Judging distribution shape using mean and standard deviation

[drawar]

Homework Statement

A recent summary for the distribution of cigarette taxes (in cents) among the 50 states and Washington, D.C. in the United States reported mean = 73 and standard deviation = 48. Based on these values, do you think that this distribution us bell-shaped? Justify your answer.

Homework Equations

Nil

The Attempt at a Solution

Basically I don't think mean and stdev have anything to do with the shape of the distribution. I'm here just to ask for some hints from you. Thanks!

 

[nate808]

Hi there,

I would approach this question by looking at the definition of a bell-shaped distribution, also known as a normal distribution. A normal distribution is a symmetrical, continuous probability distribution that follows a specific mathematical formula. It is characterized by a bell-shaped curve, with the majority of the data falling near the mean and tapering off towards the tails.

In this case, we have a mean of 73 and a standard deviation of 48 for the distribution of cigarette taxes among the 50 states and Washington, D.C. This means that the majority of the data (about 68%) would fall within one standard deviation of the mean, which would be between 25 and 121 cents. This suggests that the data is relatively spread out and not clustered closely around the mean, which is not typical of a normal distribution.

Furthermore, a normal distribution has specific properties, such as a skewness of 0 and a kurtosis of 3. Skewness is a measure of symmetry, with a skewness of 0 indicating perfect symmetry. A kurtosis of 3 indicates that the distribution has a similar shape to a normal distribution. However, it is worth noting that a kurtosis value of 3 is not a definitive indicator of a normal distribution, as other distributions can also have a kurtosis of 3.

In summary, based on the given mean and standard deviation, it is unlikely that the distribution of cigarette taxes among the 50 states and Washington, D.C. is bell-shaped. The data appears to be relatively spread out and not symmetrically distributed around the mean. However, further analysis and examination of the data would be needed to make a definitive conclusion.