normal distribution of test scores, we need to understand the characteristics of a normal distribution.
(a) The mode of a normal distribution refers to the value(s) that occur most frequently, or the peak(s) of the distribution. In a normal distribution, the mode corresponds to the value with the highest frequency.
However, in this case, the information provided does not directly give us the mode. The given information only tells us the range within which approximately 95% of the test scores fall. It does not specify if there are any specific values or frequencies associated with the mode.
To determine the mode, more information is needed, such as the specific values and frequencies of the test scores. Without that additional information, we cannot determine the mode.
(b) The standard deviation measures the dispersion or spread of the data points around the mean in a normal distribution. It provides an indication of how closely the individual scores resemble the average (mean) score.
While the information provided does not directly give us the standard deviation, we can make an approximate estimation using the empirical rule, also known as the 68-95-99.7 rule.
According to this 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 of the mean.
- Approximately 99.7% falls within three standard deviations of the mean.
Given that approximately 95% of the test scores fall between 38 and 62, we can assume that the interval from 38 to 62 represents two standard deviations around the mean.
To find the standard deviation, we can calculate the range of two standard deviations (i.e., the distance between the upper and lower bounds):
Range = Upper bound - Lower bound = 62 - 38 = 24
Since this range corresponds to two standard deviations, we can divide it by 2 to find the standard deviation:
Standard Deviation = Range / 2 = 24 / 2 = 12
Therefore, the standard deviation based on the given information is approximately 12.