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=_____
Thanks in advance!
To find the mean (this is also the mode):
(38 + 62)/2 = 50
95% equals 2 standard deviations about the mean. Therefore, to find 1 standard deviation:
(50 - 38)/2 = 6
...or...
(62 - 50)/2 = 6
I hope this helps.
Thank you very much! I am having real problems with that. Is that the formula that should be used:
ơ=√∑ (x- µ)2/N
?
(a) The mode of a normal distribution is the value that appears most frequently. In this case, the mode would be the midpoint between 38 and 62, which is:
Mode = (38 + 62) / 2 = 50
So, the mode is 50.
(b) The standard deviation measures the spread or dispersion of the distribution. In a normal distribution, approximately 95% of the data lies within two standard deviations of the mean. Since we know that approximately 95% fall between 38 and 62, we can calculate the standard deviation as follows:
2 * Standard Deviation = Range = 62 - 38 = 24
Dividing both sides by 2:
Standard Deviation = 24 / 2 = 12
So, the standard deviation is 12.
To find the mode and standard deviation of a normal distribution of test scores based on the given information, we need to make a few assumptions.
Since the distribution is approximately normal, we can assume that the mean, median, and mode are the same. Additionally, we can use the empirical rule for normal distributions to estimate the standard deviation.
Let's start by finding the mean or mode of the distribution. Since we know that the distribution is symmetric and approximately normal, the mean and mode should be right in the middle of the range. So, the mean (or mode) is the average of the lower and upper bounds:
(a) Mode = (38 + 62) / 2 = 100 / 2 = 50.
Now, let's estimate the standard deviation using the empirical rule. The empirical rule states that for a normal distribution, approximately 68% of data falls within one standard deviation from the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.
Based on the given information, we know that approximately 95% of the data falls between 38 and 62. This means that these values are approximately two standard deviations away from the mean. So, the estimated standard deviation is the range divided by four:
Range = Upper bound - Lower bound = 62 - 38 = 24.
(b) Standard deviation = Range / 4 = 24 / 4 = 6.
Therefore, in this hypothetical normal distribution of test scores, the mode is 50 and the estimated standard deviation is 6.